comportement en grand temps et intégrabilité de certaines

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Comportement en grand temps et intégrabilité decertaines équations dispersives sur l’espace de Hardy

Ruoci Sun

To cite this version:Ruoci Sun. Comportement en grand temps et intégrabilité de certaines équations dispersives surl’espace de Hardy. Equations aux dérivées partielles [math.AP]. Université Paris-Saclay, 2020.Français. NNT : 2020UPASS111. tel-03092314

https://tel.archives-ouvertes.fr/tel-03092314

https://hal.archives-ouvertes.fr

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Comportement en grand temps etintégrabilité de certaines équationsdispersives sur l’espace de Hardy

Thèse de doctorat de l’université Paris-Saclay

Ecole Doctorale de Mathématiques Hadamard (EDMH) n 574Spécialité de doctorat : Mathématiques fondamentales

Unité de recherche : Université Paris-Saclay, CNRS, Laboratoire demathématiques d’Orsay, 91405, Orsay, France

Référent : Faculté des sciences d’Orsay

Thèse présentée et soutenue à Orsay, le 26 juin 2020, par

Ruoci SUN

Composition du jury :

Sandrine GRELLIER PrésidenteProfesseure, Université d’OrléansErwan FAOU Rapporteur & ExaminateurDirecteur de recherche, (HDR), Université de Rennes 1,École Normale Supérieure de RennesDidier PILOD Rapporteur & ExaminateurProfesseur, University of BergenOana POCOVNICU ExaminatriceProfesseure, Heriot-Watt University, Maxwell Institute forMathematical SciencesFrédéric ROUSSET ExaminateurProfesseur, Université Paris-Saclay

Patrick GÉRARD Directeur de thèseProfesseur, Université Paris-Saclay

Remerciements

Je voudrais tout d’abord exprimer ma plus grande reconnaissance envers mon directeur de these, PatrickGerard. Son enthousiasme communicatif, sa rigueur de pensee, sa generosite intellectuelle m’ont pro-fondement marque. Il m’a fait decouvrir des domaines de recherche passionnants. Il m’a soutenu toujourslors de difficultes de recherche et de preparation des articles. Ses explications sont toujours hyper claires etcoherentes. J’ai pu en profiter pendant ces quatre annees de these et je suis conscient que c’est une grandechance. J’ai ete aussi particulierement impressionne par sa capacite a s’interesser a tous les domaines desmathematiques et donc a pouvoir toujours repondre a mes questions quel que soit le sujet. C’est pour moiun modele a suivre. Il m’a aide aussi enormement dans la langue francaise, je lui en suis tres reconnaissant.

Je remercie vivement Erwan Faou et Didier Pilod d’avoir accepte avec obligeance de relire ma these endetail, et d’ecrire sur elle leur rapport. L’attention de mathematiciens de leur envergure, la minutie deleur travail (dont j’ose a peine imaginer le temps qu’il leur a coute), ainsi que la pertinence de leursremarques, sont autant d’honneurs dont je mesure la portee.

Sandrine Grellier, Oana Pocovnicu et Frederic Rousset me font une vraie faveur en donnant de leur tempspour sieger a mon jury : puissent ces mots leur temoigner ma gratitude. La sollicitude des mathematiciensaccomplis pour leurs cadets est pour moi un constant motif d’emerveillement.

Je remercie toute l’equipe analyse numerique et equations aux derivees partielles (ANEDP) pour son ac-cueil et son dynamique. Merci a Frederic Paulin et Stephane Nonnenmacher pour m’accepter d’entrer al’Ecole dotorale de mathematiques Hadamard (EDMH). Je voudrais remercier aussi l’institut de Fields aToronto pour trois semaines extraordinaires que j’y ai passes, et ou pointerent les germes du chapitres 2 a4 du present manuscrit. Je voudrais exprimer mes remerciements a Catherine Sulem et Jean-Claude Sautpour leurs acceuil et pour leurs aides a obtenir mon visa de Canada. Je voudrais remercier Claude Zuilyet Laurent Thomann pour ses invitations aux colloques ’analyse asympototique’ en Italie et a Marseille.Merci egalement a David Dos Santos Ferreira, a Frederic Rousset et aux tous les organisateurs(trices)des conferences Journees EDPs qui portent toujours des exposes magnefiques chaque annee. Merci aHerbert Koch pour son accueil a Bonn en janvier 2020. Je voudrais remercier egalement Sijue Wu pourson invitation a Michigan, qui me permet de presenter mes resultats par poster.

Les voyages effectues au cours de ma these ont beaucoup stimule ma creativite. Merci a tous ceux quiles ont finances, en particulier, a l’EDMH, a l’ANR ANAE et a l’equipe ANEDP. Je voudrais exprimermes remerciements a Marie-Christine Myoupo, Clotilde d’Epenoux, Laurie Vincent, Severine Simon,Florence Rey, Estelle Savinien et Ophelie Molle pour leurs tres grande efficacite pour des affaires ad-ministratives. Merci a Mathilde Rousseau, Laurent Dang et le service informatique a LMO pour leursaides a la visio-conference de ma soutenance et a resoudre des problemes de l’ordinateur dans mon bureau.

Je profite aussi de l’occasion pour remercier tous les professeurs que j’ai pu avoir dans ma scolarite etqui m’ont donne envie de faire des mathematiques, specialement a l’Ecole Normale Superieure (ENS),a l’Universite de Paris-Sud 11, a l’Universite de Paris-Dauphine et a l’Universite Paris VI . Je voudraisd’abord remercier mes trois tuteurs a l’ENS: Bastien Mallein, Bertrand Maury et Cyril Imbert pourleurs guides et soutiens sur ma scolarite a l’ENS. Je voudrais exprimer mes grands remerciements a mesprofesseurs Thomas Alazard, Scott Armstrong, Nalini Anantharaman, Patrick Bernard, Olivier Biquard,Yann Brenier, Nicolas Burq, Raphael Cerf, Jean-Yves Chemin, Olivier Debarre, Thomas Duquesne,Jacques Fejoz, Patrick Gerard, Ilia Itenberg, Christophe Lacave, Nicolas Lerner, Mathieu Lewin, Yvan

Martel, Stephane Mischler, Stephane Nonnenmacher, Frederic Rousset, Eric Sere, Sylvia Serfaty, Em-manuel Trelat, Claude Viterbo, etc pour leur cours excellents que j’ai suivis pendant mes scolarite deLicence 3 au Master 2, qui m’ont incite a choisir EDPs hamiltoniennes comme domaine de recherche.

Je remercie egalement a Guy David, Maria Paula Gomez Aparicio, David Harari et Hans Henrik Rughpour me donner la chance d’enseigner les TDs a l’Universite Paris-Sud. L’experience de 350 heures deTDs sur mathematiques me permet de toucher d’autre domaines mathematiques dehors analyse des EDPs.

La recherche etant bien plus interessante lorsqu’elle est collective, je suis tres heureux d’avoir discutemon sujet avec Piotr Bizon, Walter Craig, Jean-Marc Delort, Alessio Figalli, Benoıt Grebert, ZaherHani, Slim Ibrahim, Thomas Kappeler, Joachim Krieger, Tai-Ping Liu, Fabricio Macia, Nader Mas-moudi, Clement Mouhot, Sohrab Shahshahani, Avraham Soffer, Gigliola Staffilani, John Francis Toland,Nikolay Tzvetkov, Michael Weinstein, Ping Zhang etc. Les discussions avec eux m’ont aide beaucoupdans ma recherche. Je les suis tres reconnaissant.

J’ai eu la chance de rencontrer des jeunes edpistes du monde: Siddhant Agrawal, Leo Bigorgne, CharlesCollot, Elek Csobo, Stefan Czimek, Yu Deng, Thibault de Poyferre, Anne-Sophie de Suzzoni, ChenjieFan, Francesco Fanelli, Olivier Graf, Chenlin Gu, Jiao He, Jiaxi Huang, Cecile Huneau, Maxime Ingre-meau, Jacek Jendrej, Yang Lan, Camille Laurent, Hugo Lavenant, Zongyuan Li, Xian Liao, BaopingLiu, Jiaqi Liu, Shuang Miao, Alexis Michelat, Quang Huy Nguyen, Chenmin Sun, Pierre Roux, JulienSabin, Annalaura Stingo, Qingtang Su, Joseph Thirouin, Victor Vilaca da Rocha, Yuexun Wang, AllenYilun Wu, Shengquan Xiang, Haiyan Xu, Pin Yu, Xu Yuan, Haitian Yue, Katherine Zhiyuan Zhang,Lei Zhao, Zhiyan Zhao, Zehua Zhao, Jiqiang Zheng, Hui Zhu etc. Merci a eux pour leur conversation,mathematique ou non, qui m’a eleve l’esprit et l’ame.

Je suis tres reconnaissant a mes collegues a l’Universite Paris-Sud, qui ont cre une atmosphere amicaletres agreable, tout particulierement Yang Cao, Zhangchi Chen, Linxiao Chen, Clementine Courtes, Lu-cile Devin, Wei-Guo Foo, Agnes Gadbled, Ning Guo, Weikun He, Zhizhong Huang, Magda Khalile, YasirAmmar Kilic, Camille Labourie, Thomas & Luc Lehericy, Bingxiao Liu, Sasha Minet, Claire Brecheteau,Jeanne Nguyen, Jingrui Niu, Davi Obeta, Gabriel Pallier, Anthony Preux, Yi Pan, Zicheng Qian, CagriSert, Julien Sedro, Changzhen Sun, Salim Tayou, Yisheng Tian, Simeng Wang, Xiaozong Wang, Bo Xia,Songyan Xie, Cong Xue, Daxin Xu, Yeping Zhang.

Je suis tres heureux de rencontrer mes camarades a l’ENS: Elie Casbi, Kaitong Hu, Jialun Li, ShengyuanZhao, Linyuan Liu, Shinan Liu, Disheng Xu, Lizao Ye, Ruxi Shi, Zhihao Duan, Hugo Federico, Censi Li,Tristan Ozuch-Meersseman, Eliot Pacherie, Jiaxin Qiao, Yichen Qin, Julien Sazadaly, Aurelien Velleret,Hua Wang, Yilin Wang, Mingchen Xia, Junyi Zhang, Yi Zhang, Tunan Zhu, Peng Zheng, etc. Leursbrillances exceptionnelles m’ont impressione beaucoup. Je les suis aussi reconnaissant.

Finalement, je voudrais exprimer mes plus grands remerciements a ma famille qui me soutient en tout letemps.

1

Resume On s’interesse dans cette these a trois modeles d’equations hamiltoniennes dispersives nonlineaires : l’equation de Schrodinger cubique defocalisante filtree par le projecteur de Szego ΠT, qui enlevetous les modes de Fourier strictement negatifs, sur le tore T := R/2πZ (NLS–Szego cubique), l’equationde Schrodinger quintique focalisante filtree par le projecteur de Szego ΠR sur la droite R (NLS–Szegoquintique) et l’equation de Benjamin–Ono (BO) sur la droite. Comme pour les deux modeles precedents,l’equation de BO peut encore s’ecrire sous la forme d’une equation de Schrodinger quadratique filtree parle projecteur de Szego ΠR. Ces trois modeles nous donnent l’occasion d’etudier les proprietes qualitativesde certaines ondes progressives, le phenomene de croissance de normes de Sobolev, le phenomene de dif-fusion non lineaire et certaines proprietes d’integrabilite de systemes dynamiques hamiltoniens. Le but decette these est de comprendre l’influence des operateurs non locaux ΠT et ΠR sur des equations de typede Schrodinger et d’adapter les outils lies a l’espace de Hardy sur le cercle et sur la droite. On appliqueaussi la methode de forme normale de Birkhoff, l’argument de concentration–compacite, qui est precisea travers le theoreme de decomposition en profils, et la transformee spectrale inverse pour resoudre cesproblemes. Dans le troisieme modele, la theorie de l’integrabilite permet de faire le lien avec certainsaspects algebriques et geometriques.

Mots− clefs : Equation de Schrodinger non lineaire, Projecteur de Szego, Equation de Benjamin–Ono,Espace de Hardy, Stabilite orbitale, Onde progressive, Turbulence d’onde, Forme normale de Birkhoff,Concentration–compacite, Seuil de diffusion, Multi-soliton, Paire de Lax, Coordonnees d’action–angle,Operateur de Toeplitz

Abstract We are interested in three non linear dispersive Hamiltonian equations : the defocusing cu-bic Schrodinger equation filtered by the Szego projector ΠT that cancels every negative Fourier modes,leading to the cubic NLS–Szego equation on the torus T := R/2πZ ; the focusing quintic Schrodingerequation, which is filtered by the Szego projector ΠR, leading to the quintic NLS–Szego equation on theline R ; and the Benjamin–Ono (BO) equation on the line. Similarly to the other two models, the BOequation on the line can be written as a quadratic Schrodinger-type equation that is filtered by the Szegoprojector ΠR. These three models allow us to study their qualitative properties of some traveling waves,the phenomenon of the growth of Sobolev norms, the phenomenon of non linear scattering and some pro-perties about the complete integrability of Hamiltonian dynamical systems. The goal of this thesis is toinvestigate the influence of the non local operators ΠT and ΠR on some one-dimensional Schrodinger-typeequations and to adapt the tools of the Hardy space on the torus and on the line. We also use the Birkhoffnormal form transform, the concentration–compactness argument, refined as the profile decompositiontheorem, and the inverse spectral transform in order to solve these problems. In the third model, theintegrability theory allows to establish the connection with some algebraic and geometric aspects.

Keywords : Non linear Schrodinger equation, Szego projector, Benjamin–Ono equation, Hardy space,Orbital stability, Traveling wave, Wave turbulence, Birkhoff normal form, Concentration–compactness,Scattering mass threshold, Multi-soliton, Lax pair, Action–angle coordinates, Toeplitz operator

2

Cette these est dediee a ma famille.

Table des matieres

1 Introduction generale 5

1.1 Presentation du contexte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 L’equation de Szego sur le tore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.2 L’equation de Szego sur la droite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Enonces des resultats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.1 L’equation de NLS–Szego cubique sur le tore . . . . . . . . . . . . . . . . . . . . . 14

1.2.2 L’equation de NLS–Szego quintique sur la droite . . . . . . . . . . . . . . . . . . . 17

1.2.3 L’equation de Benjamin–Ono sur la droite . . . . . . . . . . . . . . . . . . . . . . . 21

1.3 Perspectives de recherche . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.3.1 L’equation de NLS–Szego amortie . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.3.2 Unicite des etats fondamentaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.3 Transformation de diffusion inverse (IST) de l’equation de BO . . . . . . . . . . . 28

1.4 Preliminaires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.4.1 Decomposition en profils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.4.2 Probleme de Cauchy pour l’equation de NLS–Szego cubique sur la tore . . . . . . 29

2 Long time behavior of the NLS–Szego equation on the torus 37

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2 The cubic Szego equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2.1 The Lax pair structure and L∞-estimate . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2.2 Wave turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2.3 A special case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3 Long time behavior for small data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.1 The case α ≥ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.2 The case 0 ≤ α < 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.4 Orbital stability of the plane wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.4.1 The proof of Theorem 2.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.4.2 Long time Hs-stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.4.3 Long time Hs-stability in the case α = 0 . . . . . . . . . . . . . . . . . . . . . . . . 58

2.5 Comparison to the NLS equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3

4 TABLE DES MATIERES

3 Traveling waves of NLS–Szego equation on the line 673.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.2 Profile decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.3 Orbital stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3.1 Proof of theorem 3.1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.3.2 The special case γ = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.4.1 Persistence of regularity for scattering . . . . . . . . . . . . . . . . . . . . . . . . . 803.4.2 The open problem of uniqueness of ground states . . . . . . . . . . . . . . . . . . . 81

4 Integrability of the BO equation on the line 854.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.1.2 Organization of this paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.1.3 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2 The Lax operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2.1 Unitary equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2.2 Spectral analysis I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2.3 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.2.4 Lax pair formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3 The action of the shift semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.4 The manifold of multi-solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.4.1 Differential structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.4.2 Spectral analysis II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.4.3 Characterization theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.4.4 The stability under the Benjamin–Ono flow . . . . . . . . . . . . . . . . . . . . . . 115

4.5 The generalized action–angle coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.5.1 The associated matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.5.2 Inverse spectral formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.5.3 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.5.4 The diffeomorphism property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.5.5 A Lagrangian submanifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.5.6 The symplectomorphism property . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.6 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.6.1 The simple connectedness of UN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.6.2 Covering manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Chapitre 1

Introduction generale

1.1 Presentation du contexte

Dans cette these, on s’interesse au comportement en grand temps et a l’integrabilite de certaines equationsdispersives. L’etude des equations aux derivees d’evolution non lineaires (EDPs) est motivee par lamodelisation d’ evolutions temporelles de certaines quantites definies sur un milieu continu pour desprobemes issus des sciences physiques, naturelles, sociales et d’autres domaines relies directement auxmathematiques. Pendant un demi-siecle, l’analyse des EDPs s’est beaucoup consacree a l’etude de l’exis-tence et de l’unicite de solutions locales et globales dans des espaces fonctionnels bien choisis. Par contrasteavec les equations differentielles ordinaires dont les solutions sont a valeurs dans des espaces vectorielsou des varietes lisses de dimension finie, les equations aux derivees partielles peuvent etre vues commedes systemes dynamiques dont les orbites sont incluses dans des espaces vectoriels ou dans des varietesde dimension infinie. Grace aux outils de l’analyse fonctionnelle, des notions de solution faible et forteont pu voir le jour, permettant de definir des trajectoires dans des espaces de fonctions avec differentsniveaux de regularite.

Une equation aux derivees partielles est dite dispersive si ses solutions se decrivent sous la forme d’ondesqui se propagent a des vitesses qui varient en fonction de la longueur d’onde. Une equation dispersivesemi lineaire s’ecrit le plus souvent sous la forme

∂tw(t) + Lw(t) = N (w(t)), (1.1.1)

ou L est un operateur anti-auto-adjoint non borne et densement defini sur certain espace d’Hilbert H, etN est une fonctionnelle non lineaire definie sur le domaine de definition de L. L’equation (1.1.1) peut sereformuler a l’aide de la formule de Duhamel,

w(t) = e−(t−t0)Lu(t0) +

∫ t

t0

e−(s−t0)LN (w(s))ds. (1.1.2)

Par exemple, pour resoudre le probleme de Cauchy en basse regularite de l’equation de Schrodingersemi-lineaire

i∂tw + ∆w = N (w), (t, x) ∈ R×M (1.1.3)

ou M est Rd ou Td, avec T = R/2πZ, on utilise souvent la theorie de Strichartz–Ginibre–Velo qui permet

5

6 CHAPITRE 1. INTRODUCTION GENERALE

de convertir l’estimation dispersive

‖eit∆ϕ‖Lp(Rd) .d,p |t|d( 1p−

12 )‖ϕ‖

Lpp−1 (Rd)

, ϕ ∈ Lpp−1 (Rd), 2 ≤ p ≤ +∞ (1.1.4)

en des inegalites controlant la norme en espace–temps de Lebesgue de la solution. On renvoie a Strichartz[130], Ginibre–Velo [58, 59, 60], Tao [138] et Bahouri–Chemin–Danchin [5] etc. pour le cas M = Rd, Bour-gain [13] pour le cas M = Td, Burq–Gerard–Tzvetkov [18, 19, 20] pour le cas M est une variete compactequelconque, etc.

Apres avoir resolu le probleme de Cauchy, on s’interesse au comportement en grand temps d’une solutionde l’equation (1.1.1), en particulier la stabilite d’ondes progressives, le phenomene de turbulence faibledu point de vue de la croissance de normes de Sobolev, le phenomene de la diffusion non lineaire, etc.Par exemple, on considere l’equation de Schrodinger cubique defocalisante sur le tore Td,

i∂tw + ∆w = |w|2w, (t, x) ∈ R× Td (1.1.5)

et on pose la question : la solution t ∈ R 7→ u(t) ∈ Hs(T) est-elle uniformement bornee sur R, pour touts ≥ 0 ?

La reponse est positive lorsque d = 1 a l’aide de la suite de lois de conservation engendree par une pairede Lax decouverte par Zakharov–Shabat [152]. Ces lois de conservation controlent toutes les Hs-normesde Sobolev de la solution (dans le cas s ≥ 0 entier, voir Faddeev–Takhtajan [33], Grebert–Kappeler [65],Gerard [46], Sun [131] ; dans le cas s > 1 non entier, voir Kappeler–Schaad–Topalov [82] ; dans le cas− 1

2 < s < 0, voir Koch–Tataru [91] et Killip–Visan–Zhang [90]).

Proposition 1.1.1. Si w0 ∈ C∞(T) et w ∈ C∞(R × T) resout l’equation (1.1.5) en dimension d = 1avec la donnee initiale w(0) = w0, on a supt∈R ‖w(t)‖Hs(T) < +∞, pour tout s ≥ 0.

Lorsque d ≥ 2, Colliander–Keel–Staffilani–Takaoka–Tao [29] et Guardia–Haus–Procesi [67] ont montreque pour tout d ≥ 2, K 1 et s > 1, il existe une solution w : t 7→ w(t) ∈ Hs(Td) globale de l’equation(1.1.5) telle que ‖w(0)‖Hs < K−1 et ‖w(T )‖Hs > K pour un certain T > 0. Par consequent, il n’existepas de loi de conservation controlant la norme Hs, si s > 1. Dans le cas ou Td est remplacee par l’espacede produit R × Td, lorsque d ≥ 2, Hani–Pausader–Tzvetkov–Visciglia [69] ont montre, a l’aide d’uneprocedure de diffusion modifiee, qu’il existe des solutions dont l’orbite n’est pas bornee dans Hs pour sassez grand.

Dans le cas M = T2, l’equation (1.1.3) cubique admet le terme nonlineaire completement resonant,N = NR,

NR(w) =∑n∈Z2

∑(n1,n2,n3)∈ΓR(n)

wn1wn2

wn3

ein·x, w =∑n∈Z2

wnein·x, R ∈ Z

⋂[0,+∞), (1.1.6)

ou ΓR(n) = (n1, n2, n3) ∈ (Z2)3 : n1− n2 + n3 = n et ‖n1‖2R2 −‖n2‖2R2 + ‖n3‖2R2 −‖n‖2R2 ∈ [−R,R],et Hani [68] a trouve qu’il existe une solution globale de l’equation (1.1.3) dont l’orbite n’est pas bornee.

On s’interesse a la relation entre l’existence d’une infinite de lois de conservations lineairement independantes,et le phenomene de croissance de normes de Sobolev de la solution d’une equation du type (1.1.1) arbi-traire. On pose une autre question : existe-il une equation du type (1.1.1) qui admet simultanement ces

1.1. PRESENTATION DU CONTEXTE 7

deux phenomenes ?

La reponse est positive. Considerons l’equation de Szego cubique sur le tore T

i∂tw = ΠT(|w|2w), (t, x) ∈ R× T, (1.1.7)

ou ΠT : L2(T)→ L2(T) est projecteur de Szego qui supprime tous les modes de Fourier negatifs,

ΠTf(x) =∑n≥0

fneinx, si f ∈ L2(T), f(x) =

∑n∈Z

fneinx, (1.1.8)

avec fn = 12π

∫ 2π

0f(x)e−inxdx. On definit L2

+(T) = ΠT(L2(T)) et Hs+(T) := L2

+(T)⋂Hs(T). Introduite

par Gerard–Grellier [47, 49, 51, 52], cette equation admet simultanement une paire de Lax qui permet deconstruire des variables d’action–angle, de trouver une formule explicite pour le flot a l’aide d’une transfor-mation spectrale inverse, et des solutions turbulentes engendrees par des donnees initiales generiquementdefinies dans le sous-espace C∞+ (T) := C∞(T)

⋂L2

+(T). La structure de paire de Lax engendre unesuite de lois de conservation lineairement independantes de l’equation (1.1.7), mais aucune parmi ellesne controle la norme Hs

+(T), si s > 12 . Inspiree par ces resultats, O. Pocovnicu a trouve des resultats

similaires sur l’equation de Szego cubique sur la droite dans [119, 120]. H. Xu [148, 149] a etudie uneperturbation lineaire de l’equation de Szego cubique sur le tore et elle a trouve une solution qui admetune croissance de norme de Sobolev exactement exponentielle. Elle a aussi implemente la strategie deHani–Pausader–Tzvetkov–Visciglia [69] dans le cadre de l’equation de ’guide d’onde’ dans Xu [150]. J.Thirouin a etabli une estimation de norme de Sobolev en grand temps de l’equation de Schrodinger frac-tionnelle dans [140]. Il a mene l’etude d’une equation de Szego quadratique. Dans Thirouin [141], il atrouve l’estimation optimale de la croissance de norme de Sobolev et dans Thirouin [142] il a classifiecompletement les ondes progressives de cette equation. On renvoie aussi a Biasi–Bizon–Evnin [10] pourla croissance de norme de Sobolev dans le cas d’une autre variante de l’equation de Szego cubique.

Rappelons maintenant certains resultats sur l’equation de Szego cubique, d’abord sur la tore, ensuite surla droite, qui sont relies aux resultats principaux de cette these.

1.1.1 L’equation de Szego sur le tore

Definition 1.1.2. Sur L2(T), on introduit le produit scalaire 〈f, g〉L2(T) = 12π

∫ 2π

0f(x)g(x)dx et la forme

symplectique ωL2(T)(f, g) := Im〈f, g〉L2(T). L’energie ETSzego(w) = 1

4‖w‖4L4(T) est densement definie sur

L2+.

Alors l’equation de Szego cubique sur le tore (1.1.7) est de type hamiltonien, ∂tw = XETSzego

(w), ou le

champ hamiltonien de ETSzego par rapport a ωL2(T) est donne par

dETSzego(w)(h) = ωL2(T)(h,XET

Szego(w)), ∀w, h ∈ H

12+(T).

Notons que, malgre la forme (1.1.1), cette equation est totalement non dispersive, puisque l’operateur Lest nul !

Proposition 1.1.3. (Gerard–Grellier [47]). Soit w0 ∈ H12+(T), il existe une unique fonction w ∈

Cb(R, H12+(T)) qui resout l’equation (1.1.7) avec la donnee initiale w(0) = w0 et l’application du flot

w0 ∈ H12+(T) 7→ w ∈ L∞(R, H

12+(T)) est continue. Si s > 1

2 et w0 ∈ Hs(T), alors w ∈ C∞(R, Hs+(T)).


Soient w ∈ L2+(T) et b ∈ L∞(T), l’operateur de Hankel de symbole w et l’operateur de Toeplitz de

symbole b sont deux operateurs bornes sur L2+(T), qui sont definis par

HTw(h) = ΠT(wh), TT

b (h) = ΠT(bh), ∀h ∈ L2+(T). (1.1.9)

L’operateur de Hankel HTw est anti-lineaire et l’operateur de Toeplitz TT

b est lineaire. L’operateur dedecalage ST : L2

+(T)→ L2+(T) est defini par

STh(x) = eixh(x), ∀x ∈ T et h ∈ L2+(T). (1.1.10)

Ensuite l’operateur de Hankel decale KTw : L2

+(T)→ L2+(T) est defini par

KTw = (ST)∗Hw = HwS

T = H(ST)∗w. (1.1.11)

Alors on a le theoreme suivant.

Theoreme 1.1.4. (Gerard–Grellier [47]). Soient s > 12 et w ∈ C∞(R, Hs

+(T)) resout l’equation de Szego

cubique sur la tore i∂tw = ΠT(|w|2w), on definit ATw = −iTT

|w|2 + i2 (HT

w)2 et CTw = −iTT

|w|2 + i2 (KT

w)2.Alors on a

∂tHTw = [AT

w, HTw], ∂tK

Tw = [CT

w,KTw]. (1.1.12)

On identifie les elements w ∈ L2+(T) aux traces des fonctions holomorphes sur le disque unite w ∈

Hol(D(0, 1)), D(0, 1) = z ∈ C : |z| < 1, telles que sup0≤r<1

∫ 2π

0|w(reix)|2dx < +∞ par la correspon-

dance suivante

w ∈ L2+(T) 7→

w : z ∈ D(0, 1) 7→∑n≥0

wnzn

, w(x) = limr→1

w(reix) (1.1.13)

qui etablit un isomorphisme d’espace de Hilbert de l’espace L2+(T) vers l’espace de Hardy sur le disque

unite (voir Chapter 17 of Rudin [124]).

Theoreme 1.1.5. (Gerard–Grellier [51]). Soient w0 ∈ H12+(T) et w ∈ C(R, H

12+(T)) la solution de

l’equation (1.1.7) avec la donnee initiale w(0) = w0, alors on a la formule suivante,

w(t, z) =

⟨(IdL2

+(T) − ze−it(HTw0

)2

eit(Kw0)2

(ST)∗)−1

e−it(HTw0

)2

w0,1

⟩L2(T)

, (1.1.14)

ou la fonction 1(x) = 1 est constante.

Definition 1.1.6. L’ensemble M(N)T consiste en toutes les fractions rationnelles w ∈ L2+(T) de la

forme w(x) = A(eix)B(eix) , ou B(0) = 1, les polynomes A ∈ C≤N−1[X], B ∈ C≤N [X] sont premiers entre eux

et tels que deg(A) = N − 1 ou deg(B) = N , les racines de B sont dehors le disque unite ferme D(0, 1).Ici C≤N [X] designe l’ensemble des polynomes P ∈ C[X] de degre deg(P ) ≤ N .

Muni du produit interieur de L2+(T), l’ensemble M(N)T est une sous-variete kahlerienne de L2

+(T) dedimension complexe dimC(M(N)T) = 2N . Un theoreme qui remonte a Kronecker [94] dit qu’une fonctionw ∈M(N)T si et seulement si le rang de l’operateur de Hankel HT

w est egal a N .

Definition 1.1.7. L’ensemble M(N)Tgen consiste en toutes les fonctions w ∈ M(N)T telles que 1 /∈Im(HT

w) et les N vecteurs (HTw)2k(1)k=1,2,··· ,N sont lineairement independants.


L’ensembleM(N)Tgen est une partie ouverte deM(N)T dont le complementaire est de mesure de Lebesgue

nulle. Munie de la forme symplectique ωL2(T), M(N)Tgen est une variete symplectique de dimension

dimR(M(N)Tgen) = 4N . Les actions sont donnees par

ΩTN = (IT1 , IT2 , · · · , ITN ; JT

1 , JT2 , · · · , JT

N ) ∈ R2N : ITk > JTk > ITk+1 > 0.

La forme symplectique canonique sur la variete ΩTN × T2N est νT =

∑Nk=1(dITk ∧ dϕT

k + dLTk ∧ dθTk ).

Theoreme 1.1.8. (Gerard–Grellier [49]). Il existe un symplectomorphisme reellement analytique

χTN : (M(N)Tgen, ωL2(T))→ (ΩT

N × T2N ,νT) (1.1.15)

telle que ETSzego

(χTN

)−1(IT1 , · · · , ITN ;LT

1 , · · · , LTN ;ϕT

1 , · · · , ϕTN ; θT1 , · · · , θTN ) = 1

4

∑Nk=1(|ITk |2 − |LT

k |2).

L’application χN introduit les coordonnees action–angle pour l’equation de Szego cubique sur le tore. Si

χTN (w) = (IT1 (w), · · · , ITN (w);LT

1 (w), · · · , LTN (w);ϕT

1 (w), · · · , ϕTN (w); θT1 (w), · · · , θTN (w)) ∈ ΩT

N × T2N ,

alors l’equation (1.1.7) est linearisee sous ces coordonnees, si w : t ∈ R 7→ w(t) ∈M(N)T resout (1.1.7),

∂t(ITk w)(t) = ∂t(L

Tk w)(t) = 0, ∂t(ϕ

Tk w)(t) =

(ITk w)(t)

2, ∂t(θ

Tk w)(t) = − (LT

k w)(t)

2.

Plus precisement, la famille 12I

Tk (w)k≥1 consiste en l’ensemble des valeurs propres de l’operateur (HT

w)2

et la famille 12L

Tk(w)k≥1 consiste en l’ensemble des valeurs propres de l’operateur (KT

w)2. De plus,l’application χT

N admet une generalisation en dimension infinie. (voir Gerard–Grellier [49]). La solutionw : t 7→ w(t) ∈ M(N)T est quasi-periodique (voir Gerard–Grellier [51]). Le resultat suivant met enevidence un phenomene de turbulence faible pour l’equation (1.1.7).

Theoreme 1.1.9. (Gerard–Grellier [50]). Il existe une partie Gδ-dense V ⊂ C∞+ (T) telle que si w0 ∈ V,alors la solution w : t ∈ R 7→ w(t) ∈ C∞+ telle que la donnee initiale est w(0) = w0 verifie les proprietessuivantes :

• il existe une suite temporelle tn tendant vers +∞, telle que ∀s > 12 , ∀M ∈ N, on a

limn→+∞

‖w(tn)‖Hs(T)

tMn

= +∞;

• il existe une suite temporelle tn tendant vers +∞, telle que

limn→+∞

w(tn) = w0 dans C∞+ (T).

Par ailleurs, pour tout w0 ∈ C∞+ (T), il existe une constante Cs > 0 telle que

‖w(t)‖Hs(T) ≤ Cs‖w0‖Hs(T)eCs‖w0‖2Hs(T)|t|, ∀t ∈ R. (1.1.16)

Comme l’union⋃N≥1M(N)T est dense dans C∞+ (T) et V est Gδ-dense dans C∞+ (T), on voit que le

comportement en grand temps de l’equation de Szego cubique (1.1.7) depend de facon sensible de sadonnee initiale.


Remarque 1.1.10. On considere l’equation hamiltonienne avec la meme energie sans le projecteur deSzego ΠT : L2

+(T)→ L2+(T) sur la tore :

i∂tV = |V |2V, (t, x) ∈ R× T. (1.1.17)

Alors on a V (t, x) = eit|V0|2V0(x) et ‖V (t)‖Hs ' |t|s, pour tout s ≥ 0, si |V0| n’est pas une fonctionconstante. Donc le projecteur de Szego ΠT : L2(T)→ L2

+(T) accelere le transfert d’energie vers les hautesfrequences.

Xu [148, 149] a mene l’etude d’une equation de Szego cubique perturbee par un terme lineaire

i∂tw = ΠT(|w|2w) + α

∫Tw, α ∈ R. (1.1.18)

Dans ce cas, (HTw, A

Tw) n’est plus une paire de Lax, mais (KT

w, CTw) est encore une paire de Lax. Lorsque

α > 0 et w0(x) = eix +√α, on note w : t ∈ R 7→ w(t) ∈ Hs

+(T) la solution de l’equation (1.1.18), alorson a la croissance exponentielle exacte de la norme de Sobolev, lorsque t→ ±∞,

‖w(t)‖Hs(T) ' e(2s−1)√α|t|, ∀s > 1

2.

Thirouin [141, 142] a trouve l’estimation optimale de la croissance de norme de Sobolev et a classifiecompletement les ondes progressives de l’equation de Szego quadratique

i∂tw = 2JΠT(|w|2) + Jw2, ∀(t, x) ∈ R× T, J := J(u) =

∫T|w|2w ∈ C. (1.1.19)

1.1.2 L’equation de Szego sur la droite

Considerons l’equation cubique sur la droite (voir Pocovnicu [119, 120])

i∂tw = ΠR(|w|2w), (t, x) ∈ R× R, (1.1.20)

ou ΠR : L2(R)→ L2(R) est projecteur de Szego sur la droite,

ΠRf(x) =1

2π

∫ +∞

0

eixξ f(ξ)dx, ∀f ∈ L2(R). (1.1.21)

On definit les espaces de Sobolev filtres par L2+(R) := ΠR(L2(R)) et Hs

+(R) = L2+(R)

⋂Hs(R), pour tout

s ≥ 0. Le theoreme de Paley–Wiener (Theorem 19.2 of Rudin [124]) donne l’identification entre l’espaceL2

+ et l’espace de Hardy sur le demi-plan de Poincare C+ = z ∈ C : Imz > 0, on a l’isomorphismed’Hilbert suivant w ∈ L2

+(R) 7→ w ∈ H2(C+) = g ∈ Hol(C+) : ‖g‖H2(C+) < +∞, ou

‖g‖Hp(C+) = supy>0

(∫R|g(x+ iy)|pdx

) 1p

, si p ∈ (0,+∞), (1.1.22)

Alors l’equation (1.1.20) est aussi de type hamiltonien par rapport a la forme symplectique ωL2(R)(f, g) =

Im〈f, g〉L2(R) avec le produit scalaire 〈f, g〉L2(R) =∫R fg. L’energie de l’equation (1.1.20) est ER

Szego(w) =14

∫R |w|

4, pour tout w ∈ L4(R)⋂L2

+(R). L’equation de Szego sur la droite est globalement bien posee

sur Hs+(R), pour tout s ≥ 1

2 .


Definition 1.1.11. Soient w ∈ L2+(R) et b ∈ L∞(R), l’operateur de Hankel HR

w : L2+(R) → L2

+(R)

et l’operateur de Toeplitz TRb : L2

+(R) → L2+(R) sont definis par HR

w(h) = ΠR(wh) et TRb (h) = ΠR(bh)

respectivement.

Theoreme 1.1.12. (Pocovnicu [119]). Soient s > 12 et w ∈ C∞(R, Hs

+(R)) resout l’equation (1.1.20),

on definit un operateur ARw = −iTR

|w|2 + i2 (HR

w)2. Alors ∂tHRw = [AR

w, HRw].

Ainsi l’equation (1.1.20) admet aussi une paire de Lax. En revanche l’operation de decalage n’existe plusmais devient un semi-groupe (S(η))η≥0 avec S(η) : L2

+(R)→ L2+(R) defini S(η)f = eηf , ou eη(x) = eiηx,

x ∈ R, dont l’adjoint est donne par S(η)∗ = Te−η . Inspire par la definition 1.1.6, on introduit la variete

M(N)R, qui consiste en toutes les fractions rationnelles sous la forme w(x) = A(x)B(x) , ou B(0) = 1, les

polynomes A ∈ C≤N−1[X], B ∈ C≤N [X] etant premiers entre eux et tels que deg(A) = N − 1 oudeg(B) = N , les racines de B etant contenues dans le demi-plan inferieur de Poincare C−. Muni duproduit interieur de L2

+(R), l’ensemble M(N)R est une sous-variete kahlerienne de L2+(R) de dimension

dimC(M(N)R) = 2N . De plus M(N)R est le revetement universel de la variete M(N)T donnee dans ladefinition 1.1.6. (voir Sun [135]) Le theoreme de Kronecker [94] donne la caracterisation par le rang desoperateurs de Hankel

M(N)R = w ∈ L2+(R) : rang(HR

w) := dimC(ImHRw) = N. (1.1.23)

Definition 1.1.13. Soit w ∈M(N)R, il existe une unique fonction g ∈ Im(HRw) = Im((HR

w)2) telle queHRwg = w. L’ensemble des valeurs propres de (HR

w)2 est σpp((HRw)2) = λ2

1, λ22, · · · , λ2

N, ou λk = λk(w),tel que 0 < λk ≤ λk+1. Il existe une base orthonormee du sous-espace Im(HR

w) notee par ekNk=1 formeepar des vecteurs propres qui satisfont HR

wek = λkek, pour tout k = 1, 2, · · · , N .

Theoreme 1.1.14. (Pocovnicu [120]). Soit w0 ∈ M(N)R et notons W (t) := eit2 (HR

w0)2

, l’operateur dedecalage infinitesimal compresse T : Im(HR

w0)→ Im(HR

w0) est defini par

Tf(x) := xf(x)−(

limx→+∞

xf(x)

)(1− g0). (1.1.24)

ou la fonction g0 ∈ Im(HRw0

) verifie que HRw0g0 = w0, donnee dans la definition 1.1.13. Soit j = 1, 2, · · · , N

fixe et on note Mj ⊂ N qui consiste en l’ensemble de tous les indices k tels que HRw0ek = λjek. On definit

βj := 〈g0, ej〉L2(R) et l’operateur lineaire S(t) : Im(HRw0

)→ Im(HRw0

) par

〈S(t)ek, ej〉L2(R) =

λj

2πi(λ2k−λ

2j )eit2 (λ2

k−λ2j )(λjβjβk − λkβkβj

), si k ∈ 1, 2, · · · , N\Mj

λ2j

2πβjβkt+ 〈Tej , ek〉L2(R), si k ∈Mj .

Si on note w : t ∈ R 7→ w(t) ∈ Im(HRw0

) la solution de l’equation de Szego associee a la donnee initialew(0) = w0, alors on a la formule explicite suivante

w(t, x) =i

2π〈w0,W (t) (S(t)− x)

−1W (t)g0〉L2(R), ∀(t, x) ∈ R× R. (1.1.25)

Remarque 1.1.15. Soit s > 12 , comme l’union

⋃N≥1M(N)R est dense dans Hs

+(R), la formule (1.1.25)se generalise au cas d’une solution a donnee initiale arbitraire dans Hs

+(R) telle que x 7→ xw0(x) ∈ L∞(R)(voir Theorem 1.4 de Pocovnicu [120]).


Definition 1.1.16. Une fonction w ∈ M(N)R est dite generique si l’operateur (HRw)2 n’admet que des

valeurs propres non nulles simples 0 < λ21 < λ2

2 < · · · < λ2N . On note M(N)Rgen l’ensemble de telles

fonctions generiques, qui est une partie ouverte dense de M(N)R.

Remarque 1.1.17. Une fonction generique w ∈ M(N)Rgen verifie que 〈w, ek〉L2(R) 6= 0, pour tout k =

1, 2, · · · , N , ou les fonctions propres ekNk=1 sont donnees dans la definition 1.1.13, d’apres Gerard–Pushnitski [57].

Alors (M(N)Rgen, ωL2(R)) est une variete symplectique de dimension dimR(M(N)Rgen) = 4N . On definit

YN := (−ΩN ) × (R∗+)N × RN × TN , ou (−ΩN ) = 0 < IR1 < IR2 < · · · < IRN. Muni de la formesymplectique canonique,

νR =

N∑k=1

(dIRk ∧ dϕR

k + dLRk ∧ dθRk

), ϕR

k ∈ R, θRk ∈ T.

YN est une variete symplectique reellement analytique de dimension dimR(Y) = 4N .

Theoreme 1.1.18. (Pocovnicu [120]). Il existe un symplectomorphisme reellement analytique

χRN : (M(N)Rgen, ωL2(R))→ (YN ,ν

R),

tel que ERSzego(χR

N )−1 ne depend que des variables d’actions (IR1 , IR2 , · · · , IRN ;LR

1 , LR2 , · · · , LR

N ) ∈ (−ΩN )×(R∗+)N .

Remarque 1.1.19. L’application χRN introduit les coordonnees d’action–angle generalisees. La moitie

des actions et des angles IRk , ϕRk , k = 1, 2, · · · , N , coıncide dans le cas de l’equation de Szego cubique

sur la tore. La famille 14π I

Rk (w)k≥1 consiste en l’ensemble des valeurs propres de l’operateur de Hankel

(HRw)2. Cependant, l’autre moitie est completement differente. Dans le cas T, le deuxieme operateur de

Lax est trouve, c’est l’operateur de Hankel decale Kw = HTwS

T dont ses valeurs propres et ses fonctionspropres donnent le reste des coordonnees. Dans le cas R, l’action de decalage devient le semi-groupe(S(η)∗)η≥0. On definit LR

k (w) = 4|〈w, ek〉L2(R)|2, ou ek est la fonction propre associee a la valeur propre

λ2j = 1

4π IRk (w) de l’operateur de Hankel (HR

w)2, donnee par la definition 1.1.13. On a LRk (w) > 0, pour

tout k = 1, 2, · · · , N , d’apres la remarque 1.1.17.

Definition 1.1.20. Une onde solitaire de l’equation de Szego cubique sur la droite (1.1.20) est unesolution lisse w : t ∈ R 7→ w(t) ∈ C∞+ (R) qui s’ecrit sous la forme w(t, x) = e−iωtw0(x − ct), pourcertaines constantes c, ω ∈ R, ou w0 = w(0).

Inspiree de la classification des ondes progressives pour l’equation de Szego sur la tore (voir Gerard–Grellier [47]), Pocovnicu a classifie l’ensemble des ondes solitaires de l’equation (1.1.20) a l’aide dutheoreme de Beurling–Lax qui permet de caracteriser tous les sous-espaces translation–invariants del’espace de Hardy L2

+(R).

Theoreme 1.1.21. (Pocovnicu [119]). Une fonction w ∈ C(R, H12+(R)) est une onde solitaire si et

seulement s’il existe C, p ∈ C telles que Imp < 0 telle que

w(t, x) =Ce−iωt

x− ct− p(1.1.26)

ou les constantes c et ω dependent de C et p.

1.2. ENONCES DES RESULTATS 13

De plus, Pocovnicu a montre la stabilite orbitale de l’onde solitaire (1.1.26) par la methode de concentration–compacite (voir Lions [101, 102]), qui est amelioree comme le theoreme de decomposition en profils parGerard [45] (voir aussi Bahouri–Gerard [6]). Ce theoreme est recapitule dans la sous-section 1.4.1.

Theoreme 1.1.22. (Pocovnicu [119]). Soient a, r > 0, considere le cylindre

C(a, r) = x ∈ R 7→ α

x− p∈ C : |α| = a, Imp = −r ⊂ H∞+ (R). (1.1.27)

Pour tout ε > 0, il existe δ = δ(ε, a, r) > 0 telle que ∀w0 ∈ H12+(R) verifiant infφ∈C(a,r) ‖w0−φ‖

H12 (R)

< δ,

on note w : t ∈ R 7→ w(t) ∈ H12+(R) la solution de l’equation de Szego sur la droite (1.1.20), alors

supt∈R

infφ∈C(a,r)

‖w(t)− φ‖H

12 (R)

< ε.

Definition 1.1.23. Une donnee initiale w0 ∈ M(N) est dite fortement generique si l’operateur H2w0

a ses valeurs propres non nulles simples 0 < λ21 < λ2

2 < · · · < λ2N avec 〈w0, ek〉L2(T), pour tout

k = 1, 2, · · · , N et |〈w0, ej〉L2(T)| 6= |〈w0, ek〉L2(T)| pour tout k 6= j. L’ensemble des fonctions fortementgeneriques est note par M(N)sgen.

Theoreme 1.1.24. (Pocovnicu [120]). Soit w0 ∈ M(N)sgen une donnee initiale fortement generiquepour l’equation de Szego cubique sur la droite (1.1.20), la solution correspondante s’ecrit comme unesomme de N -solitons et d’un reste. Plus precisement, on a

u(t, x) =

N∑j=1

Cje−iωjt

x− cjt− pj+ ε(t, x), lim

t→±∞‖ε(t, x)‖Hs+(R) = 0, ∀s ≥ 0,

ou ωj = λ2j , les constantes pj ∈ C− et Cj ∈ C dependent de λj , ej et w0.

Theoreme 1.1.25. (Pocovnicu [120]). Soient s > 12 , il existe des solutions w : t ∈ R 7→ w(t) ∈ Hs

+(R)de l’equation de Szego, telles que limt→±∞ ‖w(t)‖Hs(R) = +∞.

1.2 Enonces des resultats

L’equation de Schrodinger cubique sur le tore de dimension 1 n’a aucune solution turbulente d’apresla proposition 1.1.1. D’un autre cote, le projecteur de Szego ΠT : L2(T) → L2

+(T) accelere le transfertd’energie vers les frequences hautes pour l’equation (1.1.17). On s’interesse donc a l’influence du projecteurde Szego sur l’equation de NLS cubique (1.1.5) dans le cas d = 1, en introduisant l’equation de NLS–Szegocubique defocalisante sur le tore

i∂tu+ ∂2xu = ΠT(|u|2u), (t, x) ∈ R× T. (1.2.1)

Sur la droite, on s’interesse a l’influence du projecteur de Szego ΠR : L2(R)→ L2+(R) donne par (1.1.21)

sur l’equation de Schrodinger L2-critique focalisante sur la droite

i∂tw + ∂2xw = −|w|4w, (t, x) ∈ R× R. (1.2.2)

Cette equation admet l’invariance galileenne et un seuil de masse, au–dessus duquel il existe des solutionsqui explosent en temps fini, et sous lequel la solution diffuse toujours. (voir Cazenave–Weissler [25, 26],


Cazenave–Lions [24], Weinstein [145] et Dodson [31]) En ajoutant le projecteur de Szego ΠR devant leterme non lineaire, on obtient l’equation de NLS–Szego quintique focalisante sur la droite,

i∂tu+ ∂2xu = −ΠR(|u|4u), (t, x) ∈ R× R . (1.2.3)

On voit tout de suite des grandes differences entre l’equation (1.2.2) et l’equation (1.2.3). Les resultatssur les equations de NLS–Szego sfont l’objet des chapitres 2 et 3 de cette these, voir [133, 134]. Quant auchapitre 4, il est consacre a l’equation de Benjamin–Ono (BO) sur la droite,

∂tu = H∂2xu− ∂x(u2), (t, x) ∈ R× R, (1.2.4)

ou u est a valeurs reelles et H est la transformation d’Hilbert H = −isign(D), definie par

Hf(ξ) = −isign(ξ)f(ξ), ∀f ∈ L2(R). (1.2.5)

avec sign(±ξ) = ±1, pour tout ξ > 0. On remarque que le projecteur de Szego peut s’ecrire sous la formeΠR = 1

2 (IdL2(R) + iH). On utilise l’abreviation Lp(R) = Lp(R,C). L’espace Lp(R,R) consiste en toutesles Lp-fonctions a valeurs reelles. Comme dans les travaux precedents, on peut en fait ecrire l’equationde BO sous la forme d’une equation de Schrodinger non lineaire filtree par le projecteur de Szego ΠR. Sion pose v = ΠRu, ou u resout l’equation (1.2.4), alors

i∂tv − ∂2xv + ∂x[v2 + 2ΠR(|v|2)] = 0, (t, x) ∈ R× R. (1.2.6)

1.2.1 L’equation de NLS–Szego cubique sur le tore

Ce projet correspond au chapitre 2 et a l’article [133]. L’equation de NLS-Szego cubique defocalisante surle tore (1.2.1) peut s’obtenir a partir de l’equation de Szego cubique (1.1.7) en ajoutant le terme dispersif∂2x. Afin de mesurer la gradation de dispersivite, on rajoute un parametre devant le laplacien ∂2

x. L’objetprincipal de cette sous-section est l’equation suivante

i∂tu+ εα∂2xu = ΠT(|u|2u), 0 < ε < 1, α ≥ 0. (1.2.7)

On munit encore L2(T) de la forme symplectique ωL2(T)(f, g) = Im〈f, g〉L2(T), en sorte que l’equation(1.2.7) est hamiltonienne avec l’energie definie dans H1

+(T) :

Eα,ε(u) =εα

2‖∂xu‖2L2 +

1

4‖u‖4L4 , u ∈ H1

+. (1.2.8)

L’equation (1.2.7) admet encore deux autres lois de conservation,

Q(u) = ‖u‖2L2 , I(u) = Im

∫Tu∂xu = ‖|D| 12u‖2L2(T).

Rappelons l’estimation d’interpolation

supt∈R‖u(t)‖Hs ≤ ‖u0‖1−2s

L2 ‖u0‖2sH

12, ∀s ∈ [0,

1

2].

Afin de montrer l’existence et l’unicite de la solution globale de l’equation (1.2.7), on utilise l’inegalitede Brezis–Gallouet [17], le theoreme d’Aubin–Lions–Simon (voir Theorem II.5.16 de Boyer–Fabrie [15])et l’inegalite de Trudinger (voir Yudovich [151], Vladimirov [144], Ogawa [114] et Gerard–Grellier [47]).L’existence et l’unicite dans les espaces de Sobolev en basse regularite peuvent etre obtenues comme dansBourgain [13]. Dans ce qui suit, on ne considere que les estimations dans l’espace de Sobolev en hauteregularite.


Proposition 1.2.1. Soient s ≥ 12 et u0 ∈ Hs

+(T), il existe une unique fonction u ∈ C(R, Hs+(T)) qui

resout l’equation (1.2.7) avec la donnee initiale u(0) = u0. Pour tout T > 0, l’application flot u0 ∈Hs

+(T) 7→ u ∈ C([−T, T ], Hs+(T)) est continue.

Cette proposition est recapitulee au theoreme 1.4.2 plus loin. On obtient deux ensembles de resultatsconcernant le comportement en grand temps des solutions de l’equation (1.2.7). Le premier resultatconcerne l’estimation en grand temps de la norme de Sobolev de la solution. Si la donnee initiale u0 estbornee par ε, on cherche un intervalle dans lequel la solution u(t) reste encore bornee par O(ε).

Theorem 1.2.2. Soit s > 12 , il existe deux constantes as ∈]0, 1[ et Ks > 0 telles que si 0 < ε 1 et

u0 ∈ Hs+ avec ‖u0‖Hs = ε, u resout l’equation (1.2.7) avec u(0) = u0, alors on a

sup|t|≤ asε4−α

‖u(t)‖Hs ≤ Ksε, si α ∈ [0, 2];

sup|t|≤ asε2‖u(t)‖Hs ≤ Ksε, si α > 2.

(1.2.9)

De plus, dans le cas α > 2 et s ≥ 1, l’intervalle temporel Iαε = [−asε2 ,asε2 ] est maximal au sens ou

∀ 0 < ε 1, il existe une donnee uε0 ∈ C∞+ telle que ‖uε0‖Hs ' ε et pour tout β > 0, et on a

sup|t|≤ 1

ε2+β

‖u(t)‖Hs & ε| ln ε| 12 ε, u(0) = uε0.

Remark 1.2.3. Dans le cas α ∈ [0, 2), on utilise la methode de forme normale de Birkhoff, qui estsimilaire a Bambusi [7], Grebert [64], Gerard–Grellier [48] et Faou–Gauckler–Lubich [34] etc. En revanche,on ne sait pas si l’intervalle temporel [− as

ε4−α ,asε4−α ] est optimal. Les termes de resonances a 6 indices

dans l’equation homologique ne peut pas etre supprimes par notre transformation de forme normale deBirkhoff.(voir le chapitre 2)

Le deuxieme ensemble de resultats concerne la stabilite en grand temps d’une onde plane. On definitl’onde plane em : x 7→ eimx, pour tout m ∈ N et s ≥ 1. Soit u = u(t, x) une solution de l’equation (1.2.7)telle que ‖u(0)− em‖Hs = ε. Par conservation de energie (2.1.6), on deduit l’estimation suivante :

supt∈R‖u(t)‖H1 .‖u0‖H1

ε−α2 , ∀0 < ε < 1, α ≥ 0. (1.2.10)

En revanche, aucune information sur la stabilite de l’onde plane em n’est obtenue a partir de (1.2.10)lorsque ε→ 0+. Compte tenu du phenomene de croissance de norme de Sobolev pour l’equation de Szegocubique (1.1.7) sur le tore (voir theoreme 1.1.9), le phenomene de turbulence d’onde pour (1.2.7) vadependre de la taille de sa dispersion. On commence par trois resultats sur la stabilite de l’onde planedans le cas ou la dispersion est polynomiale en ε, εα∂2

x avec 0 ≤ α ≤ 2. Le prochain theoreme indique lastabilite orbitale de l’onde plane em par rapport a la norme H1 pour l’equation (1.2.7).

Theoreme 1.2.4. Soient ε ∈]0, 1[, α ∈ [0, 2] et m ∈ N, il existe Cm > 0 telle que si ‖u(0)− em‖H1 = ε,alors on a

supt∈R

infθ∈R‖u(t)− eiθem‖H1 ≤ Cmε1−

α2 .

Pour tout t ∈ R, la borne inferieure est atteint lorsque θ = arg um(t). Des resultats similaires sont obtenuspar Zhidkov [153, Sect. 3.3] et Gallay–Haragus [43, 44] pour l’equation de Schrodinger cubique (1.1.5)avec d = 1. Dans le cas de basse dispersion, le theoreme 1.2.4 ameliore l’estimation (1.2.10). Pour tout

φ ∈ H12+ , on note u : t ∈ R 7→ Sα,ε(t)φ ∈ H

12+ la solution de l’equation (1.2.7) telle que u(0) = φ.


Corollary 1.2.5. Soit m ∈ N, on a sup 0<ε<10≤α≤2

sup‖φ−em‖H1≤ε supt∈R ‖Sα,ε(t)φ‖H1 <∞.

En comparant avec le phenomene de marguerite dans Gerard–Grellier [47, 48, 51], le terme dispersif εα∂2x

empeche la croissance de la norme H1, pour l’equation (1.2.7), si 0 ≤ α ≤ 2. En utilisant le changementde variable u(t) = ei arg um(t)(em + ε1−

α2 v(t)), un argument de bootstrap conduit a la stabilite orbitale en

grand temps de l’onde plane em par rapport aux normes de Sobolev en hautes frequences.

Proposition 1.2.6. Soient s ≥ 1 et m ∈ N, il existe deux constantes bm,s ∈]0, 1[ et Lm,s > 0 telles quesi 0 ≤ α < 2 et ‖u(0)− em‖Hs = ε ∈]0, 1[, alors on a

sup|t|≤ bm,s

ε1−α

2

infθ∈R‖u(t)− eiθem‖Hs ≤ Lm,sε1−

α2 . (1.2.11)

On aimerait trouver un intervalle temporel plus grand dans lequel l’estimation (1.2.11) soit encore vraiepar la transformation de forme normale de Birkhoff. Mais les coefficients devant les modes de Fourier dehautes frequences dans l’equation homologique peuvent etre arbitrairement grands, si α ∈]0, 2[. Donc onrevient au cas α = 0 et on considere l’equation suivante.

i∂tu+ ∂2xu = ΠT(|u|2u), (t, x) ∈ R× T.

Dans ce cas, on a le theoreme suivant.

Theorem 1.2.7. Pour tout s ≥ 1, pour tout m ∈ N, il existe trois constantes dm,s, εm,s ∈ (0, 1) etKm,s > 0 telles que si ‖u(0)− em‖Hs = ε ∈ (0, εm,s), alors

sup|t|≤ dm,s

ε2

infθ∈R‖u(t)− eiθem‖Hs ≤ Km,sε.

Des resultats similaires ont ete etablis par Faou–Gauckler–Lubich [34] pour l’equation de Schrodingercubique (1.1.5) pour tous les d ∈ N. (voir le chapitre 2 pour la comparaison entre l’equation (1.2.7) et(1.1.5)) Apres avoir etabli ces resultats de stabilite, on va construire des solutions pour l’equation (1.2.7)qui sont grandes par rapport a leur donnees initiales.

Theorem 1.2.8. Il existe une constante K > 0 telle que pour tout 0 < δ 1, on note U : t ∈ R 7→U(t) ∈ C∞+ (T) la solution de l’equation de NLS-Szego suivante

i∂tU + ν2∂2xU = ΠT(|U |2U), U(0, x) = eix + δ, (1.2.12)

ou ν = e−πK2δ2 , alors on a ‖U(tδ)‖H1 ' 1

δ avec tδ := πδ√

4+δ2.

En d’autres termes, le support de l’energie de la solution de l’equation (1.2.12) est transfere vers lesgrandes modes de Fourier. Ce resultat est similaire au cas de l’equation de Szego cubique (1.1.7) obtenupar Gerard–Grellier [50, 51, 52] et au cas l’equation de Schrodinger cubique (1.1.5) avec d = 2 obtenupar Colliander–Keel–Staffilani–Takaoka–Tao [29]. En comparant avec le theoreme 1.2.4, on montre quele phenomene de turbulence faible demeure si la dispersion de l’equation de NLS–Szego est exponentiel-lement basse par rapport a la perturbation de l’onde plane e1 : x 7→ eix.

Remarque 1.2.9. La deuxieme partie du theoreme 1.2.2 est une consequence du theoreme 1.2.8. En

fait, a chaque α > 2 fixe, on fait la dilatation u(t, x) = εU(ε2t, x) avec e−πK2δ2 = ν = ε

α−22 . Alors u resout

(1.2.7) avec la donnee initiale u(0, x) = ε(eix + δ) et on a

‖u( tδ

ε2 )‖H1 = ε‖U(tδ)‖H1 ' ε

δ' ε√

(α− 2)| ln ε| ε,


et tδ

ε2 '√

(α−2)| ln ε|ε2 1

ε2+β , pour tout β > 0. En revanche, cette methode ne marche pas dans le cascritique α = 2. Si u resout

i∂tu+ ε2∂2xu = ΠT(|u|2u), u(0, x) = ε(eix + δ),

on fait le changement de variable U(t, x) = ε−1u(ε−2t, x) et on obtient l’equation (1.2.12) avec ν = 1.Grace aux theoremes 1.2.4 et 1.2.7, on a des estimations suivantes

supt∈R‖u(t)‖H1 = O(ε), sup

|t|≤ d1,s

ε2δ2

‖u(t)‖Hs = O(ε), ∀0 < δ 1, ∀0 < ε < 1,

pour tout s > 12 . Le probleme de trouver l’intervalle temporel optimal dans le cas α = 2 du theoreme 1.2.2

reste ouvert.

1.2.2 L’equation de NLS–Szego quintique sur la droite

Ce projet correspond au chapitre 3 de la these et a l’article Sun [134]. On rappelle d’abord la definitionde la diffusion pour une solution de (1.2.3).

Definition 1.2.10. Soit s ≥ 0 fixe, une solution globale u ∈ C(R;Hs+(R)) de l’equation (1.2.3) est dite

Hs-diffuse vers l’avant en temps s’il existe une fonction u+ ∈ Hs+(R) telle que

limt→+∞

‖eit∂2xu+ − u(t)‖Hs(R) = 0.

Une solution globale u ∈ C(R;Hs+(R)) de l’equation (1.2.3) est dite Hs-diffuse vers l’arriere en temps s’il

existe une fonction u− ∈ Hs+(R) telle que

limt→−∞

‖eit∂2xu− − u(t)‖Hs(R) = 0.

Dans le cas d’une donnee petite dans L2, l’equation (1.2.3) et (1.2.2) sont globalement bien posees dansl’espace de Hardy L2

+(R) et leurs solutions L2-diffusent simultanement a l’avant et a l’arriere en temps.La preuve est basee sur l’inegalite de Strichartz, et est similaire a celle de Cazenave–Weissler [25, 26].

Proposition 1.2.11. Il existe ε0 > 0 telle que si ‖u0‖L2 ≤ ε0, il existe une unique fonction u ∈ C(R;L2+)

qui resout l’equation (1.2.3) et u L2-diffuse simultanement a l’avant et a l’arriere en temps.

Il y a trois lois de conservation communes pour (1.2.2) et (1.2.3)

M(u) = ‖u‖2L2 , P (u) = 〈Du, u〉L2 , ENLS(u) =‖∂xu‖2L2

2−‖u‖6L6

6, (1.2.13)

ou D = −i∂x et on choisit u ∈ H1+(R) pour (1.2.3), w ∈ H1(R) pour (1.2.2). Si u ∈ C(R;H1

+(R)) resout

l’equation (1.2.3), alors la quantite de mouvement P (u) = ‖|D| 12u‖2L2(R) et la masse M(u) controlent la

norme H12 de la solution, ce qui permet de resoudre le probleme de Cauchy de l’equation de NLS–Szego

L2-(sur)critique sur la droite, pour tous les donnees initiales u0 ∈ H1+(R).

Theoreme 1.2.12. Soient m ≥ 0, λ = ±1 et u0 ∈ H1+(R), il existe une unique fonction u ∈ C(R;H1

+(R))qui resout l’equation de NLS–Szego,

i∂tu+ ∂2xu = λΠR(|u|2mu), u(0, x) = u0(x), (t, x) ∈ R× R. (1.2.14)


Demonstration. L’existence et l’unicite sont obtenues par l’injection de Sobolev. Dans le cas λ = −1,pour montrer que la solution soit globale, il suffit d’utiliser l’inegalite de Gagliardo–Nirenberg suivante,

‖u‖L2m+2(R) .m ‖|D|12u‖

mm+1

L2(R)‖u‖1

m+1

L2(R), ∀m ≥ 0, ∀u ∈ H12+(R). (1.2.15)

et les trois lois de conservation (1.2.13) pour obtenir que

supt∈R‖∂xu(t)‖2L2(R) .m ‖∂xu0‖2L2(R) + ‖|D| 12u0‖2mL2(R)‖u0‖2L2(R).

En revanche, lorsque la donnee initiale w0 ∈ H1(R) telle que ENLS(w0) < 0 et w0 ∈ L2(R, x2dx), alors ilest bien connu que la solution w : t ∈ R 7→ w(t) ∈ H1(R) de l’equation (1.2.2) associee a la donnee initialew0 explose en temps fini, grace a l’identite du viriel (voir Glassey [61], Cazenave [22, 23], Ogawa–Tsutsumi[115] etc.). On fait reference a Perelman [118] et Merle–Raphael [106] pour une description detaillee dela dynamique de l’explosion pour l’equation (1.2.2).

On va montrer que l’equation (1.2.3), admet une onde progressive qui s’ecrit sous la forme u(t, x) =eiωtQ(x + ct) pour certaines constantes ω, c ∈ R. La fonction u resout l’equation (1.2.3) si et seulementsi le profil Q resout l’equation elliptique non locale suivante

∂2xQ+ ΠR(|Q|4Q) = ωQ+ cDQ. (1.2.16)

Pour m ≥ 2 et γ ≥ 0, on definit une fonctionnelle qui est invariante par la translation spatiale, par larotation de phase et par la dilatation interieure et exterieure,

I(γ)m (f) :=

‖∂xf‖mL2(R)‖f‖m+2L2(R) + γ‖|D| 12 f‖2mL2(R)‖f‖

2L2(R)

‖f‖2m+2L2m+2(R)

, ∀f ∈ H1(R)\0. (1.2.17)

on a I(γ)m (f) = I

(γ)m (fλ,µ,y,θ), ou fλ,µ,y,θ(x) = λeiθf(µx+ y), ∀x, y, θ ∈ R et ∀λ, µ > 0.

Definition 1.2.13. La borne inferieure de I(γ)m est notee par J

(γ)m = inff∈H1

+(R)\0 I(γ)m (f). L’ensemble

de ses minimiseurs est note par

G(γ)m = f ∈ H1

+(R)\0 : I(γ)m (f) = J (γ)

m =⋃a,b>0

G(γ)m (a, b), (1.2.18)

ou G(γ)m (a, b) = f ∈ G(γ)

m : ‖f‖L2(R) = a, ‖f‖L2m+2(R) = b.

Alors on a

G(γ)m (a, b) = λf(µ·) ∈ H1

+(R)\0 : f ∈ G(γ)m (1, 1), λ = a−

1m b

m+1m et µ = b

2m+2m a−

2m+2m .

Une consequence du theoreme de decomposition en profil (theoreme 1.4.1) est le resultat suivant, qui

indique l’existence de miniseurs de I(γ)m .

Theoreme 1.2.14. Soient m ≥ 2 et γ ≥ 0, si (fn)n∈N ⊂ H1+(R) est une suite minimisante de I

(γ)m telle

que ‖fn‖L2(R) = ‖fn‖L2m+2(R) = 1 et limn→+∞ I(γ)m (fn) = J

(γ)m , il existe un profil U ∈ G

(γ)m (1, 1), une

fonction strictement croissante ψ : N→ N et une suite a valeurs reelles (xn)n∈N tels que

limn→+∞

‖fψ(n) − τxnU‖H1(R) = 0, avec τyU(x) = U(x− y), ∀x, y ∈ R. (1.2.19)


Remarque 1.2.15. Si H1+(R) est remplace par H1(R), alors il existe w ∈ H1(R)\0 telle que ‖w‖L2(R) =

‖w‖L2m+2(R) = 1, I(γ)m (w) = minf∈H1(R)\0 I

(γ)m (f) et la limite (1.2.19) marche encore.

Soient m ≥ 2, γ ≥ 0 et f ∈ H1+\0 est un minimiseur de I

(γ)m , alors d

dε

∣∣∣ε=0

log I(γ)m (f + εh) = 0, pour

tout h ∈ H1+. f resout l’equation d’Euler–Lagrange

m‖f‖m+2L2 ‖∂xf‖m−2

L2 ∂2xf + 2(m+ 1)J (γ)

m ΠR(|f |2mf)

=((m+ 2)‖f‖mL2‖∂xf‖mL2 + 2γ‖|D| 12 f‖2mL2 )f + 2γm‖f‖2L2‖|D|12 f‖2m−2

L2 Df.(1.2.20)

On fixe m = 2. Ensuite l’equation (1.2.16) est identifiee comme l’equation suivante

‖Q‖4L2

3J(γ)2

∂2xQ+ ΠR(|Q|4Q) =

2‖Q‖2L2‖∂xQ‖2L2 + γ‖|D| 12Q‖4L2

3J(γ)2

Q+2γ‖Q‖2L2‖|D|

12Q‖2L2

3J(γ)2

DQ.

Un minimiseur Q(γ) ∈ G(γ)2 sera appele un etat fondamental de la fonctionnelle I

(γ)2 , si ‖Q(γ)‖4L2 = 3J

(γ)2 .

Lorsque u(t, x) = eiωtQ(γ)(x + ct) resout (1.2.3) et Q(γ) ∈ G(γ)2 (

4

√3J

(γ)2 , b) pour certaines constantes

b, ω > 0 et c, γ ≥ 0, alors c = 0 si et seulement si γ = 0.

Si γ = 0, alors on a ‖∂xQ(γ)‖2L2 = ω2

√3J

(0)2 and ‖Q(γ)‖6L6 = 3ω

2

√3J

(0)2 .

Si γ > 0, alors on a

‖∂xQ(γ)‖2L2 =

√3J

(γ)2

8γ(4γω − c2), ‖|D| 12Q(γ)‖2L2 =

√3J

(γ)2 c

2γ, ‖Q(γ)‖6L6 = 3

√3J

(γ)2 (

ω

2+c2

8γ).

De plus, l’inegalite ‖|D| 12Q(γ)‖2L2 ≤ ‖Q(γ)‖L2‖∂xQ(γ)‖L2 implique que c2 ≤ 4γ2ωγ+2 .

Meme si les etats fondamentaux ne sont pas encore tous classifies, on a une version faible de H1-stabiliteorbitale par le theoreme 1.2.14 et par la conservation de la H

12 -norme.

Theoreme 1.2.16. Soient ε, b > 0 et γ ≥ 0, il existe δ = δ(b, ε, γ) > 0 telle que si

inff∈G(γ)

2 (4√

3J(γ)2 ,b)

‖u0 − f‖H1+(R) < δ,

alors on a supt∈R infΨ∈

⋃C(γ)−1≤θ≤C(γ) G

(γ)2 (

4√

3J(γ)2 ,θb)

‖u(t)−Ψ‖H1+(R) < ε, ou u est la solution de l’equation

(1.2.3) avec la donnee initiale u(0) = u0 et C(γ) :=

(inff∈H1

+(R)\0‖|D|

13 f‖L2(R)

‖f‖L6(R)

)−16

√J

(γ)2

1+γ .

Remarque 1.2.17. Le probleme d’unicite de l’etat fondamental d’une equation elliptique non locale estdifficile a resoudre. (Voir Frank–Lenzmann [40] et Frank–Lenzmann–Silvestre [41] pour les Laplaciensfractionnaires dans R et aussi Lenzmann–Sok [99] pour un principe de rearrangement de Fourier) Pourun γ ≥ 0 general , on a seulement la stabilite orbitale avec dilatation parce que l’unicite des etats fon-

damentaux de I(γ)2 est inconnue, la norme de L6 de l’etat fondamental Ψ qui s’approche u(t) est aussi

inconnue. On a seulement un domaine‖f‖L6

C(γ) ≤ ‖Ψ‖L6 ≤ C(γ)‖f‖L6 , ou f est l’etat fondamental qui

s’approche de la donnee initiale u0.


Toutefois, dans un cas particulier, on peut montrer l’unicite des etats fondamentaux a dilatation, arotation de phase et a translation spatiale pres, en utilisant l’inegalite de Cauchy–Schwarz.

Proposition 1.2.18. Dans le cas γ = m = 2, on a

J(2)2 = min

f∈H1+\0

I(2)2 (f) =

8π2

3, G

(2)2 = x 7→ λeiθ

µx+ y + i∈ H1

+ : ∀λ, µ > 0 ∀θ, y ∈ R. (1.2.21)

Si u(t, x) = eiωtQ(x+ ct) est l’onde progressive de l’equation (1.2.3) et Q ∈ G(2)2 , alors on a

3c2 = 8ω, ‖Q‖4L2 = 8π2, ‖Q‖6L6 =3πc2√

2, ‖|D| 12Q‖2L2 =

πc√2, ‖∂xQ‖2L2 =

πc2

2√

2.

On en deduit que l’onde progressive uc(t, x) = e3c2it

8 Qc(x + ct) est H1-stable orbitalement, pour tout

c > 0, ou Qc ∈ G(2)2 (

4√

8π2, 6

√3πc2√

2).

Theoreme 1.2.19. Pour tout ε, c > 0, il existe une constante δε,c > 0 telle que

inff∈G(2)

2 (4√

8π2, 6√

3πc2√2

)

‖u0 − f‖H1 < δ,

alors on a supt∈R inff∈G(2)

2 (4√

8π2, 6√

3πc2√2

)‖u(t) − f‖H1 < ε, ou u resout l’equation (1.2.3) avec la donnee

initiale u(0) = u0.

On trouvera des resultats similaires de classification des etats fondamentaux au moyen de l’inegalite deCauchy–Schwarz dans Foschi [39], Gerard–Grellier [47, 52], Pocovnicu [119] etc.

Dans le cas γ = 0, on a

I(0)2 (f) :=

‖∂xf‖2L2‖f‖4L2

‖f‖6L6

, ∀f ∈ H1(R)\0.

Tous les etats fondamentaux dans H1(R) de I(0)2 sont completement classifies dans Weinstein [145].

On sait que minf∈H1(R)\0 I(0)2 (f) = π2

4 . Il existe une unique fonction radiale a valeurs positives qui

vaut R(x) =4√3√

cosh(2x)telle que (I

(0)2 )−1(π

2

4 ) = λeiθR(µ · −y) : λ, µ > 0 θ, y ∈ R. L’onde progrs-

sive w(t, x) = eiωtR(x) est une solution instable de l’equation de Schrodinger L2-critique focalisante

(1.2.2) au sens suivant : il existe une suite w(n)0 = (1 + 1

n )R ⊂ H1(R) telle que w(n)0 → R, lorsque

n → +∞, mais la solution maximale correspondante w(n) s’explose en temps fini. On note un etat fon-

damental par Q(0) ∈ G(0)2 (

4

√3J

(0)2 , ‖Q(0)‖L6) de I

(0)2 dans l’espace de Hardy H1

+(R). Comme R /∈ H1+ et

Q+ : x 7→ 1x+i ∈ H

1+, on a π2

4 = I(0)2 (R) < J

(0)2 = I

(0)2 (Q(0)) ≤ I(0)

2 (Q+) = 4π2

3 .

La proposition 1.2.11 implique que les solutions de l’equation (1.2.3) et de l’equation (1.2.2) L2-diffusentsimultanement a l’avant et a l’arriere en temps, lorsque leurs donnees initiales sont suffisament petites.De plus, Dodson [31] a montre que la solution globale de l’equation de Schrodinger L2-critique focalisante(1.2.2) L2-diffusent simultanement a l’avant et a l’arriere en temps, si sa donnee initiale admet la masse‖w0‖ < ‖R‖L2 . Vu le resultat d’instabilite de Weinstein [145], le seui”l de masse de l’equation (1.2.2)

pour L2-diffusion egale la masse de l’etat fondamental R ∈ H1(R) de I(0)2 .


Par contre, le theoreme de stabilite orbitale (theoreme 1.2.16) implique que le seuil de masse de l’equation

(1.2.3) pour L2-diffusion est strictement plus petit que la masse de l’etat fondamental Q(0) ∈ H1+ de I

(0)2 .

On definit E ⊂ R∗+ l’ensemble des ε > 0 telles que si ‖u0‖L2 < ε, la solution correspondante de l’equation(1.2.3) L2-diffuse simultanement a l’avant et a l’arriere en temps. Si une H1-solution de l’equation (1.2.3)L2-diffuse, alors elle H1-diffuse aussi et son Lr-norme decroıt, pour tout 2 < r ≤ +∞. Pour cette raison,les ondes progressives ne peuvent pas L2-diffuser. Les solutions qui s’approchent des ondes progressivesne L2-diffusent non plus.

Corollary 1.2.20. On a sup E < ‖Q(0)‖L2 =4

√3J

(0)2 . Plus precisement, il existe une fonction u ∈

C(R;H1+(R)) qui resout l’equation (1.2.3), telle que ‖u(0)‖L2 < ‖Q(0)‖L2 , et telle que u ne L2-diffuse ni

a l’avant ni a l’arriere en temps.

Remark 1.2.21. La masse de l’etat fondamental de I(2)2 est strictement plus grande que la masse de

l’etat fondamental de I(0)2 . ‖Q(2)‖4L2 = 8π2 > 4π2 ≥ ‖Q(0)‖4L2 .

E(Q(2)) = − πc2

4√

2< 0 = E(Q(0)).

L’identification du seuil de masse pour la diffusion des solutions de l’equation (1.2.3) demeure un problemeouvert. L’equation (1.2.3) n’a pas d’invariance galileenne, qui est necessaire pour utiliser des estimationsde type Morawetz. On fait reference a Dodson [31], Killip–Visan [88], Killip–Visan–Zhang [89] pour plusde detail dans le cas non filtre.

1.2.3 L’equation de Benjamin–Ono sur la droite

Ce paragraphe correspond au chapitre 4 et a l’article Sun [135]. L’equation de Benjamin–Ono (BO) surla droite (1.2.4)

∂tu = H∂2xu− ∂x(u2), (t, x) ∈ R× R,

a ete d’abord introduite par Benjamin [8] puis par Ono [116] pour modeliser la propagation d’ondesinternes longues non lineaires dans un fluide stratifie par deux couches de differentes densites, jointes parune region mince ou la densite varie de facon continue, la couche inferieure est infinie. L’equation (1.2.4)admet une solution globale et unique dans l’espace C(R, Hs(R,R)), pour tout s ≥ 0. (Voir Tao [137] pours ≥ 1, Burq–Planchon [21] pour s > 1

4 , Ionescu–Kenig [80], Molinet–Pilod [108] et Ifrim–Tataru [79] pours ≥ 0, etc.) Si u = u(t, x) est solution de l’equation de BO (1.2.4), alors uc,y : (t, x) 7→ cu(c2t, c(x−y)) l’estaussi. Une solution u = u(t, x) s’appelle une onde solitaire s’il existe R ∈ C∞(R) qui resout l’equationelliptique non locale suivante

HR′ +R−R2 = 0, R(x) > 0 (1.2.22)

et u(t, x) = Rc(x − y − ct), ou Rc(x) = cR(cx), pour certaines constantes c > 0 et y ∈ R. L’equation(1.2.22) admet une unique solution a translation pres,

R(x) =2

1 + x2, ∀x ∈ R, (1.2.23)

trouvee par Benjamin [8] et dont l’unicite est montree dans Amick–Toland [4]. On s’interesse aux solutionsmulti-soliton de l’equation (1.2.4).


Definition 1.2.22. Soit N un entier strictement positif, une fonction u : R → R s’appelle un N -soliton s’il existe c1, c2, · · · , cN > 0 et x1, x2, · · · , xN ∈ R tels que u(x) =

∑Nj=1Rcj (x − xj). On note

UN ⊂ L2(R,R) l’ensemble des N -solitons.

L’ensemble UN est une sous-variete reellement analytique simplement connexe de l’espace de HilbertL2(R,R). On a dimR UN = 2N . Tous les espaces tangents de UN sont inclus dans un sous-espace auxiliaire

T := h ∈ L2(R, (1 + x2)dx) : h(R) ⊂ R,∫Rh = 0, (1.2.24)

sur lequel on definit un 2-tenseur alterne ω ∈ Λ2(T ∗) par

ω(h1, h2) =i

2π

∫R

h1(ξ)h2(ξ)

ξdξ, ∀h1, h2 ∈ T . (1.2.25)

Grace a l’inegalite de Hardy, ω ∈ Λ2(T ∗) est bien defini. Muni de la 2-forme ω : u ∈ UN 7→ ω ∈Λ2(T ∗), (UN , ω) est une variete symplectique. Soit une fonction lisse f : UN → R, son champ de vecteurshamiltonien Xf ∈ X(UN ) s’ecrit sous la forme

Xf : u ∈ UN 7→ ∂x∇f(u) ∈ Tu(UN ), (1.2.26)

ou ∇f(u) est la derivee de Frechet : df(u)(h) = 〈h,∇uf(u)〉L2 , pour tout h ∈ Tu(UN ). Le crochet dePoisson entre deux fonctions lisses f, g : UN → R est defini par

f, g : u ∈ UN 7→ ωu(Xf (u), Xg(u)) = 〈∂x∇uf(u),∇ug(u)〉L2 ∈ R.

Alors l’equation de BO (1.2.4) dans la variete UN est un systeme hamiltonien

∂tu = XE(u), E(u) =1

2〈|D|u, u〉

H−12 ,H

12− 1

3

∫Ru3. (1.2.27)

Le systeme (1.2.27) admet une unique solution globale t ∈ R 7→ u(t) ∈ UN . On s’inspire de la constructiondes coordonnees de Birkhoff pour l’equation de BO sur la tore T = R/2πZ dans Gerard–Kappeler [54].On va construire les coordonnees d’action–angle generalisees pour le systeme (1.2.27).

Soit ΩN := (r1, r2, · · · , rN ) ∈ RN : rj < rj+1 < 0, ∀j = 1, 2, · · · , N − 1 l’ensemble des actions et

ν =∑Nj=1 drj ∧ dαj la forme symplectique canonique sur ΩN × RN . On a le resultat principal.

Theoreme 1.2.23. Il existe un symplectomorphisme reellement analytique ΦN : (UN , ω)→ (ΩN×RN , ν)tel que

E Φ−1N (r1, r2, · · · , rN ;α1, α2, · · · , αN ) = − 1

2π

N∑j=1

|rj |2. (1.2.28)

Remarque 1.2.24. Par consequent, la variete UN est simplement connexe. Elle est le revetement uni-versel de la variete des potentiels a N “gap” de l’equation de Benjamin–Ono sur le tore decrite parGerard–Kappeler [54]. Dans l’appendice 4.6, on fournit une preuve directe de ces resultats topologiques,qui est independante du theoreme 1.2.23.


Remarque 1.2.25. L’application ΦN : u ∈ UN 7→ (I1(u), I2(u), · · · , IN (u); γ1(u), γ2(u), · · · , γN (u)) ∈ΩN × RN introduit les coordonnees d’action–angle generalisees de l’equation de BO dans la variete deN -solitons,

Ik, E(u) = 0, γk, E(u) =Ik(u)

π, ∀u ∈ UN . (1.2.29)

Theoreme 1.2.23 donne une description complete de l’orbite du flot du systeme (1.2.27) a conjugaison reelanalytique pres. Soit u : t ∈ R 7→ u(t) ∈ UN la solution de l’equation (1.2.27), on note rk(t) = Ik u(t)les coordonnees d’actions et αk(t) = γk u(t) les coordonnees d’angles generalises αk(t) = γk u(t), ona donc

rk(t) = rk(0), αk(t) = αk(0)− rk(0)t

π, ∀k = 1, 2, · · · , N. (1.2.30)

On donnera le detail de l’application ΦN : UN → ΩN × RN dans la definition 4.5.1 et le theoreme 4.5.2.

Afin d’etablir le lien entre les coordonnees action–angle et les parametres de translation–dilatation d’unN -soliton, on introduit la matrice d’inversion spectrale associee a ΦN , definie par

M : u ∈ UN 7→ (Mkj(u))1≤j,k≤N ∈ CN×N , Mkj(u) =

2πiIk(u)−Ij(u)

√Ik(u)Ij(u) , if j 6= k,

γj(u) + πiIj(u) , if j = k,

(1.2.31)

ou Ik, γk : U → R sont donnees par remarque 4.1.3. On peut montrer la caracterisation polynomialesuivante de UN .

Proposition 1.2.26. Une fonction a valeurs reelles u appartient a UN si et seulement s’il existe unpolynome Qu ∈ C[X] de degre N dont les racines sont contenues dans le demi-plan inferieur de Poincare

C−, et u = Πu + Πu, avec Πu = iQ′uQu

. De plus, Qu est unique et est le polynome caracteristique de la

matrice M(u) ∈ CN×N donnee par (1.2.31).

Le probleme de trouver une formule explicite pour la solution d’equation (1.2.27) devient donc un

probleme algebrique. Plus precisement, un N -soliton s’ecrit sous la forme u(x) =∑Nj=1Rcj (x − xj)

si et seulement si ses parametres de translation–dilatation xj − c−1j i1≤j≤N ⊂ CN− sont des racines du

polynome caracteristique Qu(X) = det(X −M(u)) dont les coefficients sont exprimees en fonction descoordonnees action–angle (Ij(u), γj(u))1≤j≤N ∈ ΩN ×RN . La proposition 1.2.26 est recapitulee avec plusde detail dans la proposition 4.4.1 et dans le theoreme 4.4.8, qui caracterise la variete UN en termesde theorie spectrale pour un certain operateur de Lax. En particulier, la formule d’inversion spectrale

Πu = iQ′uQu

avec Qu(x) = det(x−M(u)) est recapitulee a la formule (4.5.11), dont on deduit l’injectivite

de l’application d’action–angle ΦN : UN → ΩN ×RN . Soit u : t ∈ R 7→ u(t) ∈ UN la solution de l’equationde BO (1.2.4), on a donc la formule explicite suivante

u(t, x) = 2Im〈(M(u0)− (x+ t

πV(u0)))−1

X(u0), Y (u0)〉CN , (t, x) ∈ R× R, (1.2.32)

ou la produit scalaire de CN est 〈X,Y 〉CN = XTY , et, pour tout u ∈ UN , la matrice V(u) ∈ CN×N etles vecteurs X(u), Y (u) ∈ CN sont definis par

√2πX(u)T = (

√|I1(u)|,

√|I2(u)|, · · · ,

√|IN (u)|),

√2π−1Y (u)T = (

√|I1(u)|−1,

√|I2(u)|−1, · · · ,

√|IN (u)|−1),

V(u) =

I1(u)I2(u)

. . .IN (u)

.


Resume de la preuve du theoreme 1.2.23

La construction des coordonnees action–angle est fondee sur la structure de paire de Lax pour l’equationde BO (1.2.4), decouverte par Nakamura [110], voir aussi Bock–Kruskal [12]. La sous-section 4.2 estconsacree a l’etude du spectre de l’operateur de Lax Lu : h ∈ H1

+(R) 7→ −i∂xh − ΠR(uh) ∈ L2+(R) a

symbole u ∈ L2(R,R), ou ΠR est le projecteur de Szego donnee par (1.1.21) et L2+(R) est l’espace de

Hardy defini par (4.1.13). Lu est un operateur auto-adjoint sur L2+(R) borne par le bas et son spectre

essentiel est σess(Lu) = [0,+∞). Si de plus x 7→ xu(x) ∈ L2(R), on montre que toutes les valeurs propresde Lu sont simples et negatives, grace a une identite auxiliaire, decouverte par Wu [146]. Ensuite onintroduit une fonction generatrice

Hλ(u) = 〈(Lu + λ)−1Πu,Πu〉L2 , if λ ∈ C\σ(−Lu), (1.2.33)

qui engendre l’ensemble des hierarchies BO, et fournit une suite de lois de conservation controlant toutesles normes de Sobolev.

Dans la sous-section 4.3, on etudie le semi-groupe de decalage (S(η)∗)η≥0 qui opere sur l’espace de HardyL2

+(R), ou S(η)f = eηf et eη(x) = eiηx. On obtient une version faible du theoreme de Beurling–Lax,qui se ramene a la resolution d’ une equation differentielle a coefficients constants, et etablit qu’un sous-espace invariant sous l’action du generateur infinitesimal G = i d

dη

∣∣η=0+S(η)∗ s’ecrit toujours sous la

formeC≤N−1[X]

Q , pour un certain polynome unitaire Q dont les racines sont contenues dans le demi-planinferieur de Poincare C−.

La sous-section 4.4 est consacree a la description geometrique de la variete de N -solitons. On etablit sastructure reel analytique et sa structure symplectique d’abord. Ensuite, on continue d’etudier le spectre del’operateur de Lax Lu a symbole u ∈ UN . L’operateur Lu admet exactement N valeurs propres distinctesλu1 < λu2 < · · · < λuN < 0 et l’espace de Hardy se decompose sous la forme

L2+ = Hcont(Lu)

⊕Hpp(Lu), Hcont(Lu) = Hac(Lu) = ΘuL

2+, Hpp(Lu) =

C≤N−1[X]

Qu. (1.2.34)

ou Qu est le polynome caracteristique de u donnee par la proposition 1.2.26 et Θu = QuQu

est une fonctioninterieure sur le demi-plan superieur de Poincare C+. On montre la proposition 1.2.26 et on identifiela matrice M(u) dans (1.2.31) comme la matrice de l’operateur G|Hpp(Lu) associee a la base spectraleϕu1 , ϕu2 , · · · , ϕuN, ou ϕuj ∈ Ker(λuj −Lu) est telle que ‖ϕuj ‖L2 = 1 et

∫R uϕ

uj > 0. La fonction generatrice

Hλ dans (1.2.33) s’identifie comme la transformee de Borel–Cauchy de la mesure spectrale de Lu associeeau vecteur Πu, ce qui entraıne la stabilite de UN sous le flot de l’equation de BO (1.2.4) dans H∞(R,R).Par consequent le systeme (1.2.27) admet une unique solution globale u : t ∈ R 7→ u(t) ∈ UN .

Le theoreme 1.2.23 est montre dans la sous-section 4.5. Les variables d’angles (generalisees) sont desparties reelles des elements de la matrice M(u), c’est-a-dire que γj : u ∈ UN 7→ Re〈Gϕuj , ϕuj 〉L2 ∈ R et lesvariables d’action sont Ij : u ∈ UN 7→ 2πλuj ∈ R. Grace a la structure de paire de Lax dL(u)(XHλ(u)) =

[Bλu , Lu], ou L : u ∈ UN 7→ Lu ∈ B(H1+, L

2+) est R-affine et Bλu est un operateur anti-auto-adjoint sur

L2+(R), on a les formules suivantes concernant le crochet de Poisson,

2πλj , γk = 1j=k, γj , γk = 0 on UN , 1 ≤ j, k ≤ N. (1.2.35)

ce qui implique que ΦN : u ∈ UN 7→ (I1(u), I2(u), · · · , IN (u); γ1(u), γ2(u), · · · , γN (u)) ∈ ΩN × RN estune immersion reel analytique. ΦN est alors un diffeomorphisme grace au theoreme d’inversion globale


de Hadamard. L’injectivite de ΦN peut aussi etre montree par la formule d’inversion spectrale Πu =Q′uQu

avec Qu(X) = det(X−G|Hpp(Lu)). Enfin, on montre que ΦN : (UN , ω)→ (ΩN ×RN , ν) preserve la forme

symplectique en etudiant la sous-variete lagrangienne ΛN :=⋂Nj=1 γ

−1j (0) ⊂ UN .

Dans l’appendice 4.6, on etablit la simple connexite de la variete UN et un revetement de UN vers lavariete des N -gap potentiels sans utiliser la structure d’integrabilite.

Travaux connexes

L’equation de Benjamin–Ono sur la droite et sur le tore a ete largement etudiee dans le domaine de EDPsdepuis les annees 1970. On renvoie a Saut [128] pour un excellent resume de ces resultats. Matsuno [104]a trouve l’expression explicite de multi-solitons en suivant la methode de transformation bilineaire deHirota [75] dans le cas de l’equation de KdV. Les solutions multiphase sont construites par Satsuma–Ishimori [126]. Les ondes solitaires sont classifiees par Amick–Toland [4], ce resultat est revisite par letheoreme 1.2.23 et la proposition 1.2.26. Dans le travail de Dobrokhotov–Krichever [30], les solutionsde multi-phase sont construites par une methode dite d’integration en zones finies, et ces auteurs ontegalement obtenu une formule d’inversion pour des solutions de multi-phase. Par rapport a ce dernierresultat, nous etablissons une description geometrique de la transformation d’inversion spectrale, puisquenotre application action–angle ΦN : UN → ΩN × RN est un symplectomorphisme reel analytique. Deplus, la formule d’inversion spectrale

Πu(x) = iQ′u(x)

Qu(x), Qu(x) = det(x−G|Hpp(Lu)) = det(x−M(u)), ∀x ∈ R. (1.2.36)

fournit une connexion spectrale entre l’operateur de Lax Lu et le generateur infinitesimal G. Dans le casde l’equation de BO quantique, Nazarov–Sklyanin [111] ont introduit aussi une fonction generatrice Hλ.Leur methode est developpee dans Moll [109] pour l’equation de BO classique. La stabilite asymptotiquede l’onde solitaire et de solutions ayant pour donnee initiale la somme de profiles de solitons largementsepares a ete obtenue par Kenig–Martel [85].

Concernant la theorie de l’integrabilite pour l’equation de BO, apres que la paire de Lax a ete trouvee,Ablowitz–Fokas [1], Coifman–Wickerhauser [28], Kaup–Matsuno [84] et Wu [146, 147] ont obtenu desresultats remarquables sur la transformation spectrale directe et inverse pour l’equation de BO sur ladroite. Gerard–Kappeler ont construit les coordonnees de Birkhoff pour l’equation de BO sur le toredans [54]. Notons que l’equation de KdV (voir Kappeler–Poschel [81]) et NLS (voir Grebert–Kappeler[65]) integrables sur la tore T admettent aussi de telles coordonnees de Birkhoff. Elles sont associees al’introduction de sous-varietes de potentiels “finite gap” permettant de transferer la theorie des systemeshamiltoniens en dimension finie vers le cas d’un systeme hamiltonien en dimension infinie. On renvoie aMatveev [105] pour un resume anterieur sur les travaux dans le domaine de ’finite gap potentials’.

Comme on l’a vu, l’equation de Szego cubique sur la tore T (voir Gerard–Grellier [47, 49, 51, 52]) et surla droite R (voir Pocovnicu [119, 120]) admettent des coordonnees d’action–angle globales (generalisees)sur les varietes generiques des fonctions rationnelles de rang fini. De plus la formule d’inversion spectrale,qui est similaire a la formule (1.2.32) de l’equation de Szego cubique permet d’exprimer explicitement leflot de Szego en fonction de la variable temporelle et de la variable de donnee initiale. Le semi-groupe dedecalage (S(η)∗)η≥0 et son generateur infinitesimal G = i d

dη

∣∣η=0+S(η)∗ sont aussi utilises dans Pocovnicu

[119, 120] pour etablir les coordonnees d’action–angle et la formule explicite de la solution.


Remarque 1.2.27. En comparant avec l’equation de Szego cubique sur la droite (1.1.20) (voir theoreme1.1.14 et Pocovnicu [120]), qui admet la paire de Lax (HR

w, ARw), on observe que l’operateur de Hankel HR

w

est compact. Il n’admet donc que du spectre discret. En revanche, dans le cas de l’equation de BO surla droite avec les solutions de multi-solitons (Sun [135]), l’operateur de Lax admet a la fois du spectrecontinu [0,+∞[ et du spectre purement ponctuel qui est une partie finie de ]−∞, 0[.

Remarque 1.2.28. On peut aussi comparer les theoremes 1.1.8, 1.1.18 et 1.2.23. Dans le cas de l’equationde Szego cubique (voir Gerard–Grellier [49] et Pocovnicu [120]), l’operateur de Hankel HR

w peut avoirune valeur propre degeneree. Donc les coordonnees d’action–angle (generalisees) de l’equation de Szegocubique sur la tore (resp. sur la droite) ne sont que generiquement definies dans M(N)T (resp. M(N)R).En revanche, pour l’equation de BO sur la tore (voir Gerard–Kappeler [54]) ou sur la droite (voir Wu[146], Sun [135]), l’operateur de Lax n’admet que des valeurs propres simples dans les regimes appropries.Ainsi les coordonnees de Birkhoff (resp. d’action–angle) sont globalement bien definies dans la varietedes N -gap potentiels (resp. des N -solitons).

Enfin, contrairement aux equations de type Szego, l’equation de BO admet une suite de lois de conser-vation qui controlent toutes les normes de Sobolev Hs (Voir Ablowitz–Fokas [1], Coifman–Wickerhauser[28] dans le cas 2s ∈ N et Talbut [136] dans le cas − 1

2 < s < 0 et les lois de conservation controlant lesnormes de Besov etc).

1.3 Perspectives de recherche

Dans ce paragraphe, on decrit trois perspectives de recherche ulterieure.

1.3.1 L’equation de NLS–Szego amortie

On s’interesse au phenomene de turbulence faible de l’equation de NLS–Szego amortie sur la tore

i∂tu+ ∂2xu+ iα〈u,1〉L2(T) = ΠT(|u|2u), (t, x) ∈ R× T, α > 0. (1.3.1)

La motivation pour etudier ce modele est basee sur les deux resultats suivant. D’une part, l’equation deSzego cubique amortie

i∂tV + iα〈V,1〉L2(T) = ΠT(|V |2V ), (t, x) ∈ R× S1, α > 0, (1.3.2)

introduite dans Gerard–Grellier [53] admet un phenomene de turbulence faible different du comportementen grand temps de l’equation de Szego cubique sur la tore (1.1.7). Il existe une partie ouverte non vide

U ⊂ H12+(T) telle que pour tout s > 1

2 , si la donnee initiale V (0) ∈ U⋂Hs

+(T), alors on a

limt→+∞

‖V (t)‖Hs = +∞.

D’autre part, l’equation de NLS–Szego cubique defocalisante sur la tore (1.2.7), qui est rappelee ici

i∂tu+ ∂2xu = ΠT(|u|2u), (t, x) ∈ R× T. (1.3.3)

se comporte comme l’equation de Schrodinger cubique (1.1.5) (d=1) lorsque la dispersion est grande,se comporte comme l’equation de Szego cubique lorsque la dispersion est petite. On s’interesse donc al’influence des resultats de stabilites par le terme amorti iα〈u,1〉L2(T). Il faut d’abord etudier l’existencede la solution turbulente de l’equation (1.3.2), c’est-a-dire que limt→+∞ ‖u(t)‖Hs(T) = +∞, pour tout

1.3. PERSPECTIVES DE RECHERCHE 27

s > 12 . On note t ∈ R 7→ Sα(t)φ (respectivement t 7→ S(t)φ) la solution de l’equation de NLS–Szego

amortie (1.3.2) (respectivement l’equation de NLS–Szego (1.3.3)) avec la donnee initiale u(0) = φ. Alorsla masse de la solution est une fonctionnelle de Lyapunov de l’equation (1.3.2). Plus precisement, pour

tout φ ∈ H12+(S1), on a

∂t‖Sα(t)φ‖2L2(T) + 2α|〈Sα(t)φ,1〉L2(T)|2 = 0. (1.3.4)

Alors ‖Sα(t)φ‖L2(T) decroıt et converge, lorsque t → +∞. En outre, la quantite de mouvement P (u) =

〈Du, u〉L2 = ‖|D| 12u‖2L2(T) est conservee le long du flot de l’equation (1.3.2). Par consequent, lorsque

limt→+∞ ‖Sα(t)φ‖L2(T) = 0, alors on a

limt→+∞

‖Sα(t)φ‖Hs(T) = +∞, ∀s > 1

2

compte tenu de l’inegalite d’interpolation

‖|D| 12φ‖L2(T) = ‖|D| 12Sα(t)φ‖L2(T) ≤ ‖Sα(t)φ‖1−12s

L2(T)‖|D|sSα(t)φ‖

12s

L2(T).

Pour mieux comprendre le comportement en temps long des solutions de (1.3.2), on utilise le principe deLaSalle, qui conduit a la proposition suivante.

Proposition 1.3.1. Soient u0 ∈ H12+(S1), u∞ est une limite H

12−faible de (Sα(t)u0) lorsque t→ +∞,

alors on a (Sα(t)u∞|1) = 0. C’est-a-dire S(t)u∞ = Sα(t)u∞ est solution commune de l’equation (1.3.2)et de l’equation (1.3.3).

1.3.2 Unicite des etats fondamentaux

La deuxieme perspective de recherche concerne l’unicite a rotation de phase, a translation spatiale et a

dilatation pres, des etats fondamentaux de la fonctionnelle I(γ)2 dans la sous-section 1.2.2, lorsque γ ≥ 0

est arbitraire. Rappelons que

I(γ)2 (f) :=

‖∂xf‖2L2(R)‖f‖4L2(R) + γ‖|D| 12 f‖4L2(R)‖f‖

2L2(R)

‖f‖6L6(R)

, D = −i∂x, ∀f ∈ H1(R)\0. (1.3.5)

Un premier resultat intermediaire est le suivant.

Proposition 1.3.2. Soient m ∈ N et m ≥ 2, si f ∈ G(γ)m , alors P (f) ∈ G

(γ)m , ou P (f)(x) =

12π

∫ +∞0|f(ξ)|eixξdξ. Plus precisement, il existe deux constantes a, b ∈ R telles que f(ξ) = |f(ξ)|ei(aξ+b),

for every ξ ≥ 0.

Par consequent, il suffit de montrer l’unicite a dilatation pres des etats fondamentaux qui ont une trans-

formee de Fourier strictement positive. Si h(ξ) = |Q(ξ)| = Q(ξ), pour un certain Q ∈ G(γ)2 , apres la

dilatation h 7→ Ah(B·), alors l’equation d’Euler–Lagrange (1.2.20) devient l’equation suivante

(EL(δ)) (1 + δ(γ)ξ + ξ2)h(ξ) = P (h, h, h, h, h)(ξ), (1.3.6)

ou

P (h1, h2, h3, h4, h5)(ξ) =

∫ξ1+ξ3+ξ5=ξ2+ξ4+ξ

ξj≥0

Π5j=1hj(ξj)dξ1dξ2dξ3dξ4


et δ : [0,+∞)→ [0,+∞) est une fonction continue. Lorsque γ = 2, on a l’unicite de la solution. Lorsqueδ s’approche de δ(2), si on suppose que hδ, gδ verifient l’equation (1.3.6), alors on a

hδ − gδ =3P (hδ − gδ, hδ, hδ, hδ, hδ) + 2P (hδ, hδ − gδ, hδ, hδ, hδ) +O(‖hδ − gδ‖2L1)

1 + δ(γ)ξ + ξ2

On introduit ensuite l’operateur Lδ : L1(R+)→ L1(R+) tel que

Lδ(ϕ)(ξ) := ϕ(ξ)− 3P (ϕ, hδ, hδ, hδ, hδ) + 2P (hδ, ϕ, hδ, hδ, hδ)

1 + δ(γ)ξ + ξ2, ∀ξ ≥ 0.

Alors on a ‖Lδ(hδ − gδ)‖L1 = O(‖hδ − gδ‖2L1). Si l’operateur Lδ(2) est inversible, alors on peut espererutiliser le theoreme d’inversion locale pour montrer l’unicite des etats fondamentaux a rotation de phase,a translation spatiale et a dilatation pres est valable, au moins lorsque γ est au voisinage de 2.

1.3.3 Transformation de diffusion inverse (IST) de l’equation de BO

Les transformations spectrales directes et inverses pour l’equation de Benjamin–Ono ont ete formellementdecrites dans Ablowitz–Fokas [1]. Le probleme de Cauchy par des methodes de transformation spectraleinverse (IST) a ete rigoureusement resolu d’abord par Coifman–Wickerhauser [28] pour des donneespetites et suffisamment decroissantes. Ces auteurs ont montre en particulier que, pour un tel potentiel,le spectre de l’operateur de Lax est absolument continu. Y. Wu a resolu completement le probleme dediffusion direct pour des donnees arbitraires mais suffisamment decroissantes dans [146, 147]. Wu a etudiele spectre de Lu et a etabli l’existence, l’unicite et les proprietes asymptotiques de solutions de Jostassociees au probleme de diffusion direct, ce qui fournit un cadre precis pour aborder le probleme dediffusion inverse. Une solution u ∈ L2(R,R) de l’equation de BO sur la droite s’ecrirait alors comme lasomme d’une partie L2-diffusive et d’une partie multi-soliton. On s’interesse a l’interaction de ces deuxparties. On rappelle que l’operateur de Lax s’ecrit sous la forme

Lu : h ∈ H1+(R) 7→ Dh−ΠR(uh) ∈ L2

+(R), u ∈ L2(R,R), D = −i∂x. (1.3.7)

Alors Lu est un operateur auto-adjoint non borne, qui est une perturbation relativement compacte deD : h ∈ H1

+(R) 7→ Dh ∈ L2+(R).

Proposition 1.3.3. (Wu [146] et Sun [135]). Si u ∈ L2(R, (1 + x2)dx) est a valeurs reelles, λ ∈ R etϕ ∈ Ker(λ− Lu), alors on a ∣∣∣ ∫

Ruϕ∣∣∣2 = −2πλ

∫R|ϕ|2. (1.3.8)

Si u ∈ L∞(R,R) en plus, le spectre purement ponctuel de Lu est une partie finie dans ]−∞, 0[.

Une question interessante est aussi de savoir si la proposition 1.3.3 reste vraie, lorsque la conditionu ∈ L∞(R,R) et la condition x 7→ xu(x) ∈ L2(R) sont enlevees.

1.4 Preliminaires

Ce paragraphe etablit quelques resultats preliminaires. On rappelle le theoreme de decomposition enprofils de Gerard [45] et on detaille la preuve de l’existence et l’unicite des solutions globales de l’equation(1.4.3).

1.4. PRELIMINAIRES 29

1.4.1 Decomposition en profils

L’injection de Sobolev H1(R) → L∞(R) n’est pas compacte. En revanche, toute suite bornee dans H1(R)s’ecrit, a une sous-suite pres, comme la somme presque orthogonale d’une suite tendant vers zero dansl’espace de Lebesgue Lp(R) pour tout 2 < p ≤ +∞, et d’une superposition de suites de translatees–dilateesde profiles fixes (voir Gerard [45]). On obtient ainsi des versions precisees du principe de concentration–compacite de Lions [101, 102]. L’enonce qui suit transfere le resultat de Hmidi–Keraani [74] au cadre del’espace de Hardy.

Theoreme 1.4.1. (Gerard [45], Hmidi–Keraani [74]). Pour toute suite (fn)n∈N+ bornee dans H1+, il

existe une sous-suite, notee par (fφ(n))n∈N+ , une suite des profils (U (j))j∈N+ ⊂ H1+ et une suite a indice

double (x(j)n )n,j∈N+ ⊂ R telle que si j 6= k, on a |x(j)

n − x(k)n | → +∞, telles que, pour tout l ∈ N+, on ait

fφ(n)(x) =

l∑j=1

U (j)(x− x(j)n ) + r(l)

n (x), ∀x ∈ R, (1.4.1)

avec liml→+∞ lim supn→+∞ ‖r(l)n ‖Lp = 0, ∀p ∈]2,+∞]. Pour tout l ∈ N+ et s ∈ [0, 1], alors on a la

propriete de presque orthogonalite

∣∣∣‖|D|sfφ(n)‖2L2 −l∑

j=1

‖|D|sU (j)‖2L2 − ‖|D|sr(l)n ‖2L2

∣∣∣→ 0, D = −i∂x, lorsque n→ +∞. (1.4.2)

Ce theoreme est souvent utilise pour etablir l’existence de minimiseurs de certaines fonctionnelles, pourla stabilite orbitale d’ondes progressives, pour la diffusion non lineaire etc. (Voir par exemple Bahouri–Gerard [6], Gallagher–Gerard [42], Hmidi–Keraani [73], Pocovnicu [119], Krieger–Luhrmann [92], Gerard–Lenzmann–Pocovnicu–Raphael [56], Sun [134] etc.)

1.4.2 Probleme de Cauchy pour l’equation de NLS–Szego cubique sur la tore

On va montrer que l’equation de NLS–Szego cubique est globalement bien posee dans les espaces Hs,pour tout s ≥ 1

2 . i∂tu(t, x) + ∂2

xu(t, x) = Π[|u|2u](t, x) t ∈ R, x ∈ Tu(0, ·) = u0 ∈ L2

+

(1.4.3)

avec Π = ΠT : u =∑n∈Z une

inx ∈ L2(T) 7→∑n≥0 une

inx 7→ L2+(T). On rappelle que Hs

+ :=

Hs(T)⋂L2

+(T).

Theoreme 1.4.2. Soient s ≥ 12 , u0 ∈ Hs

+, il existe une unique fonction u ∈ C(R;Hs+)⋂C1(R;Hs−2

+ )qui resout l’equation (1.4.3). L’application flot u0 7→ u est continue de Hs

+ vers C(R;Hs+). En outre, la

norme ‖ · ‖H

12+

est une loi de conservation de l’equation (1.4.3).

La preuve se divise en deux parties. La premiere partie est de montrer le cas s > 12 par un theoreme du

point fixe et la loi de conservation u 7→ ‖u‖H

12

. La deuxieme partie traite le cas s = 12 par un argument

de compacite.


Cas s > 12

Soit s > 12 , Hs(T) est une algebre de Banach et on a l’injection de Sobolev compacte Hs(T) → L∞(T).

Il faut d’abord prouver l’existence et l’unicite locale de l’equation (1.4.3) par le theoreme du point fixe.

Lemme 1.4.3. Soit s ≥ 0, on a l’inegalite suivante

‖fg‖Hs(T) .s ‖f‖L∞(T)‖g‖Hs(T) + ‖g‖L∞(T)‖f‖Hs(T),

pour tout f, g ∈ Hs(T)⋂L∞(T). En particulier, si s > 1

2 , Hs(T) → L∞(T) et on a

‖fg‖Hs(T) .s ‖f‖Hs(T)‖g‖Hs(T),

pour tout f, g ∈ Hs(T).

Demonstration. Voir Alinhac–Gerard [3] et Bahouri–Chemin–Danchin [5] dans le cas ou T est remplaceepar R.

Proposition 1.4.4. Soient s > 12 et ε > 0, il existe T := T (ε) > 0 telle que ∀u0 ∈ Hs

+, si ‖u0‖Hs(T) < ε,l’equation (1.4.3) admet une unique solution dans l’espace C([−T, T ];Hs

+). De plus l’application du flot

u0 7→ u est lipchitzienne continue Bε −→ C([−T, T ];Hs+), ou Bε est la boule fermee de l’espace Hs

+

centree en l’origine de rayon ε.

Demonstration. Il suffit de montrer que l’application K : u 7→ (t 7→ eit∆u0− i∫ t

0ei(t−s)∆Π[|u(s)|2u(s)]ds)

est contractante de BT,2ε dans lui-meme, ou BT,2ε est la boule fermee de rayon 2ε centree sur l’originedans l’espace C([−T, T ];Hs

+), pour un certain T > 0. Grace au lemme 1.4.3, un tel T existe toujours caron a les estimations suivantes

‖Ku(t)‖Hs ≤ ‖eit∆u0‖Hs +

∫ T

−T‖|u(τ)|2u(τ)‖Hsdτ ≤ ‖u0‖Hs + 2CsT‖u‖3L∞t Hsx ,

‖Ku(t)−Kv(t)‖Hs ≤∫ T

−T‖|u(τ)|2u(τ)− |v(τ)|2v(τ)‖Hsdτ

≤ 4Cs

∫ T

−T‖u(τ)− v(τ)‖Hs(‖|u(τ)‖2Hs + ‖v(τ)‖2Hs)dτ

≤ 8CsT‖u− v‖L∞(−T,T ;Hs)

[‖u‖2L∞(−T,T ;Hs) + ‖v‖2L∞(−T,T ;Hs)

],

pour tout t ∈ [−T, T ] et ∀u, v ∈ BT,2ε. L’existence de la solution locale est donc obtenue.

Soient deux solutions u, v ∈ C([−T, T ];Hs+) de l’ equation (1.4.3) telles que u(σ) = v(σ), pour un certain

σ ∈ [−T, T ], alors pour tout intervalle I ⊂ [−T, T ] qui contient σ tel que

8Cs|I|(‖u‖2C([−T,T ];Hs(T)) + ||v||2C([−T,T ];Hs(T))) < 1,

on a ‖u− v‖L∞(I;Hs) ≤ 4Cs∫I‖u(τ)− v(τ)‖Hs(‖u(τ)‖2Hs+‖v(τ)‖2Hs)dτ ≤

12‖u− v‖L∞(I;Hs). Alors u = v

sur l’intervalle I et l’unicite globale est obtenue.


Soient u0, v0 ∈ Bε, on note u, v leur flots associes dans l’espace C([−T, T ];Hs+). Plus precisement,

u, v ∈ BT,2ε. Pour tout t ∈ [0, T ], on a

‖u(t)− v(t)‖Hs ≤‖u0 − v0‖Hs + 4Cs

∫ t

0

‖u(τ)− v(τ)‖Hs(‖u(τ)‖2Hs + ‖v(τ)‖2Hs)dτ

≤‖u0 − v0‖Hs exp[4Cs

(‖u‖2L∞(−T,T ;Hs) + ‖v‖2L∞(−T,T ;Hs)

)T],

grace a l’inegalite de Gronwall. La continuite lipchitzienne de l’application du flot est donc obtenue.

Soit s > 12 , on note Ismax,u0

l’intervalle maximal associe a la donnee initiale u0. On a donc le critere del’explosion suivante.

Proposition 1.4.5. En ecrivant Imax,u0=]− T−, T+[, on a T+ < +∞ =⇒ limt↑T+ ||u(t)||Hs = +∞ et

T− < +∞ =⇒ limt↓−T− ||u(t)||Hs = +∞.

Afin d’obtenir l’existence de la solution globale, il suffit de montrer que ‖u(t)‖Hs n’explose jamais en touttemps. On introduit deux lois de conservation de l’equation (1.4.3).

Proposition 1.4.6. Soient s > 12 , u ∈ C(I,Hs

+) une solution de l’equation (1.4.3) definie sur un

certain intervalle I ⊂ R. Alors ‖u(t)‖L2 = ‖u(0)‖L2 et ‖|D| 12u(t)‖L2 = ‖|D| 12u(0)‖L2 pour tout t ∈ I, ou

‖|D| 12ϕ‖L2 =∑k≥0 k|ϕk|2, pour tout ϕ ∈ H

12+ .

Demonstration. Si u0 ∈ C∞+ (T), alors son flot associe u ∈ C∞(I × T). Dans ce cas,

d

dt‖u(t)‖2L2 = 2Re〈u, ∂tu〉 = −2Im〈u, ∂2

xu−Π[|u|2u]〉 = 2Im[‖∂xu(t)‖2L2 + ‖u(t)‖2L4

]= 0

d

dt‖u(t)‖2

H12

= 2Re〈1i∂xu, ∂tu〉 = −2Im〈1

i∂xu, ∂

2xu−Π[|u|2u]〉 = 2Im‖u(t)‖2

H32− 1

2

∫T∂x|u(t, x)|4dx = 0

Si u0 ∈ Hs+ pour un certain s > 1

2 , soit t0 ∈ I fixe, alors l’intervalle

I ′ := t ∈ I : ‖u(t)‖L2 = ‖u(t0)‖L2 et ‖u(t)‖H

12

= ‖u(t0)‖H

12

est fermee dans I. Il suffit de montrer que I ′ est ouverte. Soit t1 ∈ I ′, il existe une suite un(t0) ∈ C∞+ (T)telle que un(t0) −→ u(t0) dans Hs

+. On prend ε := supn∈N ‖u0‖Hs , il existe donc une famille des flotsun ∈ C∞([−T, T ] × T) associes a un0 par la proposition 1.4.4 pour un certain T = T (ε). On a donc

‖un(t)‖L2 = ‖un(t0)‖L2 et ‖|D| 12un(t)‖L2 = ‖|D| 12un(t0)‖L2 , ∀t ∈ [−T, T ]. Grace a la continuite del’application du flot u(t0) ∈ Hs

+ 7→ u ∈ C(−T, T ;Hs+), on a donc [−T, T ] ⊂ I ′. Alors I ′ = I.

On utilise alors l’inegalite de Brezis–Gallouet [17] pour finir la preuve du cas s > 12 du theoreme 1.4.2.

Lemme 1.4.7. Soit s > 12 , il existe une constante Cs > 0 telle que

‖u‖L∞(T) ≤ Cs‖u‖H 12 (T)

√√√√log

(1 +‖u‖Hs(T)

‖u‖H

12 (T)

), (1.4.4)

pour tout u ∈ Hs(T).


Demonstration. Supposons que u =∑|k|≤N uke

ikx +∑|k|>N uke

ikx, pour tout N ∈ N, on a

‖u‖L∞(T) ≤∑|k|≤N

|uk|+∑|k|>N

|uk|

≤

∑|k|≤N

1

1 + |k|

12 ∑|k|≤N

(1 + |k|)|uk|2 1

2

+

∑|k|>N

1

k2s

12 ∑|k|>N

k2s|uk|2 1

2

.s‖u‖H

12

(ln(1 +N))12 + ‖u‖HsN

12−s.

Ensuite, on choisit Ns− 12 = b ‖u‖Hs‖u‖

H12

c+ 1, alors ‖u‖HsN12−s ≤ ‖u‖

H12

et

1 + (ln(1 +N))12 ≤ 1 +

(ln 2 +

2

2s− 1ln(1 +

‖u‖Hs‖u‖

H12

)

) 12

.s

(ln 2 + ln(1 +

‖u‖Hs‖u‖

H12

)

) 12

.

Alors ‖u‖L∞(T) .s ‖u‖H 12

(1 + (ln(1 +N))12 ) .s

(ln(1 + ‖u‖Hs

‖u‖H

12

)

) 12

.

Grace aux lemmes 1.4.3 et 1.4.7, on peut montrer que la solution construite dans la proposition 1.4.4 estglobale.

Preuve du theoreme 1.4.2 lorsque s > 12 . En utilisant la formule de Duhamel, on a ∀t ∈ Imax,u0

⋂R+

‖u(t)‖Hs ≤ ‖u0‖Hs +

∫ t

0

‖|u(τ)|2u(τ)‖Hsdτ.

Comme Hs⋂L∞(T) est une algebre de Banach, on a

‖|u(τ)|2u(τ)‖Hs .s ‖u(τ)‖2L∞‖u(τ)‖Hs .s ‖u(τ)‖2H

12‖u(τ)‖Hs ln

(1 +‖u(τ)‖Hs‖u0‖

H12

),

ou on applique l’inegalite de Brezis–Gallouet (1.4.4) dans la deuxieme estimation ci-dessus. Grace a la loide conservation ‖u‖

H12

, on a ‖|u(τ)|2u(τ)‖Hs .s,‖u0‖H

12

‖u(τ)‖Hs ln (2 + ‖u(τ)‖Hs). Alors il existe une

constante C = C(‖u0‖12

H , s) > 0 telle que

‖u(t)‖Hs ≤‖u0‖Hs +

∫ t

0

‖|u(τ)|2u(τ)‖Hsdτ.

≤‖u0‖Hs + C

∫ t

0

‖u(τ)‖Hs ln (2 + ‖u(τ)‖Hs) dτ

≤ exp[ln(2 + ‖u0‖H

12

) exp(Ct)]

ou on utilise une version logarithmique de l’inegalite de Gronwall dans la derniere etape. Donc

‖u‖L∞(0,T ;Hs) < +∞,

pour tout T ∈ Imax,u0

⋂R+. Alors Imax,u0

= R.

Pour tout T > 0, l’application flot est donc lipschitzienne de Hs+ vers C([R]−T, T ];Hs

+), grace a la loi deconservation ‖u‖

H12

. L’application flot est donc continue de Hs+ vers l’espace de Frechet C(R;Hs

+).


Cas s = 12

La preuve se decompose en trois etapes. Il faut d’abord construire la solution faible de l’equation (1.4.3).Ensuite on montre l’unicite de la solution faible. En utilisant les lois de conservation construites dans laproposition 1.4.6, on verifie que la solution faible est fortement continue en temps a la fin. D’abord, onintroduit le theoreme d’Aubin–Lions–Simon et un corollaire du theoreme d’Ascoli.

Lemme 1.4.8 (Aubin–Lions–Simon). Soient X0, X1, X2 trois espaces de Banach tels que l’injectionX0 → X1 est compacte et l’injection X1 → X2 est continue. L’espace

W := u ∈ L∞(0, T ;X0) : ∂tu ∈ L∞(0, T ;X2)

muni de la norme ‖u‖L∞(0,T ;X0) + ‖∂tu‖L∞(0,T ;X2). Alors l’injection W → C([0, T ];X1) est compacte.

Demonstration. Cf le Theoreme II.5.16 de la reference [15].

On designe par Cw(R;Hs(T)) l’ensemble des fonctions x : t ∈ R 7→ x(t) ∈ Hs(T) telle que la fonctiont 7→ 〈x(t), ϕ〉Hs,H−s est continue sur R, pour tout v ∈ H−s(T).

Lemme 1.4.9. Soient s ∈ R, k > 0, une suite de fonctions (xn)n∈N ⊂ C(R;Hs+)⋂C1(R;H−k+ ) telles

que

supn∈N

[‖xn‖L∞(R;Hs+) + ‖∂txn‖L∞(R;H−k+ )

]< +∞.

Alors il existe une sous-suite nk → +∞ et x ∈ Cw(R;Hs+) telles que 〈xnk , ϕ〉L2 −→ 〈x, ϕ〉L2 uni-

formement sur tout intervalle compact de R, pour tout ϕ ∈ C∞(T).

Demonstration. On definit un sous-ensemble de C∞(T) :

D := f ∈ C∞(T) : ∃N ∈ N, λj ∈ Q, −N ≤ j ≤ N tels que f(x) =

N∑n=−N

λneinx

Alors ∀s ∈ R fixe, D est un sous-espace denombrable dense de Hs(T). Pour tout fonction v ∈ D, lafamille (〈xn(t), v〉Hs,H−s)n∈N est ponctuellement bornee

supn∈N|〈xn(t), v〉Hs,H−s | ≤ ‖v‖H−s · sup

ε>0‖xn(t)‖Hs < +∞, ∀t ∈ R.

De plus, la famille (〈xn(t), v〉Hs,H−s)n∈N est equi-continue,

|〈xn(t), v〉Hs,H−s − 〈xn(t′), v〉Hs,H−s | ≤ ‖v‖Hk supε>0

supt∈R‖∂txn(t)‖H−k · |t− t′|, ∀t, t′ ∈ R

D’apres theoreme d’Ascoli et le processus diagonal, on suppose que t 7→ 〈xn(t), v〉Hs,H−s converge uni-formement sur tout les intervalles compacts dans R, pour tout v ∈ D, quitte a extraire une sous-suite.Alors on definit

gv(t) := limn→+∞

〈xn(t), v〉Hs,H−s , ∀t ∈ R.

La fonction v ∈ D 7→ gv(t) ∈ C est lineaire continue par rapport a la topologie de Hs. Alors elle admetun unique prolongement dans H−s(T) qui est representee par la forme v 7→ 〈x(t), v〉, pour une certainex(t) ∈ Hs(T), grace au theoreme de Riesz. Donc ∀t ∈ R, xn(t) x(t) dans Hs(T), et

‖x(t)‖Hs ≤ supn∈N‖xn(t)‖L∞(R;Hs(T)).


Apres avoir construit x, il reste a verifier sa continuite faible. Pour tout ϕ ∈ H−s(T;C2) I ⊂ R intervallecompact et η > 0, ∃v ∈ D tel que ‖v − ϕ‖H−s < η

2 supε>0 supt∈R ‖xε(t)‖Hs. Alors

supt∈I|〈xn(t)− x(t), ϕ〉Hs,H−s | ≤ sup

t∈I|〈xn(t)− x(t), v〉Hs,H−s |+ ‖v − ϕ‖H−s · sup

t∈I[‖xn(t)‖Hs + ‖x(t)‖Hs ]

On a lim supn→+∞ supt∈I |〈xn(t) − x(t), ϕ〉Hs,H−s | ≤ η, ∀η > 0. Donc 〈xn, ϕ〉 −→ 〈x, ϕ〉 uniformementen I et x ∈ Cw(R;Hs(T)).

Proposition 1.4.10. Soit u0 ∈ H12+ , il existe une fonction u ∈ Cw(R;H

12+)⋂C1w(R;H

− 32

+ ) qui resoutl’equation (1.4.3) faiblement, i.e.

d

dt〈u(t), ϕ〉

H12 ,H−

12

+ 〈∂2xu(t), ϕ〉

H−32 ,H

32

= 〈Π[|u(t)|2u(t)], ϕ〉L2 , ∀t ∈ R. (1.4.5)

pour tout ϕ ∈ C∞(T). De plus, ‖u(t)‖H

12≤ ‖u0‖

H12

pour tout t ∈ R.

Demonstration. Soient un0 ∈ C∞(T) −→ u0 dans H12+ . Alors M := 1 + supn∈N ‖un0‖H 1

2 (T)< +∞. Il existe

donc un ∈ C∞(R×R) qui est la solution de l’equation (1.4.3) qui associe a un0 , pour tout n ∈ N. Comme‖ · ‖

H12 (T)

est une loi de conservation de l’equation (1.4.3), on a supn∈N ‖un‖L∞(R;H12 (T))

= M < +∞.

Par l’injection de Sobolev H12 (T) → Lp(T), pour tout p ∈ [1,∞[, on a

‖Π[|un(t)|2un(t)]‖L2(T) ≤ ‖un(t)‖3L6(T) ≤ C‖un(t)‖3

H12 (T)≤ CM3,

pour tout t ∈ R. Alors supn∈N ‖∂tu‖L∞(R;H

− 32

+ ).M3 < +∞.

Grace a l’injection compacte H12+ → Hs

+, pour tout s < 12 , quitte a extraire une suite, on suppose que

un → u dans l’espace C([0, T ];Hs+) pour tout T > 0 et s ∈] 1

3 ,12 [, d’apres le lemme de Aubin–Lions–Simon

(Lemme 1.4.8) et le processus diagonal. Comme s > 13 , on a l’injection Hs(T) → L

21−2s (T) → L6(T).

Alors

supt∈[0,T ]

‖Π[|un(t)|2un(t)− |u(t)|2u(t)]‖L2(T)

≤C‖un − u‖L∞([0,T ];L6(T))

[‖un‖2L∞([0,T ];L6(T)) + ‖u‖2L∞([0,T ];L6(T))

].M6‖un − u‖L∞([0,T ];Hs(T)) → 0,

(1.4.6)

lorsque n→ +∞, ∀T > 0.

D’autre part, en utilisant le lemme 1.4.9 , on a u ∈ Cw(R;H12+) et pour tout ϕ ∈ C∞(T), on suppose

〈un, ϕ〉L2 = 〈un, ϕ〉H

12 ,H−

12→ 〈u, ϕ〉

H12 ,H−

12

= 〈u, ϕ〉L2 (1.4.7)

converge uniformement sur tous les intervalles compacts de R, quitte a extraire une sous-suite. Grace auxformules (1.4.6) et (1.4.7), on a la convergence uniforme de 〈∂tun, ϕ〉

H−32 ,H

32−→ d

dt 〈u(t), ϕ〉H

12 ,H−

12

sur

tout compact de R. Alors u ∈ C1w(R;H

− 32

+ ) et la formule (1.4.5) est obtenue. Comme un(t) u(t) dans

H12+ , pour tout t ∈ R. On a

‖u(t)‖H

12≤ lim inf

n→+∞‖un(t)‖

H12

= lim infn→+∞

‖un0‖H 12

= ‖u0‖H

12.


Apres avoir construit la solution faible de l’equation (1.4.3), on va montrer qu’elle est unique associee a

u0 ∈ H12+ en utilisant la formule Duhamel et l’inegalite de Trudinger [143].

Lemme 1.4.11. Il existe une constante universelle C > 0 telle que

‖u‖Lp(T) ≤ C√p‖u‖

H12 (T)

, (1.4.8)

pour tout p ∈ [2,+∞[ et u ∈ H 12 (T).

Demonstration. Il suffit d’adapter la methode dans Chemin–Xu [27] pour les injections de Sobolevgenerales. On suppose que ‖u‖

H12

= 1. Pour tout λ > 0, on decompose u comme u = u≤λ + u>λ,

ouu≤λ(x) =

∑|k|≤λ

ukeikx, u>λ(x) =

∑|k|>λ

ukeikx.

‖u≤λ‖2H1 =∑|k|≤λ(1 + k2)|uk|2 ≤ (1 + λ)2‖u≤λ‖2

H12

. l’inegalite de Brezis–Gallouet implique qu’il existe

une constante C > 0 telle que

‖u≤λ‖L∞ . ‖u≤λ‖2H

12

(ln(1 +

‖u≤λ‖H1

‖u≤λ‖H

12

)

) 12

≤ C(ln(1 + λ))12 .

On remplace λ par λ(t) := e( t2C )2−1, alors ‖u≤λ(t)‖L∞ ≤ t

2 , for every t > 0. On a donc

x ∈ T : |u(x)| > t ⊂ x ∈ T : |u>λ(t)(x)| > t

2, ∀t > 0.

On note σ comme la mesure de Haar classique sur la tore T. Alors pour tout p ≥ 1, on a

‖u‖pLp = p

∫ +∞

0

tp−1σx ∈ T : |u(x)| > tdt ≤ 4p

∫ +∞

0

tp−3‖u>λ(t)‖2L2dt.

Comme λ(t) < |n| si et seulement si t < 2C(ln(1 + |n|)) 12 , on applique le theoreme de Fubini,

∫ +∞

0

tp−3‖u>λ(t)‖2L2dt =∑k∈Z|uk|2

∫ 2C(ln(1+|n|))12

0

tp−3dt =(2C)p−2

p− 2

∑k∈Z|uk|2(ln(1 + |k|))

p−22 .

Comme ex ≥ xm

m! , ∀x ≥ 0, on a (ln(1 + |k|))m ≤ m!(1 + |k|) ≤ mm(1 + |k|), ∀m, k ∈ N. On choisit

m = bp2c >p−2

2 . Alors ‖u‖pLp ≤4pp−2 (2C)p−2‖u‖2

H12

(p2 )p2 . Comme la fonction p ∈ [2,+∞[7→ ( 4p

p−2 )1p ∈ R

est bornee, on a

‖u‖Lp ≤ 2C(4p

p− 2)

1p

√p

2.√p.

La demonstration de l’unicite globale de la solution faible est similaire a l’argument introduit par Yudovich[151] et est utilisee dans Vladimirov [144] et Ogawa [114].

Proposition 1.4.12. Soient u1, u2 ∈ C(R;H12+) deux solutions faibles de l’equation (1.4.3) telles que

u1(0) = u2(0), alors on a u1 = u2.


Demonstration. Comme on a i∂tu1 + ∂2xu1 = Π[|u1|2u1] et i∂tu2 + ∂2

xu2 = Π[|u2|2u2], on definit ladifference v := u1 − u2. Alors

i∂tv + ∂2xv = Π[|u1|2u1 − |u2|2u2], v(0) = 0.

Grace a la formule de Duhamel, on a

‖v(t)‖L2 ≤∫ t

0

‖Π[|u1|2u1 − |u2|2u2](τ)‖L2dτ ≤∫ t

0

‖|u1|2u1(τ)− |u2|2u2(τ)‖L2dτ =: Z(t)

pour presque partout t ∈ R+. On calcule la derivee de Z.

Z ′(t)2 ≤C∫T|v(t, x)|2(|u1(t, x)|+ |u2(t, x)|)4dx ≤ C

∫T|v(t, x)|2(1− 1

p )(|u1(t, x)|+ |u2(t, x)|dx)4+ 2p

pour tout p ≥ 2, parce que |v(t, x)| ≤ |u1(t, x)|+ |u2(t, x)|. En utilisant l’inegalite de Holder, on a∫T|v(t, x)|2(1− 1

p )(|u1(t, x)|+ |u2(t, x)|dx)4+ 2p ≤ ‖v(t)‖

2p−2p

L2 ‖|u1(t)|+ |u2(t)|‖4p+2p

L4p+2

Grace a l’inegalite de Trudinger (1.4.8), ‖|u1(t)| + |u2(t)|‖L4p+2 ≤ C√

4p+ 2(‖u1(t)‖H

12

+ ‖u2(t)‖H

12

).

On a donc

Z ′(t) ≤ ‖v(t)‖p−1p

L2 ‖|u1(t)|+ |u2(t)|‖2p+1p

L4p+2 ≤ CpZ(t)p−1p .

Alors Z(t) ≤ (Ct)p pour tout p ≥ 2. Alors Z(t) = 0 pour presque partout t ∈ [0, 12C ]. Alors u1 = u2 sur

[0, 12C ]. Ensuite on peut montrer que u1 = u2 sur [0, n

2C ] par recurrence par rapport a n ∈ N.

Corollary 1.4.13. Soit u ∈ Cw(R;H12+)⋂C1w(R;H

− 32

+ ) l’unique solution faible de l’equation (1.4.3)

associee a la donnee initiale u0 ∈ H12+ . Alors ‖u(t)‖

H12

= ‖u0‖H

12

, pour tout t ∈ R.

Demonstration. Soit t 6= 0, il existe une solution faible v ∈ Cw(R;H12+)⋂C1w(R;H

− 32

+ ) qui resout l’equation(1.4.3) associee a la donnee initiale u(t) et ‖v(−t)‖

H12≤ ‖v(0)‖

H12

= ‖u(t)‖H

12

par la proposition 1.4.10.

Mais v(t′) = u(t+ t′) pour tout t′ ∈ R, par la proposition 1.4.12. On a donc

‖u(0)‖H

12

= ‖v(−t)‖H

12≤ ‖v(−t)‖

H12.

Preuve du theoreme 1.4.2 lorsque s = 12 . Il suffit de verifier que la solution faible construite dans la pro-

position 1.4.10 est fortement continue en temps. En effet, soient tn −→ t ∈ R, on a u(tn) u(t) dans

l’espace H12+ . Comme ‖u(t)‖

H12

= ‖u0‖H

12

= ‖u(tn)‖H

12

, pour tout n ∈ N et H12+ est un espace d’Hilbert,

on a donc u(tn) → u(t) dans H12+ . Alors u ∈ C(R;H

12+). L’application du flot est une isometrie de H

12+

vers L∞(R;H12+)⋂C(R;H

12+) par rapport a la topologie uniforme.

Chapitre 2

Long time behavior of theNLS–Szego equation on the torus

Ce chapitre est une reprise de l’article Sun [133].

Resume Dans la premiere partie de la these, on s’interesse au comportement en grand temps de l’equationde NLS–Szego cubique sur le tore.

i∂tu+ εα∂2xu = Π(|u|2u), Π = ΠT = ΠS1

, 0 < ε < 1, α ≥ 0.

On obtient deux ensembles des resultats. D’abord, on etablit une estimation de Sobolev en grand tempspour les donnees petites a l’aide d’une transformation de forme normale de Birkhoff. Le deuxieme ensemblede resultats concerne la stabilite orbitale d’ondes planes. Grace a la propriete d’instabilite de l’equationde Szego cubique sur le tore decouverte par Gerard–Grellier [50], on met en evidence un phenomene deturbulence faible lorsque la dispersion est petite.

Mots− clefs : Equation de Schrodinger cubique, Projecteur de Szego, Dispersion petite, Stabilite,Turbulence d’onde, Forme normale de Birkhoff

Abstract We are interested in the influence of filtering the positive Fourier modes in the integrablenon linear Schrodinger equation. Equivalently, we want to study the effect of dispersion added to thecubic Szego equation, leading to the NLS-Szego equation on the circle S1

i∂tu+ εα∂2xu = Π(|u|2u), Π = ΠT = ΠS1

, 0 < ε < 1, α ≥ 0.

There are two sets of results in this paper. The first result concerns the long time Sobolev estimatesfor small data. The second set of results concerns the orbital stability of plane wave solutions. Someinstability results are also obtained, leading to some wave turbulence phenomenon.

Keywords Cubic Schrodinger equation, Szego projector, Small dispersion, Stability, Wave turbulence,Birkhoff normal form

37

38 CHAPITRE 2. LONG TIME BEHAVIOR OF THE NLS–SZEGO EQUATION ON THE TORUS

2.1 Introduction

We consider the NLS-Szego equation defined on the circle S1

i∂tu+ ∂2xu = Π(|u|2u), u(0, ·) = u0. (2.1.1)

Here Π : L2(S1)→ L2(S1) denotes the orthogonal projector from L2(S1) onto the space of L2 boundaryvalues of holomorphic functions on the unit disc,

Π :∑k∈Z

ukeikx 7−→

∑k≥0

ukeikx.

We denote by L2+ := Π(L2(S1)) ⊂ L2(S1), Hs

+ := Hs(S1)⋂L2

+, for all s ≥ 0, and C∞+ := C∞(S1)⋂L2

+.

2.1.1 Motivation

The NLS-Szego equation can be seen as the combination of two completely integrable systems : thedefocusing cubic Schrodinger equation

i∂tu+ ∂2xu = |u|2u, (t, x) ∈ R× S1, (2.1.2)

and the cubic Szego equation

i∂tV = Π(|V |2V ), (t, x) ∈ R× S1. (2.1.3)

They have both a Lax Pair structure and the action-angle coordinates, which can be used to obtain theirexplicit formulas with the inversed spectral method(see Zakharov–Shabat [152], Faddeev–Takhtajan [33],Grebert–Kappeler [65], Gerard [46], for the NLS equation and Gerard–Grellier [47, 49, 51, 52] for the cubicSzego equation). However, these two Lax pairs cannot be combined in order to give a Lax pair for (2.1.1).Moreover, the long time behaviors of these two equations are totally different.

The NLS equation (2.1.2) has a sequence of conservation laws controlling every Sobolev norms(seeFaddeev–Takhtajan [33], Grebert–Kappeler [65], Gerard [46]), so all the solutions are uniformly boundedin every Hs space. Moreover, Grebert and Kappeler [65] have proved the existence of the global Birkhoffcoordinates for the NLS equation. So the solutions of (2.1.2) are actually almost periodic on R valuedinto Hs(S1).

Compared to (2.1.2), the cubic Szego equation, which stands for a non-dispersive model, has both theLax pair structure and the wave turbulence phenomenon. Its long time behavior is extremely sensibleaccording to the different initial data. P.Gerard and S.Grellier have shown that(in [50, 51, 52]) for a Gδdense subset of initial data in C∞+ , the solutions may blow up inHs, for every s > 1

2 with super–polynomialgrowth on some sequence of times, while they go back to their initial data on another sequence of timestending to infinity. For another dense subset of initial data in C∞+ , the solutions are quasi-periodic. (seealso Theorem 2.2.3).

Remark 2.1.1. Consider the following equation without the Szego projector Π on S1 :i∂tV = |V |2V,V (0, ·) = V0.

(2.1.4)

2.1. INTRODUCTION 39

Then V (t, x) = eit|V0|2V0(x) and we have ‖V (t)‖Hs ' |t|s, for all s ≥ 0, if |V0| is not a constant function.Hence, the Szego projector both accelerates the energy transfer to high frequencies, and facilitates thetransition to low frequencies for (2.1.4).

One wonders about whether filtering the positive Fourier modes can change the long time Sobolev esti-mates of the cubic defocusing Schrodinger equation. So we introduce equation (2.1.1). On the other hand,it can also be obtained from the cubic Szego equation by adding the dispersive term ∂2

x to its linear part.In order to see the gradual change of the dispersion, we add the parameter εα in front of the Laplacian∂2x to get a more general model, the NLS-Szego equation (with small dispersion) :

i∂tu+ εα∂2xu = Π(|u|2u), u(0, ·) = u0, 0 < ε < 1, α ≥ 0. (2.1.5)

Equation (2.1.1) is the special case α = 0 for (2.1.5).

We endow L2+ with the canonical symplectic form ω(u, v) = Im

∫S1

uv2π . Equation (2.1.5) has the Hamil-

tonian formalism with the energy functional

Eα,ε(u) =εα

2‖∂xu‖2L2 +

1

4‖u‖4L4 , u ∈ H1

+. (2.1.6)

Besides Eα,ε, equation (2.1.5) has two other conservation laws,Q(u) = ‖u‖2L2 ,

I(u) = Im∫S1 u∂xu = ‖u‖2

H12,

which give the estimate of the solution for low frequencies :

supt∈R‖u(t)‖Hs ≤ ‖u0‖1−2s

L2 ‖u0‖2sH

12, ∀s ∈ [0,

1

2].

Proceeding as in the case of equation (2.1.2), one can prove the global existence and uniqueness ofthe solution of the NLS-Szego equation in high frequency Sobolev spaces, by using the Brezis–Gallouettype estimate [17], the Aubin–Lions–Simon theorem (see Theorem II.5.16 in Boyer–Fabrie [15]) and theTrudinger type inequality (see Yudovich [151], Vladimirov [144], Ogawa [114] and Gerard–Grellier [47]).Its well-posedness problem in low frequency Sobolev spaces can be dealt with Strichartz’s inequalityintroduced in Bourgain [13]. Only the high frequency Sobolev estimates are considered in this paper.

Proposition 2.1.2. For every s ≥ 12 , given u0 ∈ Hs

+, there exists a unique solution u ∈ C(R, Hs+) of

(2.1.5) such that u(0) = u0. For every T > 0, the mapping u0 ∈ Hs+ 7→ u ∈ C([−T, T ], Hs

+) is continuous.

2.1.2 Main results

The first result concerns the long time stability around the null solution of the NLS-Szego equation(2.1.5). If the initial data u0 is bounded by ε, we look for a time interval Iαε , in which the solution u(t)is still bounded by O(ε). Now we state the first result of this paper.

Theorem 2.1.3. For every s > 12 , there exist two constants as ∈ (0, 1) and Ks > 0 such that for all

0 < ε 1 and u0 ∈ Hs+, if ‖u0‖Hs = ε and u denotes the solution of (2.1.5) with u(0) = u0, then

sup|t|≤ asε4−α

‖u(t)‖Hs ≤ Ksε, if α ∈ [0, 2];

sup|t|≤ asε2‖u(t)‖Hs ≤ Ksε, if α > 2.

(2.1.7)


Moreover, the time interval Iαε = [−asε2 ,asε2 ] is maximal for the case α > 2 and s ≥ 1 in the following

sense : for every 0 < ε 1, there exists uε0 ∈ C∞+ such that ‖uε0‖Hs ' ε and for every β > 0, we have

sup|t|≤ 1

ε2+β

‖u(t)‖Hs & ε| ln ε| 12 ε, u(0) = uε0.

Remark 2.1.4. In the case α ∈ [0, 2), the proof is based on the Birkhoff normal form method, similarlyto Bambusi [7], Grebert [64], Gerard–Grellier [48] and Faou–Gauckler–Lubich [34] for instance. However,the time interval [− as

ε4−α ,asε4−α ] may not be optimal. The resonant term of 6 indices in the homological

equation can not be cancelled by the Birkhoff normal form transform.(see subsubsection 2.3.2)

The second set of results concerns the long time Hs-estimates for the solutions of (2.1.5), if its initialdatum is a perturbation of the plane wave em : x 7→ eimx, for some m ∈ N and s ≥ 1. Let u = u(t, x)be the solution of equation (2.1.5) such that ‖u(0) − em‖Hs = ε. Its energy functional (2.1.6) gives thefollowing estimate :

supt∈R‖u(t)‖H1 .‖u0‖H1

ε−α2 , ∀0 < ε < 1, α ≥ 0. (2.1.8)

However, no information on the stability of the plane waves em is obtained from (2.1.8) during theprocess ε → 0+. Consider the super-polynomial growth of Sobolev norms in the cubic Szego equationcase (see Gerard–Grellier [50, 52] and Proposition 2.2.4 in this paper), the occurence of wave turbulencephenomenon for (2.1.5) depends on the level of its dispersion. We begin with three long time stabilityresults for the polynomial dispersion εα∂2

x case with 0 ≤ α ≤ 2. The following theorem indicates H1-orbital stability of the traveling waves em for equation (2.1.5).

Theorem 2.1.5. For all ε ∈ (0, 1), α ∈ [0, 2] and m ∈ N, there exists Cm > 0 such that if ‖u(0)−em‖H1 =ε, then we have

supt∈R

infθ∈R‖u(t)− eiθem‖H1 ≤ Cmε1−

α2 .

For each t ∈ R, the infimum can be attained when θ = arg um(t). A similar result is established byZhidkov [153, Sect. 3.3] and Gallay–Haragus [43, 44] for the 1D cubic Schrodinger equation. In smalldispersion case, Theorem 2.1.5 gives a significant improvement of estimate (2.1.8). We denote by Sα,ε the

non linear evolution group defined by (2.1.5) on H12+ . In other words, for every φ ∈ H

12+ , t 7→ Sα,ε(t)φ is

the solution u ∈ C(R, H12+) of equation (2.1.5) such that u(0) = φ.

Corollary 2.1.6. For every m ∈ N, we have

sup0<ε<10≤α≤2

sup‖φ−em‖H1≤ε

supt∈R‖Sα,ε(t)φ‖H1 <∞.

Compared to Proposition 2.2.4 (see Gerard–Grellier [47, 48, 51]), the dispersive term εα∂2x counteracts the

wave turbulence phenomenon in H1 norm for equation (2.1.5), if 0 ≤ α ≤ 2. After the change of variableu(t) = ei arg um(t)(em + ε1−

α2 v(t)), we use a bootstrap argument to get long time orbital stability of the

traveling waves em with respect to higher Sobolev norms.

Proposition 2.1.7. For all s ≥ 1 and m ∈ N, there exist two constants bm,s ∈ (0, 1) and Lm,s > 0 suchthat if 0 ≤ α < 2 and ‖u(0)− em‖Hs = ε ∈ (0, 1), then we have

sup|t|≤ bm,s

ε1−α

2

infθ∈R‖u(t)− eiθem‖Hs ≤ Lm,sε1−

α2 . (2.1.9)


We also look for a larger time interval in which the estimate (2.1.9) holds, by using the Birkhoff normalform transformation. But the coefficients in front of the high frequency Fourier modes in the homologicalequation may be arbitrarily large, if α ∈ (0, 2). For this reason, we return to the case α = 0 and considerequation (2.1.1).

i∂tu+ ∂2xu = Π(|u|2u).

Then the time interval can be enlarged as [−dm,sε2 ,dm,sε2 ] in this case.

Theorem 2.1.8. In the case α = 0, for all s ≥ 1 and m ∈ N, there exist three constants dm,s, εm,s ∈ (0, 1)and Km,s > 0 such that if ‖u(0)− em‖Hs = ε ∈ (0, εm,s), then we have

sup|t|≤ dm,s

ε2

infθ∈R‖u(t)− eiθem‖Hs ≤ Km,sε.

A similar result is obtained in Faou–Gauckler–Lubich [34] for the focusing or defocusing cubic Schrodingerequation on the arbitrarily dimensional torus. (see Section 2.5 for the comparison between (2.1.1) and(2.1.2))

After stating the stability results, we turn to construct some large solutions for (2.1.5) with respect totheir initial data, if the level of dispersion is exponentially small with respect to the level of perturbationof the plane wave e1 : x 7→ eix. We state the last result of this paper.

Theorem 2.1.9. There exists a constant K > 0 such that for all 0 < δ 1, we denote by U the solutionof the following NLS-Szego equation with small dispersion

i∂tU + ν2∂2xU = Π(|U |2U), U(0, x) = eix + δ, (2.1.10)

where ν = e−πK2δ2 , then we have ‖U(tδ)‖H1 ' 1

δ with tδ := πδ√

4+δ2.

This H1-instability result indicates that the support of the energy functional of equation (2.1.10) is trans-ferred to higher Fourier modes. This phenomenon is similar to the cubic Szego equation case (see Gerard–Grellier [50, 51, 52]) and the 2D cubic NLS equation case (see Colliander–Keel–Staffilani–Takaoka–Tao

[29]). Compared to Theorem 2.1.5, adding the low-level dispersion e−πKδ2 ∂2

x fails to change the quality ofwave turbulence phenomenon (Proposition 2.2.4) for the cubic Szego equation.

The second part of Theorem 2.1.3 is a consequence of Theorem 2.1.9. Indeed, if α > 2 is fixed, we rescale

u(t, x) = εU(ε2t, x) with e−πK2δ2 = ν = ε

α−22 . Then u solves (2.1.5) with u(0, x) = ε(eix + δ) and

‖u(tδ

ε2)‖H1 = ε‖U(tδ)‖H1 ' ε

δ' ε√

(α− 2)| ln ε| ε,

while tδ

ε2 '√

(α−2)| ln ε|ε2 1

ε2+β , for all β > 0. However, this method does not work in the critical caseα = 2. If u solves

i∂tu+ ε2∂2xu = Π(|u|2u), u(0, x) = ε(eix + δ),

after rescaling U(t, x) = ε−1u(ε−2t, x), we get equation (2.1.10) with ν = 1, leading to (2.1.1) with initialdatum U(0, x) = eix + δ. Theorem 2.1.5 and Theorem 2.1.8 yield the following two estimates

supt∈R‖u(t)‖H1 = O(ε), sup

|t|≤ d1,s

ε2δ2

‖u(t)‖Hs = O(ε), ∀0 < δ 1, ∀0 < ε < 1,


for every s > 12 . The problem of the optimal time interval in the case α = 2 of Theorem 2.1.3 remains open.

This paper is organized as follows. In Section 2.2, we recall some basic facts of the cubic Szego equationand its consequences. In Section 2.3, we study long time behavior for (2.1.5) with small data and proveTheorem 2.1.9 and Theorem 2.1.3. In Section 2.4, we study the orbital stability of the plane waves emfor (2.1.5) for every m ∈ N and give the proof of Theorem 2.1.5, Proposition 2.1.7 and Theorem 2.1.8.We compare the NLS-Szego equation with the NLS equation in Section 2.5.

2.2 The cubic Szego equation

In this section, we recall some results of the cubic Szego equation

i∂tV = Π(|V |2V ), V (0, ·) = V0. (2.2.1)

2.2.1 The Lax pair structure and L∞-estimate

Given V ∈ H12+ , the Hankel operator HV : L2

+ → L2+ is defined by

HV (h) = Π(V h).

Given b ∈ L∞(S1), the Toeplitz operator Tb : L2+ → L2

+ is defined by

Tb(h) = Π(bh).

Theorem 2.2.1. (Gerard–Grellier [47]) Set V ∈ C(R;Hs+) for some s > 1

2 . Then V solves the cubicSzego equation if and only if HV satisfies the following evolutive equation

∂tHV = [BV , HV ]. (2.2.2)

where BV := i2H

2V − iT|V |2 . In other words, (LV , BV ) is a Lax pair for the cubic Szego equation.

The equation (2.2.2) yields that the spectrum of the Hankel operator HV is invariant under the flow ofthe cubic Szego equation. Thus the quantity Tr|HV | is conserved. A theorem of Peller ([117] Theorem 2p. 454) states that

‖V ‖B11,1' Tr|HV |.

Using the embedding theorem Hs → B11,1 → L∞, for any s > 1, we have the following L∞ estimate of

the Szego flow.

Corollary 2.2.2. (Gerard–Grellier [47]) Assume V0 ∈ Hs+ for some s > 1, then we have

supt∈R‖V (t)‖L∞ .s ‖V0‖Hs

2.2.2 Wave turbulence

The following theorem indicates its chaotic long time behavior with turbulence phenomenon for generalinitial data.

2.2. THE CUBIC SZEGO EQUATION 43

Theorem 2.2.3. (Gerard–Grellier [50, 51, 52]) 1.There exists a Gδ−dense set U ⊂ C∞+ such that ifV0 ∈ U , then there exist two sequences (tn)n∈N and (t′n)n∈N tending to infinity such that

limn→+∞‖V (tn)‖Hs|tn|p = +∞, ∀s > 1

2 , ∀p ≥ 1,

limn→+∞ V (t′n) = V0.

2. If V0 is rational, then V (t) is also rational for every t ∈ R and the mapping t ∈ R 7→ V (t) ∈ C∞+ isquasi-periodic.

2.2.3 A special case

Set V0(x) = V δ0 (x) := δ + eix, we denote V δ the solution of (2.2.1). Refering to Gerard–Grellier [47 Sect.6.1, 6.2 ; 48 Sect. 3 ; 51 Sect. 4], we have the following explicit formula

V δ(t, x) =aδ(t)eix + bδ(t)

1− pδ(t)eix, (2.2.3)

where

aδ(t) = e−it(1+δ2), bδ(t) = e−it(1+ δ2

2 )(δ cos(ωt)− i 2 + δ2

√4 + δ2

sin(ωt)),

pδ(t) = − 2i√4 + δ2

sin(ωt)e−itδ2

2 , ω = δ

√1 +

δ2

4.

Proposition 2.2.4. (Gerard–Grellier [47, 48, 51]) For 0 < δ 1, set tδ := π2ω = π

δ√

4+δ2∼ π

2δ . Let V δ

be the solution of (2.2.1) with V δ(0, x) = eix + δ, then we have the following estimate

‖V δ(t)‖Hs .s ‖V δ(tδ)‖Hs 's1

δ2s−1, ∀t ∈ [0, tδ].

for every s > 12 .

Demonstration. Expanding formula (2.2.3) as Fourier series, we have

‖V δ(t)‖2Hs 's|aδ(t) + bδ(t)pδ(t)|2

(1− |pδ(t)|2)2s+1=

‖V δ(t)‖2H

12

(1− |pδ(t)|2)2s−1

with ‖V δ(t)‖2H

12

= ‖V δ(0)‖2H

12

= 1. By the explicit formula of pδ, we have

‖V δ(t)‖2Hs 's(

4 + δ2

4 cos2(ωt) + δ2

)2s−1

≤ ‖V δ(tδ)‖2Hs 'Csδ4s−2

with tδ := π2ω = π

δ√

4+δ2.


2.3 Long time behavior for small data

2.3.1 The case α ≥ 2

For all s > 12 , consider the NLS-Szego equation with small dispersion and small data.

i∂tu+ εα∂2xu = Π(|u|2u), ‖u(0)‖Hs = ε, 0 < ε < 1, α ≥ 0. (2.3.1)

At first, we show how to find the time interval Iαε = [−asε2 ,asε2 ], in which the solutions of (2.3.1) are

bounded by O(ε), for all α ≥ 0. Then we prove the maximality of Iαε in the case α > 2.

The bootstrap argument

The time interval Iαε = [−asε2 ,asε2 ] is given by a bootstrap argument.

Lemma 2.3.1. Let a, b, T > 0, q > 1 and M : [0, T ] −→ R+ be a continuous function satisfying

M(τ) ≤ a+ bM(τ)q, for all τ ∈ [0, T ]

Assume that (qb)1q−1M(0) ≤ 1 and (qb)

1q−1 a ≤ q−1

q . Then

M(τ) ≤ q

q − 1a

for all τ ∈ [0, T ].

Demonstration. The function fq : z ∈ R+ 7−→ z − bzq attains its maximum at the critical point zc =

(qb)−1q−1 . fq(zc) = q−1

q (qb)−1q−1 . Since a ≤ maxz≥0 fq(z) = fq(zc), there exists z− ≤ zc ≤ z+ such that

z ≥ 0 : fq(z) ≤ a = [0, z−] ∪ [z+,+∞[

and fq(z±) = a. Since fq(M(τ)) ≤ a, ∀0 ≤ τ ≤ T and M(0) ≤ zc, we have M([0, T ]) ⊂ [0, z−]. By the

concavity of fq on [0,+∞[, we have fq(z) ≥ fq(zc)zc

z for all z ∈ [0, zc]. Consequently, M(τ) ≤ z− ≤ qq−1a,

for all 0 ≤ τ ≤ T .

Proof of of estimate (2.1.7). For all α ≥ 0 and ε ∈ (0, 1) fixed, we rescale u as u = εµ, equation (2.3.1)becomes

i∂tµ+ εα∂2xµ = ε2Π(|µ|2µ),

‖µ(0)‖Hs = 1.(2.3.2)

Duhamel’s formula of equation (2.3.2) gives the following estimate :

sup0≤τ≤t

‖µ(τ)‖Hs ≤ ‖µ(0)‖Hs + Csε2t sup

0≤τ≤t‖µ(τ)‖3Hs (2.3.3)

Here Cs denotes the Sobolev constant in the inequality ‖|µ|2µ‖Hs ≤ Cs‖µ‖3Hs . We choose as = 427Cs

andthe following estimate holds

sup|t|≤ as

ε2

‖u(t)‖Hs ≤3

2ε, (2.3.4)

by using Lemma 2.3.1 with q = 3, T = asε2 , a = M(0) = 1, b = Csε

2T and M(t) = sup0≤τ≤t ‖µ(τ)‖Hs .The case t < 0 is similar.

2.3. LONG TIME BEHAVIOR FOR SMALL DATA 45

Optimality of the time interval if α > 2

In order to prove the optimality of Iαε in which estimate (2.3.4) holds, we set u(0, x) = ε(eix + δ) andrescale u(t, x) = εU(ε2t, x). Then, we have

i∂tU + ν2∂2xU = Π(|U |2U), U(0, x) = eix + δ,

where ν := εα−2

2 . Since the optimality is a consequence of Theorem 2.1.9, we prove at first Theorem 2.1.9by comparing U to the solution of the cubic Szego equation with the same initial datum,

i∂tV = Π(|V |2V ), V (0, x) = eix + δ.

Proof of Theorem 2.1.9. We shall estimate their difference r(t, x) := U(t, x)− V (t, x), which satisfies thefollowing equation

i∂tr + ν2∂2xr = −ν2∂2

xV + Π(V 2r + 2|V |2r) +Q(r), r(0) = 0, (2.3.5)

with Q(r) := Π(V r2 + 2V |r|2 + |r|2r). Thus, we can calculate the derivative of ‖r(t)‖2H1 ,

∂t‖r(t)‖2H1

=∂t‖r(t)‖2L2 + ∂t‖∂xr(t)‖2L2

=2Im〈i∂tr(t), r(t)〉L2 + 2Im〈∂x(i∂tr(t)), ∂xr(t)〉L2

=2Im

∫S1

ν2∂xV ∂xr + V 2r2 − V |r|2r

+ 2Im

∫S1

−ν2∂3xV ∂xr + V 2(∂xr)

2 + 2V ∂xV r∂xr + 4Re(V ∂xV )r∂xr

+ 2Im

∫S1

∂xV r2∂xr + 2V r|∂xr|2 + 2∂xV |r|2∂xr + 4V Re(r∂xr)∂xr + r2(∂xr)

2.

Then, we have the following estimate∣∣∣∂t‖r(t)‖2H1

∣∣∣≤2ν2‖∂xr‖L2(‖∂xV ‖L2 + ‖∂3

xV ‖L2) + 2‖V ‖2L∞‖r‖2H1 + 2‖V ‖L∞‖r‖L∞‖r‖2L2

+ 12‖V ‖L∞‖r‖L∞‖∂xV ‖L2‖∂xr‖L2 + 6‖r‖2L∞‖∂xV ‖L2‖∂xr‖L2

+ 12‖V ‖L∞‖r‖L∞‖∂xr‖2L2 + 2‖r‖2L∞‖∂xr‖2L2

≤ν2(2‖∂xr‖2L2 + ‖∂xV ‖2L2 + ‖∂3xV ‖2L2) + 2‖V ‖2L∞‖r‖2H1

+ 12‖V ‖L∞‖∂xV ‖L2‖r‖L∞‖∂xr‖L2 +O(‖r‖3H1).

L∞-estimate of V is given by Corollary 2.2.2 and Hs-growth of V is given by Proposition 2.2.4, for alls > 1

2 . Thus, we have

M∞ := sup0<δ<1

supt∈R‖V (t)‖L∞ < +∞,

and there exist C1, C3 > 0 such that

‖∂xV (t)‖L2 ≤ C1

δ, ‖∂3

xV (t)‖L2 ≤ C3

δ5,


for all 0 ≤ t ≤ tδ = πδ√

4+δ2. We use a bootstrap argument to estimate the term O(‖r‖3H1). Set

T := supt > 0 : sup0≤τ≤t

‖r(τ)‖H1 ≤ 1,

then we havesup

0≤t≤T‖r(t)‖L∞ ≤ C sup

0≤t≤T‖r(t)‖H1 ≤ C,

where C denotes the Sobolev constant H1(S1) → L∞. Consequently, for all 0 ≤ t ≤ min(T, tδ), we have∣∣∣∂t‖r(t)‖2H1

∣∣∣≤ν2(‖∂xV ‖2L2 + ‖∂3

xV ‖2L2)

+ ‖r‖2H1(2ν2 + 2‖V ‖2L∞ + 12C‖V ‖L∞‖∂xV ‖L2 + 6C‖r‖L∞‖∂xV ‖L2 + 12‖V ‖L∞‖r‖L∞ + 2‖r‖2L∞)

≤ν2(C2

1

δ2+C2

3

δ10) + (2 + 2M2

∞ + 12CM∞ + 2C2 + (12CM∞ + 6C2)C1

δ)‖r‖2H1

≤K(ν2

δ10+‖r(t)‖2H1

δ), ∀0 < δ, ν < 1,

with K := max(C21 + C2

3 , 2 + 2M2∞ + 12CM∞ + 2C2 + (12CM∞ + 6C2)C1). We set

ν = εα−2

2 = e−πK2δ2 ⇐⇒ δ =

√πK

(α−2)| ln ε| .

Using Gronwall’s inequality, we deduce that

‖r(t)‖2H1 ≤ν2

δ9eπK2δ2 = δ−9e−

πK2δ2 1 δ−2, ∀0 ≤ t ≤ tδ, ∀0 < δ 1.

Since ‖V (tδ)‖H1 ' 1δ by Theorem 2.2.4, we have ‖U(tδ)‖H1 = ‖V (tδ) + r(tδ)‖H1 ' 1

δ .

Fix α > 2, for every 0 < ε 1, we set

δ = δα,ε :=√

πK(α−2)| ln ε| 1, Tα,ε := tδ

ε2 = π

2ε2δ

√1+ δ2

4

'√

(α−2)| ln ε|ε2 .

Then we have ‖u(Tα,ε)‖H1 ' ε√

(α− 2)| ln ε| ε, while u(0, x) = ε(eix + δ). Then the optimality ofIαε = [−asε2 ,

asε2 ] is obtained.

2.3.2 The case 0 ≤ α < 2

We assume at first that u(0) ∈ C∞+ so that the energy functional of (2.3.1)

Eα,ε(u) =εα

2‖∂xu‖2L2 +

1

4‖u‖4L4

is well defined. For general initial data u(0) ∈ Hs+, if s ∈ ( 1

2 , 1), we use the density argument C∞+ = Hs+

and the continuity of the mapping u(0) ∈ Hs+ 7→ C([− as

ε4−α ,asε4−α ];Hs

+) .

We rescale u(t, x) 7→ ε−α2 u(ε−αt, x), then (2.3.1) is reduced to the case α = 0. It suffices to prove the

following estimatesup|t|≤ as

ε4

‖u(t)‖Hs = O(ε)

for the NLS-Szego equation i∂tu+ ∂2xu = Π(|u|2u) with ‖u(0)‖Hs = ε.


Identifying the resonances

The study of the resonant set of the NLS-Szego equation is necessary before the Birkhoff normal formtransformation. We refer to Eliasson–Kuksin [32] to see the analysis of resonances for a more general nonlinear term and the KAM theorem for the NLS equation.

We use again the change of variable u = εµ and the Duhamel’s formula of µ with η(t) =∑k≥0 ηk(t)eikx :=

e−it∂2xµ(t). Then we have

ηk(t) = µ0(t)− iε2∑

k1−k2+k3−k=0

∫ t

0

e−iτ(k21−k

22+k2

3−k2)ηk1

(τ)ηk2(τ)ηk3

(τ)dτ,

for all k ≥ 0. Recall the classical identification of the resonant setk1 − k2 + k3 − k4 = 0

k21 − k2

2 + k23 − k2

4 = 0.⇐⇒

k1 = k2

k3 = k4

or

k1 = k4

k2 = k3

.

In order to cancel all the resonances, we apply the transformation v(t) := e2itε2‖µ(0)‖2L2µ(t). As ‖µ‖L2 is

a conservation law, we have the following nonlinear Schrodinger equation without resonances

i∂tv(t) + ∂2xv(t) = ε2

[Π(|v(t)|2v(t))− 2‖v(t)‖2L2v(t)

], ∀t ∈ R. (2.3.6)

Its energy functional is

Hε(v) =1

2‖∂xv‖2L2 +

ε2

4

(‖v‖4L4 − 2‖v‖4L2

)=: H0(v) + ε2R(v). (2.3.7)

Consider the Fourier modes of v =∑n≥0 vne

inx, then we have

4R(v) =∑

k1−k2+k3−k4=0k2

1−k22+k2

3−k24 6=0

vk1vk2

vk3vk4−∑k≥0

|vk|4.

The Birkhoff normal form

Equation (2.3.6) is transferred to another Hamiltonian equation which is closer to the linear Schrodingerequation by the Birkhoff normal form transformation. We look for a symplectomorphism Ψε such thatthe energy functional Hε is reduced to the following Hamiltonian

Hε Ψε(v) = H0(v) + ε2R(v) +O(ε4−α),

where R(v) = − 14

∑k≥0 |vk|4. Ψε is chosen as the value at time 1 of the Hamiltonian flow of some energy

ε2F .

We fix the value s > 12 . Recall that, given a smooth real valued function H, we denote XH the Hamiltonian

vector field, i.e,dH(v)(h) = ω(h,XH(v)).

Given two smooth real-valued functions F and G on Hs+, their Poisson bracket F,G is defined by

F,G(v) := ω(XF (v), XG(v)) =2

i

∑k≥0

(∂vkF∂vkG− ∂vkG∂vkF ) (v), (2.3.8)


for all v =∑k≥0 vke

ikx ∈ Hs+. In particular, if F and G are respectively homogeneous of order p and q,

then their Poisson bracket is homogeneous of order p+ q − 2.

Lemma 2.3.2. Set F (v) :=∑k1−k2+k3−k4=0 fk1,k2,k3,k4

vk1vk2

vk3vk4

, with the coefficients

fk1,k2,k3,k4=

i

4(k21−k2

2+k23−k2

4), if k2

1 − k22 + k2

3 − k24 6= 0,

0, otherwise.

Thus, F is real-valued and its Hamiltonian field XF is smooth on Hs+ such that F,H0 + R = R and

the following estimates hold for all v ∈ Hs+.

‖XF (v)‖Hs .s ‖v‖3Hs ,‖dXF (v)‖B(Hs) .s ‖v‖2Hs .

Demonstration. F is well defined because sup(k1,k2,k3,k4)∈Z4 |fk1,k2,k3,k4| ≤ 1

4 , the Sobolev embedding

yields that |F (v)| ≤ 14

(∑k≥0 |vk|

)4

.s ‖v‖4Hs . The Sobolev estimates of ‖XF (v)‖Hs and ‖dXF (v)‖B(Hs)

are given by the Young’s convolution inequality l1 ∗ l1 ∗ l2 → l2. Using (2.3.8) and the definition offk1,k2,k3,k4

, we have

F,H0(v) =i∑

k1−k2+k3−k4=0

(k21 − k2

2 + k23 − k2

4)fk1,k2,k3,k4vk1

vk2vk3

vk4

=− 1

4

∑k1−k2+k3−k4=0k2

1−k22+k2

3−k24 6=0

vk1vk2vk3vk4

=−R(v) + R(v).

Set χσ := exp(ε2σXF ) the Hamiltonian flow of ε2F , i.e.,

d

dσχσ(v) = ε2XF (χσ(v)), χ0(v) = v.

We perform the canonical transformation Ψε := χ1 = exp(ε2XF ). The next lemma will prove the localexistence of χσ, for |σ| ≤ 1 and give the estimate of the difference between v and Ψ−1

ε (v)

Lemma 2.3.3. For s > 12 , there exist two constants ρs, Cs > 0 such that for all v ∈ Hs

+, if ε‖v‖Hs ≤ ρs,then χσ(v) is well defined on the interval [−1, 1] and the following estimates hold :

supσ∈[−1,1]

‖χσ(v)‖Hs ≤3

2‖v‖Hs ,

supσ∈[−1,1]

‖χσ(v)− v‖Hs ≤ Csε2‖v‖3Hs ,

‖dχσ(v)‖B(Hs) ≤ exp(Csε2‖v‖2Hs |σ|), ∀σ ∈ [−1, 1].


Demonstration. The Lipschitz coefficient of the mapping v 7−→ ε2XF (v) is bounded by Csε2‖v‖2Hs ≤

Csρ2s, by Lemma 2.3.2. If ρs is sufficiently small, the Hamiltionian flow (σ, v) 7→ χσ(v) exists on the

maximal interval (−σ∗, σ∗), by the Picard-Lindelof theorem. Assume that σ∗ < 1, then Lemma 2.3.2 andthe following integral formula

χσ(v) = v + ε2∫ σ

0

XF (χτ (v))dτ, ∀0 ≤ σ < σ∗. (2.3.9)

yield thatsup

0≤τ≤σ‖χτ (v)‖Hs ≤ ‖v‖Hs + Csσε

2 sup0≤τ≤σ

‖χτ (v)‖3Hs , ∀0 ≤ σ < σ∗ < 1.

By Lemma 2.3.1 with M(t) = sup0≤τ≤t ‖χτ (v)‖Hs , q = 3, a = M(0) = ‖v‖Hs and b = Csε2, we have

sup|σ|≤σ∗

‖χσ(v)‖Hs ≤3

2‖v‖Hs ,

if ε‖v‖Hs ≤ 23√

3Cs. This is a contradiction to the blow-up criterion. Hence σ∗ ≥ 1, and we have

sup|σ|≤1 ‖χσ(v)‖Hs ≤ 32‖v‖Hs , if ε‖v‖Hs ≤ ρs := 2

3√

3Cs. For all σ ∈ [−1, 1], by using Lemma 2.3.2,

we have

‖χσ(v)− v‖Hs ≤ |σ|ε2 sup0≤t≤|σ|

‖XF (χt(v))‖Hs ≤ Csε2 sup0≤t≤|σ|

‖χt(v)‖3Hs ≤ Csε2‖v‖3Hs .

if ε‖v‖Hs ≤ ρs. We differentiate equation (2.3.9) and use again Lemma 2.3.2 to obtain

‖dχσ(u)‖B(Hs) =‖IdHs + ε2∫ σ

0

dXF (χt(u))dχt(u)dt‖B(Hs)

≤1 + ε2∣∣∣ ∫ σ

0

‖dXF (χt(v))‖B(Hs)‖dχt(v)‖B(Hs)dt∣∣∣

≤1 + Csε2‖v‖2Hs

∣∣∣ ∫ σ

0

‖dχt(v)‖B(Hs)dt∣∣∣

≤eCsε2|σ|‖v‖2Hs , ∀σ ∈ [−1, 1].

Gronwall’s inequality is used in the last step.

Composed with the symplectomorphism Ψε = χ1, the energy functional Hε can be reduced to the normalform.

Lemma 2.3.4. For s > 12 , there exists a smooth mapping Y : Hs

+ −→ Hs+ and a constant C ′s > 0 such

that XHεΨε = XH0 + ε2XR + ε4Y,

‖Y (v)‖Hs ≤ C ′s‖v‖5Hs ,

for all v ∈ Hs+ such that ε‖v‖Hs ≤ ρs. Set w(t) := Ψ−1

ε (v(t)), then we have∣∣∣ d

dt‖w(t)‖2Hs

∣∣∣ ≤ C ′sε4‖w(t)‖6Hs , (2.3.10)

if ε‖w(t)‖Hs ≤ ρs.


Demonstration. We expand the energy Hε χ1 = H0 χ1 + ε2R χ1 with Taylor’s formula at time σ = 1around 0. Since χ0 = IdHs+ , we have

Hε χ1 =

(H0 +

d

dσ[H0 χσ]|σ=0 +

∫ 1

0

(1− σ)d2

dσ2[H0 χσ]dσ

)+ ε2R+ ε2

∫ 1

0

d

dσ[R χσ]dσ

=H0 + ε2 [F,H0+R] + ε4∫ 1

0

(1− σ)F, F,H0 χσ + F,R χσdσ

=H0 + ε2R+ ε4∫ 1

0

[(1− σ)F, R+ σF,R

] χσdσ

We set G(σ) := (1 − σ)F, R + σF,R, ∀σ ∈ [0, 1]. Since XF,R and XF,R(u) are homogeneous ofdegree 5 with uniformly bounded coefficients, we have

‖XG(σ)(v)‖Hs ≤ (1− σ)‖XF,R(v)‖Hs + σ‖XF,R(v)‖Hs .s ‖v‖5Hs , ∀v ∈ Hs+,

By the chain rule of Hamiltonian vector fields :

XG(σ)χσ (v) = dχ−σ(χσ(v)) XG(σ)(χσ(v)), ∀v ∈ Hs+, ∀σ ∈ [0, 1], (2.3.11)

and Lemma 2.3.3, we have

‖XG(σ)χσ (v)‖Hs ≤ ‖dχ−σ(χσ(v))‖B(Hs)‖XG(σ)(χσ(v))‖Hs .s eCsε2‖v‖2Hs ‖χσ(v)‖5Hs .s ‖v‖5Hs ,

for all v ∈ Hs+ such that ε‖v‖Hs ≤ ρs. Thus we define Y :=

∫ 1

0XG(σ)χσdσ and we have

XHεΨε = XHεχ1= XH0

+ ε2XR + ε4Y.

If ε‖v‖Hs ≤ ρs, then ‖Y (v)‖Hs .s ‖v‖5Hs .

Since XR(w)(k) = −i|wk|2wk, ∀k ≥ 0 and w(t) = Ψ−1ε (v(t)), w =

∑k≥0 wke

ikx solves the followinginfinite dimensional Hamiltonian system of Fourier modes :

i∂twk(t)− k2wk(t)− ε2|wk(t)|2wk(t) = iε4 Y (w(t))(k), ∀k ≥ 0. (2.3.12)

If ε‖w(t)‖Hs ≤ ρs, then we have∣∣∣∂t‖w(t)‖2Hs∣∣∣ ≤ 2ε4‖Y (w(t))‖Hs‖w(t)‖Hs .s ε4‖w(t)‖6Hs .

End of the proof of the case 0 ≤ α < 2

Demonstration. Recall that w(t) = χ−1(v(t)) and ‖v(0)‖Hs = 1. Lemma 2.3.3 yields that if ε ≤ ρs, thenwe have

‖v(0)− w(0)‖Hs = ‖v(0)− χ−1(v(0))‖Hs ≤ Csε2‖v(0)‖3Hs ≤ Cs.


Set Ks := 3Cs + 1. Then ‖w(0)‖Hs ≤ Ks3 . We define

εs := min

(ρs

3Ks,

1√8CsK2

s

)and

T := supt ≥ 0 : sup0≤τ≤t

‖v(τ)‖Hs ≤ 2Ks.

For all ε ∈ (0, εs) and t ∈ [0, T ], we have ε‖v(t)‖Hs ≤ ρs. Hence Lemma 2.3.3 gives the following estimate

‖w(t)‖Hs ≤‖v(t)‖Hs + ‖χ−1(v(t))− v(t)‖Hs≤‖v(t)‖Hs + Csε

2‖v(t)‖3Hs≤2Ks + 8CsK

3s ε

2

≤3Ks.

So we have ε sup0≤t≤T ‖w(t)‖Hs ≤ ρs and∣∣∣ d

dt‖w(t)‖2Hs∣∣∣ ≤ C ′sε

4‖w(t)‖6Hs , by Lemma 2.3.4. Set as :=1

37K4sC′s. We can precise the estimate of ‖w(t)‖Hs by limiting 0 ≤ t ≤ asε−4 :

‖w(t)‖2Hs ≤‖w(0)‖2Hs + C ′stε4‖w(t)‖6Hs ≤

K2s

9+ 36C ′sK

6s tε

4 ≤ 4K2s

9,

for all 0 ≤ t ≤ min(T, asε4 ). Then we have

‖v(t)‖Hs ≤ ‖w(t)‖Hs + ‖χ1(w(t))− w(t)‖Hs ≤ ‖w(t)‖Hs + Csε2‖w(t)‖3Hs ≤ Ks,

for all t ∈ [0, asε4 ]. Consequently, we have

sup0≤t≤ as

ε4

‖u(t)‖Hs = ε sup0≤t≤ as

ε4

‖v(t)‖Hs ≤ Ksε.

In the case t < 0, we use the same procedure and we replace t by −t.

The open problem of optimality

Recall that Hε = H0 + ε2R is the energy functional of the equation

i∂tu+ ∂2xu = Π(|u|2u), ‖u(0, ·)‖Hs = ε. (2.3.13)

with H0(v) = 12‖∂xv‖

2L2 and R(v) = 1

4 (‖v‖4L4 − 2‖v‖4L2). In order to get a longer time interval in whichthe solution is uniformly bounded by O(ε), we expand the Hamiltonian Hε χ1 by using the Taylorexpansion of higher order to see whether the resonances can be cancelled by the Birkhoff normal formmethod.

Hε χ1

=H0 χ1 + ε2R χ1

=H0 + ∂σ(H0 χσ)∣∣σ=0

+1

2∂2σ(H0 χσ)

∣∣σ=0

+1

2

∫ 1

0

(1− σ)2∂3σ(H0 χσ)dσ

+ ε2(R+ ∂σ(R χσ)

∣∣σ=0

+

∫ 1

0

(1− σ)∂2σ(H0 χσ)dσ

)=H0 + ε2R+

ε4

2F,R+ R+

ε6

2

∫ 1

0

(1− σ)F, F, (1− σ)R+ (1 + σ)R χσdσ


We try to cancel the term ε4

2 F,R+R by using a canonical transform to Hε1 = Hεχ1 with the following

functional

G(v) =∑

k1−k2+k3−k4+k5−k6=0

gk1,k2,k3,k4,k5,k6vk1

vk2vk3

vk4vk5

vk6.

Then we should solve the homological equation G,H0+ 12F,R+ R = 0.

G,H0(v)

=i∑

k1−k2+k3−k4+k5−k6=0

(k21 − k2

2 + k23 − k2

4 + k25 − k2

6)gk1,k2,k3,k4,k5,k6vk1

vk2vk3

vk4vk5

vk6.

We can calculate that

1

2F,R+ R(v)

=2Im∑

k1−k2+k3−k4+k5−k6=0ki,k:=k1−k2+k3≥0

fk1,k2,k3,kvk1vk2vk3vk4vk5vk6

− 4Im∑

k1−k2+k3−k4+k5−k6=0ki,k:=k1−k2+k3≥0,k5=k6

fk1,k2,k3,kvk1vk2

vk3vk4

vk5vk6

− 2Im∑

k1−k2+k3−k4+k5−k6=0ki,k:=k1−k2+k3≥0,k4=k5=k6

fk1,k2,k3,kvk1vk2

vk3vk4

vk5vk6

In the first term of the preceding formula, there is a resonant set k21 − k2

2 + k23 − k2

4 + k25 − k2

6 = 0 thatcannot be cancelled by the other two terms. Thus the resonant subset

k1 − k2 + k3 − k4 + k5 − k6 = 0

k21 − k2

2 + k23 − k2

4 + k25 − k2

6 = 0

should be cancelled before the Birkhoff normal form transform, just like the step µ 7→ v = e2itε2‖µ(0)‖2L2µ(t),

which can cancel all the resonances k1 − k2 + k3 − k4 = 0

k21 − k2

2 + k23 − k2

4 = 0

before we do the canonical transformation Hε 7→ Hε χ1. We only know that fk1,k2,k3,k1−k2+k3=

fk4,k5,k6,k4−k5+k6if

k1 − k2 + k3 − k4 + k5 − k6 = 0

k21 − k2

2 + k23 − k2

4 + k25 − k2

6 = 0.

This resonant subset contains the case k5 6= k6. The optimality of the time interval for the case 0 ≤ α < 2remains open. We can see Grebert–Thomann [66] and Haus–Procesi [71] for instance to analyse theresonant set for 6 indices for the quintic NLS equation.

2.4. ORBITAL STABILITY OF THE PLANE WAVE 53

2.4 Orbital stability of the plane wave

Consider the following NLS-Szego equation

i∂tu+ εα∂2xu = Π(|u|2u), 0 < ε < 1, 0 ≤ α ≤ 2. (2.4.1)

We shall prove at first H1-orbital stability of the traveling waves em, for all m ∈ N. Then, we study theirlong time Hs-stability, for all s ≥ 1.

2.4.1 The proof of Theorem 2.1.5

We follow the idea of using conserved quantities mentioned in Gallay–Haragus [43] for equation (2.4.1).

Demonstration. For all m ∈ N, 0 < ε < 1 and 0 ≤ α ≤ 2, we denote u(0, x) = eimx + εf(x) with‖f‖H1 ≤ 1. The NLS-Szego equation has three conservation laws :

Q(u(t)) := ‖u(t)‖2L2 = ‖u(0)‖2L2 ;

P (u(t)) := (Du(t), u(t)) = P (u(0));

Eα,ε(u(t)) := εα

2 ‖∂xu(t)‖2L2 + 14‖u(t)‖4L4 = Eα,ε(u(0)),

with D = −i∂x and (u, v) := Re∫S1 uv. Thus the following quantity is conserved,

εα

2‖Du(t)−mu(t)‖2L2 +

1

4‖|u(t)|2 − 1‖2L2

=Eα,ε(u(t))− εαmP (u(t)) +|m|2εα − 1

2Q(u(t)) +

1

4

=ε2∫S1

|Ref(x)e−imx|2dx+ε2+α

2‖Df −mf‖2L2 + ε3

∫S1

|f(x)|2Re(f(x)e−imx)dx+ ε4‖f‖4L4

.mε2.

Then, we have supt∈R ‖Du(t)−mu(t)‖L2 .m ε1−α2 . Recall that em(x) = eimx, then the following estimate

holds,

‖u(t)− um(t)em‖2H1 =∑n 6=m

(1 + n2)|un(t)|2 .m ‖Du(t)−mu(t)‖2L2 .m ε2−α.

We have

infθ∈R‖u(t)− um(0)eiθem‖2H1

=‖u(t)− um(0)ei(arg um(t)−arg um(0))em‖2H1

=(1 +m2)∣∣∣|um(t)| − |um(0)|

∣∣∣2 + ‖u(t)− um(t)em‖2H1

and by the conservation of ‖u(t)‖L2 , we have∣∣∣|um(t)| − |um(0)|∣∣∣2 ≤∣∣∣|um(t)|2 − |um(0)|2

∣∣∣=∣∣∣ ∑n6=m

|un(0)|2 −∑n 6=m

|un(t)|2∣∣∣

= max(‖u(0)− um(0)em‖2L2 , ‖u(t)− um(t)em‖2L2)

.mε2−α.


Thus supt∈R ‖u(t) − um(0)ei(arg um(t)−arg um(0))em‖H1 .m ε1−α2 . The proof can be finished by um(0) =

1 + εfm = 1 +O(ε).

The preceding theorem also holds for the defocusing NLS equation on Td, for d = 1, 2, 3 (in the energysub-critical case) with T = R/Z ' S1.(see Gallay–Haragus [43, 44]) We refer to Zhidkov [153 Sect. 3.3]for a detailed analysis of the stability of plane waves.

Remark 2.4.1. Obtaining the estimate supt∈R ‖u(t)− um(t)em‖L∞ .m ε1−α2 by the Sobolev embedding

H1(S1) → L∞, we can also proceed by using the following estimate, which is uniform on x and t,

|u(t, x)|2 − 1 = |um(t)|2 − 1 +Om(ε1−α2 ).

The notation Om means that sup(t,x)∈R×S1

∣∣∣|u(t, x)|2−|um(t)|2∣∣∣ .m ε1−

α2 . Integrating the preceding term

with respect to x, we have

||um(t)|2 − 1| ≤ ‖|u(t)|2 − 1‖L2 + ‖Om(ε1−α2 )‖L2 .m ε1−

α2

Thus |um(t)| = 1 +Om(ε1−α2 ) and um(t) = ei arg um(t) +Om(ε1−

α2 ). Then we have

supt∈R‖u(t)− ei arg um(t)em‖2H1 .m ε2−α. (2.4.2)

Recall that if z = 1 +O(ε) then ei arg z = 1 +O(ε).

2.4.2 Long time Hs-stability

For every s ≥ 1, we suppose that ‖u(0)− em‖Hs ≤ ε. Thanks to estimate (2.4.2), we change the variableu 7→ v = vm,α,ε(t, x) =

∑n≥0 vn(t)einx ∈ C∞(R× S1) such that

u(t, x) = ei arg um(t)(eimx + ε1−α2 v(t, x)) (2.4.3)

to study Hs-stability of plane waves em and we have

Im := sup0≤α≤2

sup0<ε<1

supt∈R‖v(t)‖H1 < +∞. (2.4.4)

Proposition 2.4.2. For every s ≥ 1, m ∈ N, ε ∈ (0, 1) and α ∈ [0, 2), if u is smooth and solves (2.4.1)with u(0, x) = eimx + εf(x) and ‖f‖Hs ≤ 1, v is defined by formula (2.4.3), then we have

vm(t) ∈ R, ∀t ∈ R,supt∈R |vm(t)| .m εmin(α2 ,1−

α2 ),

supt∈R |∂tvm(t)| .m ε1−α2 ,

‖v(0)‖Hs .m,s εα2 .

Moreover, there exists a smooth function ϕ = ϕm : R→ R/2πZ ' S1 and ε∗m ∈ (0, 1) such that for every1 < 2− < 2, we have

arg um(t) = −(1 +m2εα)t+ εmin(1,2−α)ϕ(t),

sup0≤α≤2− sup0<ε<ε∗msupt∈R |ϕ′(t)| < +∞.


The parameter v satisfies the following equation

i∂tv + εα∂2xv −He2imx(v)− (1−m2εα + εmin(1,2−α)ϕ′(t))v

=εmin(α2 ,1−α2 )ϕ′(t)eimx + ε1−

α2 Π(e−imxv2 + 2eimx|v|2) + ε2−αΠ(|v|2v),

(2.4.5)

where He2imx(v) := Π[e2imxv] denotes the Hankel operator of symbol e2m.

Demonstration. Since um(t) = ei arg um(t)(1 + ε1−α2 vm(t)), we have 1 + ε1−

α2 vm(t) = |um(t)| ∈ R. So

vm(t) ∈ R, for all t ∈ R. By using the conservation law ‖ · ‖L2 and estimate (2.4.4), we have

1 + 2εRefm + ε2 = ‖u(0)‖2L2 = ‖u(t)‖2L2 = 1 + 2ε1−α2 vm(t) + ε2−α‖v(t)‖2L2 ,

which yields that supt∈R |vm(t)| .m εmin(α2 ,1−α2 ). Recall that

u(0, x) =∑n≥0

un(0)einx = eimx + εf(x).

Then we have um(0) = 1 + εfm = 1 +O(ε) and |ei arg um(0) − 1| . ε. Thus we have

ε1−α2 ‖v(0)‖Hs . (1 +m2)

s2 |ei arg um(0) − 1|+ ε‖f‖Hs .m,s ε.

We define θ(t) := arg(um(t)). Combing the following two formulas

i∂tu+ εα∂2xu = eiθ(t)

[ε1−

α2 (i∂tv + εα∂2

xv − θ′(t)v)− (m2εα + θ′(t))eimx]

Π[|u|2u] = eiθ(t)[eimx + ε1−

α2 (2v + Π(e2imxv)) + ε2−αΠ(e−imxv2 + 2eimx|v|2) + ε3(1−α2 )Π(|v|2v)

]we obtain that

ε1−α2 [i∂tv + εα∂2

xv −He2imx(v)− (2 + θ′(t))v]

=(1 +m2εα + θ′(t))eimx + ε2−αΠ(e−imxv2 + 2eimx|v|2) + ε3(1−α2 )Π(|v|2v),(2.4.6)

where He2imx(v) := Π[e2imxv]. The Fourier mode vm(t) satisfies the following equation

ε1−α2

[i∂tvm(t)−m2εαvm(t)− vm(t)− (2 + θ′(t))vm(t)

]=1 +m2εα + θ′(t) + ε2−αΠ(e−imxv2 + 2eimx|v|2)m(t) + ε3(1−α2 )Π(|v|2v)m(t).

Estimate (2.4.4) yields that

supt∈R|ε2−αΠ(e−imxv2 + 2eimx|v|2)m(t) + ε3(1−α2 )Π(|v|2v)m(t)| .m ε2−α.

Thus, we have

ε1−α2

[i∂tvm(t)− (3 +m2εα + θ′(t))vm(t)

]= 1 +m2εα + θ′(t) +Om(ε2−α). (2.4.7)

The imaginary part and the real part of (2.4.7) give respectively the two following estimates :

supt∈R|∂tvm(t)| .m ε1−

α2 ;

1 +m2εα + θ′(t) =−2ε1−

α2 vm(t) +Om(ε2−α)

1 + ε1−α2 vm(t)

=Om(εmin(1,2−α))

1 +Om(εmin(1,2−α))= Om(εmin(1,2−α)).


for all 0 < ε 1. Then we define ϕ(t) := (1+m2εα)t+θ(t)εmin(1,2−α) . Consequently, there exists ε∗m ∈ (0, 1) such that

arg um(t) = −(1 +m2εα)t+ εmin(1,2−α)ϕ(t)

sup0≤α≤2− sup0<ε<ε∗msupt∈R |ϕ′(t)| < +∞.

We replace θ′(t) by −1−m2εα + εmin(1,2−α)ϕ′(t) in (2.4.6) and we obtain (2.4.5).

Proof of Proposition 2.1.7

For every n ∈ N, we define the projector Pn : L2+ → L2

+ such that

Pn(∑k≥0

vkeikx) =

n∑j=0

vkeikx.

Now we prove Proposition 2.1.7 by using a bootstrap argument.

Demonstration. At the beginning, we suppose that u(0) ∈ C∞+ . In the general case u(0) ∈ Hs+, the proof

can be completed by using the continuity of the mapping u(0) 7→ u from Hs+ to C([− bs,m

ε1−α2,bs,m

ε1−α2

], Hs+).

We use the same transformation u 7→ v as (2.4.3). Proposition 2.4.2 yields that there exists Am,s ≥ 1such that ‖v(0)‖Hs .m,s ε

α2 ≤ Am,s. By using estimate (2.4.4), we have

supε∈(0,1)

supt∈R‖P2m(v(t))‖Hs ≤ (1 + 4m2)

s2 Im

We define that Lm,s := max(2(1 + 4m2)s2 Im, 2Am,s + 1) and

T := supt > 0 : sup0≤τ≤t

‖v(τ)‖Hs ≤ 2Lm,s.

Rewrite equation (2.4.5) with Fourier modes and we have

i∂tvn − (1 + (n2 −m2)εα + εmin(1,2−α)ϕ′(t))vn = ε1−α2 [Z(v(t))]n, ∀n ≥ 2m+ 1,

with Z(v) =∑n≥0[Z(v)]ne

inx = Π(e−imxv2 + 2eimx|v|2) + ε1−α2 Π(|v|2v). Then we have

‖Z(v)− P2m(Z(v))‖Hs .s ‖v‖2Hs + ε1−α2 ‖v‖3Hs . (2.4.8)

Then we have

|∂t‖v(t)− P2m(v(t))‖2Hs | ≤2ε1−α2

∑n≥2m+1

(1 + n2)s|vn(t)||[Z(v(t))]n|

≤2ε1−α2 ‖v(t)‖Hs‖Z(v(t))− P2m(Z(v(t)))‖Hs

.sε1−α2 ‖v(t)‖3Hs + ε2−α‖v(t)‖4Hs .

For all t ∈ [0, T ], we have

‖v(t)‖2Hs =‖P2mv(t)‖2Hs + ‖v(t)− P2m(v(t))‖2Hs≤(1 + 4m2)sI2

m + Cs(ε1−α2 ‖v(t)‖3Hs + ε2−α‖v(t)‖4Hs)t+ ‖v(0)‖2Hs

≤1

4L2m,s + 32CsL

4m,sε

1−α2 t+A2m,s.

Define bm,s = 164CsL2

m,sand we have ‖v(t)‖Hs ≤ Lm,s, for all t ∈ [0,

bs,m

ε1−α2

]. The case t < 0 is similar.


Homological equation

We try to improve Proposition 2.1.7 and get a longer time interval in which the solution v is stillbounded by Om,s(1), by using the Birkhoff normal form transformation. Recall the symplectic formω(u, v) = Im

∫S1 uv

dθ2π on the energy space H1

+ and the Poisson bracket for two smooth real-valuedfunctionals F,G : C∞+ → R

F,G(v) =2

i

∑k≥0

(∂vkF∂vkG− ∂vkG∂vkF ) (v),

for all v =∑k≥0 vke

ikx ∈ C∞+ . For all 0 ≤ α < 2 and 0 < ε 1, equation (2.4.5) is a non autonomousHamiltonian equation. Its energy functional is

Hm,α,ε(t, v)

=Hm,α,ε0 (v) + ε1−α2Hm1 (v) +

ε2−α

4N4(v) + εmin(α2 ,1−

α2 )ϕ′(t)(Lm(v) +

ε1−α2

2N2(v)),

with Hm,α,ε0 (v) = εα

2 ‖∂xv‖2L2 + 1−m2εα

2 ‖v‖2L2 + 12

∫S1 Re(e2imxv2),

Hm1 (v) = Re(∫S1 e−imx|v|2v),

Np(v) = ‖v‖pLp , p = 2 or 4,

Lm(v) = Revm.

We want to cancel all the high frequencies in the term Hm1 (v) by composing Hm,α,ε with the Hamiltonianflow of some auxiliary functional ε1−

α2 Fm. In order to get the appropriate Fm, we need to solve the

following homological systemFm,Hm,α,ε0 (v) +Hm1 (v) = Rm(v)

Fm,Lm(v) = −Nm2 (v) := −

∑n≥2m+1 |vn|2

such that Rm depends only on finitely many Fourier modes of v. The remaining coefficient in front ofε1−

α2 would be Rm + ϕ′(t)εmin(α2 ,1−

α2 )(−Nm

2 + N2

2 ). One can prove the following proposition.(see alsoProposition 2.4.4 and Appendix for the proof in the special case α = 0)

Proposition 2.4.3. For every m ∈ N, the homogenous functional Fm of degree 3 is defined as

Fm(v) =∑

j−l+k=mj,k,l∈N

Re(aj,l,kvjvlvk), ∀v ∈⋃n≥0

Pn(C∞+ ),

for some ak,l,j = aj,l,k ∈ C. Then we have the following formula

Fm,Hm,α,ε0 (v) +Hm1 (v) = Re(

Resonlow(v) + Reson≥2m+1(v)),

where

Resonlow(v)

=∑

0≤j,k≤2m

cj,j+k−m,kvjvj+k−mvk −∑

j−l+k=mj,l,k∈N,l≤2m

iaj,l,kvjvk[(1 + (l2 −m2)εα)vl + v2m−l

],


for some cj,j+k−m,k = ck,j+k−m,j ∈ C and

Reson≥2m+1(v)

=∑

k≥2m+1

m−1∑j=0

2 (−ia2m−j,m+k−j,k + (1 + 2(m− j)(k − j)εα)iaj,k,k+m−j + 1) vjvkvk+m−j

+∑

k≥2m+1

m−1∑j=0

2((1− 2(k −m)(m− j)εα)ia2m−j,m+k−j,k − iaj,k,k+m−j + 1)v2m−jvm+k−jvk

+∑

k≥2m+1

(2(iam,k,k − iam,k,k + 1)vm|vk|2 + ((1− 2(k −m)2εα)iak,2k−m,k + 1)v2

kv2k−m)

+∑

k≥2m+1

∑j≥k+1

2((1− 2(j −m)(k −m)εα)iak,j+k−m,j + 1)vkvj+k−mvj .

The term Resonlow depends only on the small Fourier modes v1, v2, · · · , v3m. We try to find a boundedsequence (aj,l,k)j−l+k=m such that Reson≥2m+1 = 0 in order to cancel the term Hm1 . However, the coeffi-cient (1−2(j−m)(k−m)εα) in front of the parameter ak,j+k−m,j may have an arbitrarily small absolutevalue if α > 0. Such sequence does not exist if ε−α ∈ 2N

⋂[2(m+ 1)2,+∞).

We suppose that ε−α /∈ Q, then Reson≥2m+1 = 0 is equivalent to a linear system, which has a uniquesolution

a2m−j,m+k−j,k = ak,m+k−j,2m−j = i(k−j)(m−j)(1−2(k−m)(k−j)εα) , ∀0 ≤ j ≤ m− 1,

aj,k,k+m−j = ak+m−j,k,j = i(m−k)(m−j)(1−2(k−m)(k−j)εα) , ∀0 ≤ j ≤ m− 1,

am,k,k = ak,k,m = i2 ,

ak,k+j−m,j = aj,k+j−m,k = i1−2(j−m)(k−m)εα , ∀j ≥ 2m+ 1,

(2.4.9)

for all k ≥ 2m+ 1. In the case m = 0, (2.4.9) has only the last two formulas. When α > 0, the sequence(aj,l,k)j−l+k=m can be arbitrarily large, for 0 < ε 1. We suppose that εα is an irrational algebraicnumber of degree d ≥ 2. Then we have the Liouville estimate [100]

|aj,j+k−m,k| ≤1

cε,α(2(j −m)(k −m))d−1, ∀j, k ≥ 2m+ 1,

which loses the regularity of v in the estimates of XFm(v). It is difficult to find the same kind of estimatefor the transcendental numbers, which can preserve the regularity. So we return to the case α = 0.

2.4.3 Long time Hs-stability in the case α = 0

For α = 0 and every m ∈ N and s ≥ 1, assume that u is the smooth solution of the NLS-Szego equation

i∂tu+ ∂2xu = Π(|u|2u), u(0, x) = eimx + εf(x), ‖f‖Hs ≤ 1,

and u(t, x) = ei arg um(t)(eimx + εv(t, x)). Then v is solves the following Hamiltonian equation

i∂tv + ∂2xv −He2imx(v)− (1−m2 + εϕ′(t))v

=ϕ′(t)eimx + εΠ(e−imxv2 + 2eimx|v|2) + ε2Π(|v|2v).


Its energy functional is

Hm,ε(t, v) = Hm0 (v) + ϕ′(t)Lm(v) + ε(Hm1 (v) +

ϕ′(t)

2N2(v)) +

ε2

4N4(v),

with Hm0 (v) = 1

2‖∂xv‖2L2 + 1−m2

2 ‖v‖2L2 + 12

∫S1 Re(e2imxv2)dx,

Lm(v) = Revm,

Hm1 (v) = Re(∫S1 e−imx|v|2v)dx,

Np(v) = ‖v‖pLp , p = 2 or 4.

We define Nm2 (v) := ‖v − P2mv‖2L2 =

∑n≥2m+1 |vn|2 and the following proposition holds.

Proposition 2.4.4. For every s > 12 and m ∈ N, there exists a sequence (aj,l,k)j−l+k=m such that

aj,l,k = ak,l,j, supm≥1 supj−l+k=m |aj,l,k| = 12 and the functional Fm : Hs

+ → R, defined by

Fm(v) =∑

j−l+k=mj,k,l∈N

Re(aj,l,kvjvlvk), ∀v ∈ C∞+ ,

satisfies that Fm,Lm = −Nm2 and Rm := Fm,Hm0 + Hm1 is a finite sum of the Fourier modes

v1, · · · , v3m. Moreover, for all v, h ∈ Hs+, we have

‖XFm(v)‖Hs .m,s ‖v‖2Hs , ‖dXFm(v)h‖Hs .m,s ‖v‖Hs‖h‖Hs .

Demonstration. For the convenience of the reader, the detailed calculus for Rm = Fm,Hm0 + Hm1and formula (2.4.9) in the case α = 0 are postponed in Appendix. We define aj,j+k−m,k = 0, for all0 ≤ j, k ≤ 2m and an,m+1+n,2m+1 = a2m+1,m+1+n,n = 0, for all 0 ≤ n ≤ m − 1. Combing Proposition2.4.3 and (2.4.9) with α = 0, we have

Reson≥2m+1(v)∣∣∣α=0

= 0, Re(

Resonlow(v)∣∣∣α=0

)= Rm(v),

Fm,Lm(v) = 2Im(∑k≥0 am,k,k|vk|2 + 1

2

∑j+k=2m aj,m,kvjvk) = −

∑n≥2m+1 |vn|2.

By (2.4.9) with α = 0, we have |aj,j+k−m,k| ≤ 12 , for all j, k ≥ 0. By the definition of Fm, we have

[XFm(v)](n) = −2i∑k−l+n=m ak,l,nvkvl − i

∑k−n+l=m ak,n,lvkvl

[dXFm(v)h](n) = −2i∑k−l+n=m ak,l,n(vkhl + vlhk)− 2i

∑k−n+l=m ak,n,lvkhl,

for all n ≥ 0. The last two estimates are obtained by Young’s convolution inequality for l1 ∗ l2 → l2.

The Birkhoff normal form

Set χmσ := exp(εσXFm) the Hamiltonian flow of εFm, i.e.,

d

dσχmσ (v) = εXFm(χmσ (v)), χm0 (v) = v.


We perform the canonical transformation Ψm,ε := χm1 . We want to reduce the energy functional Hm,ε tothe following normal form

Hm,ε(t) Ψm,ε = Hm0 + ϕ′(t)Lm(v) + ε

(Rm + ϕ′(t)(−Nm

2 +N2

2)

)+O(ε2).

Since Rm depends only on low-frequency Fourier modes v1, · · · , v3m, the high-frequency filtering Hs-norm of Ψ−1

m,ε(v) is appropriately estimated by the Birkhoff normal form transformation. The estimateof ‖P3m(v)‖Hs is given by (2.4.4). The next lemma will give the local existence of χmσ , for |σ| ≤ 1 andestimate the difference between v and Ψ−1

m,ε(v).

Lemma 2.4.5. For every s > 12 and m ∈ N, there exist two constants γm,s, Cm,s > 0 such that for all

v ∈ Hs+, if ε‖v‖Hs ≤ γm,s, then χmσ (v) is well defined on the interval [−1, 1] and the following estimates

hold :

supσ∈[−1,1]

‖χmσ (v)‖Hs ≤ 2‖v‖Hs ,

supσ∈[−1,1]

‖χmσ (v)− v‖Hs ≤ Cm,sε‖v‖2Hs ,

‖dχmσ (v)‖B(Hs) ≤ exp(Cm,sε‖v‖Hs |σ|), ∀σ ∈ [−1, 1].

The proof is based on a bootstrap argument, which is similar to Lemma 2.3.3, given by Lemma 2.3.1with q = 2. The energy functional Hm,ε is reduced to the normal form in the following lemma.

Lemma 2.4.6. For all s > 12 , m ∈ N and 0 < ε < ε∗m, there exists a smooth mapping Ym : R×Hs

+ −→Hs

+ and a constant C ′m,s > 0 such that for all t ∈ R, we have

XHm,ε(t)Ψm,ε = XHm0 + ϕ′(t)XLm + ε

(XRm + ϕ′(t)(−XNm2 +

1

2XN2)

)+ ε2Ym(t),

and supt∈R ‖Ym(t, v)‖Hs ≤ C ′m,s‖v‖2Hs(1 + ‖v‖Hs), for all v ∈ Hs+ such that ε‖v‖Hs ≤ γm,s. Set w(t) :=

Ψ−1m,ε(v(t)), then we have∣∣∣ d

dt‖w(t)− P3m(w(t))‖2Hs

∣∣∣ ≤ C ′m,sε2‖w(t)‖3Hs(1 + ‖w(t)‖Hs), (2.4.10)

if ε‖w(t)‖Hs ≤ γm,s.

Demonstration. For every t ∈ R, we expand the energy Hm,ε(t) Ψm,ε = Hm,ε(t) χm1 with Taylor’sformula at time σ = 1 around 0. Since χm0 = IdHs+ , we have

(Hm0 + ϕ′(t)Lm) χm1

=Hm0 + ϕ′(t)Lm +d

dσ[(Hm0 + ϕ′(t)Lm) χmσ ]|σ=0 +

∫ 1

0

(1− σ)d2

dσ2[(Hm0 + ϕ′(t)Lm) χmσ ]dσ

=Hm0 + ϕ′(t)Lm + εFm,Hm0 + ϕ′(t)Lm+ ε2∫ 1

0

(1− σ)Fm, Fm,Hm0 + ϕ′(t)Lm χmσ dσ

and (Hm1 +

ϕ′(t)

2N2

) χm1 =Hm1 +

ϕ′(t)

2N2 +

∫ 1

0

d

dσ[(Hm1 +

ϕ′(t)

2N2) χmσ ]dσ

=Hm1 +ϕ′(t)

2N2 + ε

∫ 1

0

Fm,Hm1 +ϕ′(t)

2N2 χmσ dσ.


Since Fm solves the homological system

Fm,Hm0 +Hm1 = RmFm,Lm+ Nm

2 = 0in Proposition 2.4.4, we have

Hm,ε(t) χm1

=Hm0 + ϕ′(t)Lm + ε

(Fm,Hm0 +Hm1 + ϕ′(t)(Fm,Lm+

1

2N2)

)+ ε2

[∫ 1

0

Fm, (1− σ)Fm,Hm0 + ϕ′(t)Lm+Hm1 +ϕ′(t)

2N2 χmσ dσ +

N4 χm14

]=Hm0 + ϕ′(t)Lm + ε

(Rm + ϕ′(t)(−Nm

2 +1

2N2)

)+ ε2

[∫ 1

0

Gm(t, σ) χmσ dσ +N4 χm1

4

],

where Gm(t, σ) = Fm, (1− σ)Rm + σHm1 + ϕ′(t)((σ − 1)Nm2 + 1

2N2). We set

Ym(t, v) :=

∫ 1

0

XGm(t,σ)χmσ (v)dσ +1

4XN4χm1 (v),

then we get

XHm,ε(t)χm1 = XHm0 + ϕ′(t)XLm + ε

(XRm + ϕ′(t)(−XNm2 +

1

2XN2

)

)+ ε2Ym(t).

Since Fm, H1m and Rm are homogeneous series of order 3 with uniformly bounded coefficients, N2 and

Nm2 are homogeneous series of order 2 with uniformly bounded coefficients, we have

‖XFm,H1m(v)‖Hs + ‖XFm,Rm(v)‖Hs .s ‖v‖3Hs ,

‖XFm,N2(v)‖Hs + ‖XFm,Nm2 (v)‖Hs .s ‖v‖2Hs ,

because for Jm(v) =∑j−l+k=m Re(bj,l,kvjvlvk) with supj−l+k=m |bj,l,k| < +∞, we have

Fm,Jm(v)

=4Im∑n≥0

∂vnFm(v)∂vnJm(v)

=∑

k1+k2=l1+l2

Im(4ak1,l1,m+l1−k1bl2,k2,m+k2−l2 + al1,l1+l2−m,l2bk1,k1+k2−m,k2)vk1vk2vl1vl2

+∑

k1+k2+l1−l2=2m

2Im(ak1,k1+l1−m,l1bk2,l2,m+l2−k2− ak2,l2,m+l2−k2

bk1,k1+l1−m,l1)vk1vk2

vl1vl2

and Fm,N2(v) = −2Im(∑j−l+k=m aj,l,kvjvlvk). Recall that sup0<ε<ε∗m

supt∈R |ϕ′(t)| < +∞ andXN4(v) =

−4iΠ(|v|2v), then we have

sup0≤σ≤1

supt∈R‖XGm(t,σ)(v)‖Hs + ‖XN4

(v)‖Hs .m,s ‖v‖2Hs(1 + ‖v‖Hs).

By using Lemma 2.4.5 and (2.3.11), for all v ∈ Hs+ such that ε‖v‖Hs ≤ γm,s, we have

sup0≤σ≤1

supt∈R‖XGm(t,σ)χmσ (v)‖Hs ≤ sup

0≤σ≤1supt∈R‖dχm−σ(χmσ (v))‖B(Hs)‖XGm(t,σ)(χ

mσ (v))‖Hs

.m,s sup0≤σ≤1

eCm,sε‖χmσ (v)‖Hs ‖χmσ (v)‖2Hs(1 + ‖χmσ (v)‖Hs)

.m,s‖v‖2Hs(1 + ‖v‖Hs)


and supt∈R ‖XN4χm1 (v)‖Hs .m,s ‖v‖2Hs(1 + ‖v‖Hs). Then we obtain the estimate of Ym.

Since w(t) = χm−1(v(t)), w =∑n≥0 wne

inx solves the following infinite dimensional Hamiltonian systemof Fourier modes :

i∂twn(t) = (1 + n2 −m2 − εϕ′(t))wn(t) + iε2 Ym(t, w(t))(n), ∀n ≥ 3m+ 1.

because for all n ≥ 3m+ 1, we haveHe2imx(w(t))(n) = XRm(w(t))(n) = XLm(w(t))(n) = 0,

XN2(w(t))(n) = XNm2(w(t))(n) = −2iwn(t).

Consequently, if ε‖w(t)‖Hs ≤ γm,s, then we have∣∣∣∂t‖w(t)− P3m(w(t))‖2Hs∣∣∣ ≤2ε2

∑n≥3m+1

(1 + n2)s| Ym(t, w(t))(n)||wn(t)|

≤2ε2‖Ym(t, w(t))‖Hs‖w(t)‖Hs.m,sε

2‖w(t)‖3Hs(1 + ‖w(t)‖Hs).

End of the proof of Theorem 2.1.8

The proof is completed by a bootstrap argument and estimate (2.4.10), obtained by the Birkhoffnormal form transformation. It suffices to prove the case u(0) ∈ C∞+ by the same density argument inthe proof of Proposition 2.1.7.

Demonstration. For all m ∈ N and s ≥ 1, we recall that ∂tv(t) = XHm,ε(t)(v(t)) and w(t) = χm−1(v(t)). InProposition 2.4.2, there exists Am,s ≥ 1 such that sup0<ε<1 ‖v(0)‖Hs ≤ Am,s. By using (2.4.4), we have

sup0<ε<1

supt∈R‖P3m(v(t))‖Hs ≤ (1 + 9m2)

s2 Im.

Set Km,s := max(6Am,s, 6(1 + 9m2)s2 Im), εm,s := min(ε∗m,

γm,s3Km,s

, 148Cm,sKm,s

) and

Tm,s := supt ≥ 0 : sup0≤τ≤t

‖v(τ)‖Hs ≤ 2Km,s.

We choose ε ∈ (0, εm,s). Since ε = ε‖v(0)‖Hs ≤ εAm,s ≤ γm,s, Lemma 2.4.5 yields that

‖v(0)− w(0)‖Hs = ‖v(0)− χm−1(v(0))‖Hs ≤ εA2m,sCm,s ≤ Am,s.

So we have ‖w(0)‖Hs ≤ Km,s3 . For all t ∈ [0, Tm,s], we have ε‖v(t)‖Hs ≤ γm,s. Hence Lemma 2.4.5 gives

the following estimate

‖w(t)‖Hs ≤‖v(t)‖Hs + ‖χm−1(v(t))− v(t)‖Hs≤‖v(t)‖Hs + Cm,sε‖v(t)‖2Hs≤2Km,s + 4Cm,sK

2m,sε

≤3Km,s.

2.5. COMPARISON TO THE NLS EQUATION 63

So we have ε sup0≤t≤Tm,s ‖w(t)‖Hs ≤ γm,s, which implies that∣∣∣ d

dt‖w(t)− P3m(w(t))‖2Hs

∣∣∣ ≤ C ′m,sε2‖w(t)‖3Hs(1 + ‖w(t)‖Hs),

in Lemma 2.4.6. Set dm,s := 1486K2

m,sC′m,s

. We can obtain the following estimate :

‖w(t)− P3m(w(t))‖2Hs≤‖w(0)‖2Hs + C ′m,s|t|ε2 sup

0≤τ≤Tm,s‖w(τ)‖3Hs(1 + sup

0≤τ≤Tm,s‖w(τ)‖Hs)

≤K2m,s

9+ 162C ′m,sK

4m,s|t|ε2

≤4K2

m,s

9,

for all 0 ≤ t ≤ min(Tm,s,dm,sε2 ). We use Lemma 2.4.5 again to obtain that

‖v(t)‖Hs ≤2‖w(t)− v(t)‖Hs + ‖w(t)− P3m(w(t))‖Hs + ‖P3m(v(t))‖Hs

≤2Cm,sε‖v(t)‖2Hs +2Km,s

3+Km,s

6≤Km,s,

for all t ∈ [0,dm,sε2 ]. In the case t < 0, we use the same procedure and we replace t by −t. Consequently,

we have

sup|t|≤ dm,s

ε2

‖u(t)− ei(·+arg um(t))‖Hs = ε sup|t|≤ dm,s

ε2

‖v(t)‖Hs ≤ Km,sε.

2.5 Comparison to the NLS equation

Although we have some similar results for the NLS equation, there are still some differences between theNLS equation and the NLS-Szego equation. We denote by u = u(t, x) =

∑n≥0 un(t)einx the solution of

the NLS equation

i∂tu+ ∂2xu = |u|2u. (2.5.1)

In Fourier modes, we have i∂tun = n2un+∑k1−k2+k3=n uk1

uk2uk3 . Fix m ∈ Z, for every n ∈ Z, we define

vn := un+mei(m2+2nm)t. Then ‖v(t)‖L2 = ‖u(t)‖L2 and we have

i∂tvn = n2vn +∑

k1−k2+k3=n

vk1vk2

vk3. (2.5.2)

If u is localized in the m-th Fourier mode, then v is localized on the zero mode. Thus the orbital stabilityof the traveling wave em can be reduced to the case m = 0. In Faou–Gauckler–Lubich [34], long timeHs-orbital stability of plane wave solutions is established by limiting the mass of the initial data to a cer-tain full measure subset of (0,+∞) for the defocusing cubic Schrodinger equation with the time interval[−ε−N , ε−N ], for all N ≥ 1 and s 1. However, this above transformation u 7→ v does not preserve the


L2 norm for the NLS-Szego equation and formula (2.5.2) fails too.

On the other hand, the approach that we use to prove Theorem 2.1.8 does not work for (2.5.1). In fact,the negative high frequency Fourier modes in the term Hm1 (v) = Re

∫S1 e−imx|v|2v can not be cancelled

by the homological equation. The energy functional of the equation of v can not be reduced as Hm0 +O(ε2)by using the same method in this paper.

The Szego filtering to (2.5.1) makes it possible to cancel all the high frequency resonances in the termRm = Fm,Hm0 + Hm1 . Then we use a bootstrap argument to deal with the equation ∂tw(t) =XHm,ε(t)χm1 (w(t)) after the Birkhoff normal form transformation.

2.6 Appendix

We give the details of the homological equation in Proposition 2.4.4 and prove (2.4.9) and Proposition2.4.3 in the case α = 0.

For all v ∈⋃n∈N Pn(C∞+ ), Hm0 (v) = 1

2‖∂xv‖2L2 + 1−m2

2 ‖v‖2L2 + 12

∫S1 Re(e2imxv2)dx and Fm(v) =∑

j−l+k=mj,k,l∈N

Re(aj,l,kvjvlvk). With the convention vn = 0, for all n < 0, we have

∂vnHm0 (v) =∂vnHm0 (v) = 1+n2−m2

2 vn + 12v2m−n,

∂vnFm(v) =∂vnFm(v) =∑

k−l+n=m

ak,l,nvkvl + 12

∑k−n+l=m

ak,n,lvkvl.

Combing (2.3.8), we have the Poisson bracket of Fm and Hm0 ,

Fm,Hm0 (v)

=− 2i∑n≥0

(∂vnFm∂vnHm0 − ∂vnHm0 ∂vnFm) (v)

=Im(∑

k−l+n=mk,l,n∈N

2(1 + n2 −m2)ak,l,nvkvlvn +∑

k−n+l=mk,l,n∈N

(1 + n2 −m2)ak,n,lvkvnvl)

+ Im(∑

k−l+n=mk,l,n∈N,n≤2m

2ak,l,nvlvkv2m−n +∑

k−n+l=mk,l,n∈N,n≤2m

ak,n,lvkvlv2m−n)

=Im(A1 +A2 +A3 +A4).

The term A4 =∑

k−n+l=mk,l,n∈N,n≤2m

ak,n,lvkvlv2m−n has only finite terms depending on low-frequency Fou-

rier modes v1, · · · , v3m. We divide A1,A2,A3 into two parts. The first part consists of low-frequencyresonances, the second part consists of high-frequency resonances.

A1 = −∑

k−l+n=mk,l,n∈N

2(1 + n2 −m2)ak,l,nvkvlvn = Alow1 +A≥2m+11 ,

2.6. APPENDIX 65

where Alow1 consists of all the resonances vjvj+k−mvk such that j, k ≤ 2m,

Alow1 =− (

bm−12 c∑j=0

2m∑k=m−j

+

2m−1∑j=bm+1

2 c

2m∑k=j+1

)2(2 + j2 + k2 − 2m2)aj,j+k−m,kvjvj+k−mvk

−m−1∑j=0

2m∑k=j+m+1

2(2 + j2 + (k +m− j)2 − 2m2)aj,k,k+m−jvjvkvk+m−j

−2m∑

k=bm+12 c

2(1 + k2 −m2)ak,2k−m,kv2kv2k−m,

and A≥2m+11 contain other resonances vjvj+k−mvk such that at least one of j, k is strictly larger than

2m.

A≥2m+11 =−

m∑j=0

∑k≥2m+1

2(2 + j2 + (k +m− j)2 − 2m2)aj,k,k+m−jvjvkvk+m−j

−2m∑

j=m+1

∑k≥2m+1

2(2 + j2 + k2 − 2m2)aj,j+k−m,kvjvj+k−mvk

−∑

k≥2m+1

∑j≥k+1

2(2 + k2 + j2 − 2m2)ak,k+j−m,jvkvk+j−mvj

−∑

k≥2m+1

2(1 + k2 −m2)ak,2k−m,kv2kv2k−m.

Then, we calculate A2 =∑

j−l+k=mj,l,k∈N

(1 + l2 −m2)aj,l,kvjvlvk = Alow2 +A≥2m+12 . Alow2 consists of all the

resonances vjvlvk such that j, k ≤ 2m or l = j + k −m ≤ 2m.

Alow2 =∑

j−l+k=mj,l,k∈N,l≤2m

(1 + l2 −m2)aj,l,kvjvlvk +∑

j−l+k=mm+1≤j≤2m−1

j+1≤k≤2m,l≥2m+1

2(1 + l2 −m2)aj,l,kvjvlvk

+

2m∑k=b 3m

2 c+1

(1 + (2k −m)2 −m2)ak,2k−m,kv2kv2k−m;

A≥2m+12 plays the same role as A≥2m+1

1 .

A≥2m+12 =

m∑j=0

∑k≥2m+1

2(1 + k2 −m2)aj,k,m+k−jvjvkvm+k−j

+

2m∑j=m+1

∑k≥2m+1

2(1 + (j + k −m)2 −m2)aj,j+k−m,kvjvj+k−mvk

+∑

k≥2m+1

∑j≥k+1

2(1 + (k + j −m)2 −m2)ak,k+j−m,nvkvk+j−mvj

+∑

k≥2m+1

(1 + (2k −m)2 −m2)ak,2k−m,kv2kv2k−m.


At last, A3 =∑

k−l+n=mk,l,n∈N,n≤2m

2ak,l,nvlvkv2m−n =∑

j−l+k=mj,k,l,n∈N,j≤2m

2al,k,2m−jvkvlvj . Using the same idea,

we have A3 = Alow3 +A≥2m+13 , with

Alow3 =

m∑j=0

2m∑k=0

2a2m−j,m+k−j,kvjvkvm+k−j +

2m∑j=m+1

2m∑k=0

2a2m−j,k,k+j−mvjvk+j−mvk;

and

A≥2m+13 =

m∑j=0

∑k≥2m+1

2a2m−j,m+k−j,kvjvkvm+k−j +

2m∑j=m+1

∑k≥2m+1

2a2m−j,k,k+j−mvjvk+j−mvk.

Recall that Hm1 (v) = Re(∫S1 e−imx|v|2v) =

∑k−l+n=m Re(vkvlvn). A similar calculus as in the case of

A1 shows that Hm1 (v) = Re(Blow +B≥2m+1) with

Blow

=(

bm−12 c∑j=0

2m∑k=m−j

+

2m−1∑j=bm+1

2 c

2m∑k=j+1

)2vjvj+k−mvk +

2m∑k=bm+1

2 c

v2kv2k−m +

m−1∑j=0

2m∑k=j+m+1

2vjvkvk+m−j ,

B≥2m+1 =∑

k≥2m+1

m∑j=0

2vjvkvk+m−j +

2m∑j=m+1

2vjvj+k−mvk + v2kv2k−m +

∑j≥k+1

2vkvk+j−mvj

.

At last we define Resonlow(v)∣∣∣α=0

= −i(Alow1 +Alow2 +Alow3 +A4) +Blow and


= −i(A≥2m+11 +A≥2m+1

2 +A≥2m+13 ) +B≥2m+1.

Then we have Fm,Hm0 (v)+Hm1 (v) = Re(Resonlow(v)∣∣∣α=0

+Reson≥2m+1(v)∣∣∣α=0

). Since Resonlow(v)∣∣∣α=0

contains only finite terms and depends only on v1, · · · , v3m, so is Rm = Re(Resonlow(v)∣∣∣α=0

). For high-

frequency resonances, we compute


=∑

k≥2m+1

m−1∑j=0

2 (−ia2m−j,m+k−j,k + (1 + 2(m− j)(k − j))iaj,k,k+m−j + 1) vjvkvk+m−j

+∑

k≥2m+1

2m∑j=m+1

2((1− 2(k −m)(j −m))iaj,j+k−m,k − ia2m−j,k,k+j−m + 1)vjvj+k−mvk

+∑

k≥2m+1

[2(iam,k,k − iam,k,k + 1)vm|vk|2 + ((1− 2(k −m)2)iak,2k−m,k + 1)v2

kv2k−m]

+∑

k≥2m+1

∑j≥k+1

2((1− 2(j −m)(k −m))iak,j+k−m,j + 1)vkvj+k−mvj .

After replacing j by 2m−j in the summ+1 ≤ j ≤ 2m, we have the equivalence between Reson≥2m+1

∣∣∣α=0

=

0 and (2.4.9) in the case α = 0.

Chapitre 3

Traveling waves of NLS–Szegoequation on the line


Resume Dans la seconde partie, afin de comprendre mieux l’action du projecteur de Szego sur l’equationde Schrodinger uni-dimensionnelle, on introduit l’equation de NLS–Szego quintique focalisante sur ladroite,

i∂tu+ ∂2xu+ Π(|u|4u) = 0, (t, x) ∈ R× R, u(0, ·) = u0.

On constate trois differences entre l’equation de NLS–Szego quintique et l’equation de SchrodingerL2-critique focalisante. La quantite de mouvement controle la norme H

12 (R) de la solution, de sorte

que l’equation de NLS–Szego quintique est globalement bien posee dans l’espace d’energie H1+(R) =

ΠR(H1(R)), pour des donnees initiales arbitraires. Ensuite, en adaptant l’argument de concentration–compacite de Lions [101, 102] precise par Gerard [45] a travers le theoreme de decomposition en profils,on peut obtenir une version faible de la stabilite des ondes progressives et classifier un certain type d’etatsfondamentaux. Enfin, le seuil inferieur de de masse pour la diffusion de l’equation de NLS–Szego quin-tique est strictement plus petit que la masse de l’etat fondamental associe, ce qui est en contraste avecle resultat de Dodson [31] pour l’equation de Schrodinger L2-critique focalisante.

Mots− clefs : Equation de Schrodinger L2-critique focalisante, Projecteur de Szego, Stabilite orbitale,Seuil de diffusion, Principe de concentration–compacite

Abstract We study the influence of Szego projector on the L2−critical one-dimensional non linearfocusing Schrodinger equation, leading to the quintic focusing NLS–Szego equation

i∂tu+ ∂2xu+ Π(|u|4u) = 0, (t, x) ∈ R× R, u(0, ·) = u0.

This equation has no Galilean invariance but the momentum P (u) = 〈−i∂xu, u〉L2 becomes the H12−norm.

Thus this equation is globally well-posed in H1+ = Π(H1(R)), for every initial datum u0. The solu-

tion L2-scatters both forward and backward in time if u0 has sufficiently small mass. By using theconcentration–compactness principle, we prove the orbital stability of some weak type of the traveling

67

68 CHAPITRE 3. TRAVELING WAVES OF NLS–SZEGO EQUATION ON THE LINE

wave : uω,c(t, x) = eiωtQ(x + ct), for some ω, c > 0, where Q is a ground state associated to Gagliardo–Nirenberg type functional

I(γ)(f) =‖∂xf‖2L2‖f‖4L2 + γ〈−i∂xf, f〉2L2‖f‖2L2

‖f‖6L6

, ∀f ∈ H1+\0,

for some γ ≥ 0. Its Euler–Lagrange equation is a non local elliptic equation. The ground states arecompletely classified in the case γ = 2, leading to the actual orbital stability for appropriate travelingwaves. As a consequence, the scattering mass threshold of the focusing quintic NLS–Szego equation isstrictly below the mass of ground state associated to the functional I(0), unlike the recent result byDodson [31] on the usual quintic focusing non linear Schrodinger equation.

Keywords L2−critical focusing Schrodinger equation, Szego projector, orbital stability, scatteringthreshold, concentration–compactness principle

3.1 Introduction

We consider the following L2−critical 1-dimensional NLS–Szego equation on the line R

i∂tu+ ∂2xu = −Π(|u|4u), (t, x) ∈ R× R, u(0, ·) = u0, (3.1.1)

where Π : L2(R)→ L2(R) denotes the Szego projector that cancels all negative Fourier modes

Πf(ξ) = 1[0,+∞)(ξ)f(ξ), ∀ξ ∈ R, ∀f ∈ L2(R). (3.1.2)

Set L2+ = Π(L2(R)), then L2

+ can be identified as the Hardy space that consists of all the holomorphicfunctions on the Poincare half-plane H+ := z ∈ C : Imz > 0 with L2 boundary

L2+ = f holomorphic on H+ : ‖f‖2L2

+:= sup

y>0

∫R|f(x+ iy)|2dx < +∞.

Since Π = id+iH2 , where H = −isign(D) is the Hilbert transform with D = −i∂x, Π : Lp(R) → Lp(R)

is a bounded operator, for every 1 < p < +∞ (Stein [129]). We define the filtered Sobolev spacesHs

+ = Hs(R)⋂L2

+, for every s ≥ 0.

The motivation to study this equation is based on the following two results. On the one hand, theL2−critical focusing non linear Schrodinger equation

i∂tU + ∂2xU = −|U |4U, (t, x) ∈ R× R, U(0, ·) = U0. (3.1.3)

marks the transition between the global existence (see Cazenave–Weissler [25, 26] for small data case) andthe blow-up phenomenon (see Glassey [61] for viriel identity method, Perelman [118] and Merle–Raphael[106] for blow-up dynamics). The instability of traveling waves U(t, x) = eiωtR(x) of equation (3.1.3) andthe classification of its ground states R associated to the Gagliardo–Nirenberg inequality

‖f‖6L6 . ‖∂xf‖2L2‖f‖4L2 , ∀f ∈ H1(R)

are established in Weinstein [145]. The ground states are unique up to scaling, phase rotation and spatialtranslation. It has been proved that the scattering mass threshold of equation (3.1.3) is equal to the mass


of ground state ‖R‖L2 in Dodson [31].

On the other hand, it has been shown in Gerard–Grellier [47, 52] that filtering the positive Fourier modescould accelerate the transition to high frequencies in a Hamiltonian evolution PDE, leading to the super-polynomial growth of Sobolev norms of solutions of the cubic Szego equation on the torus S1. So weintroduce the cubic defocusing NLS–Szego equation on the torus S1 in Sun [133] in order to understandhow applying a filter keeping only positive Fourier modes modifies the long time dynamics of the nonlinear Schrodinger equation.

We continue this topic in this paper and we put the NLS–Szego equation on the line R. The travelingwaves and the classification of ground states of the cubic Szego equation on the line R

i∂tV = Π(|V |2V ), (t, x) ∈ R× R, V (0, ·) = V0 (3.1.4)

are studied in Pocovnicu [119]. Now we consider the quintic focusing NLS–Szego equation on the line Rin order to understand how Π modifies the global wellposedness result, the scattering mass threshold andthe stability result of traveling waves of the L2−critical non linear Schrodinger equation.

Definition 3.1.1. Fix s ≥ 0, a global solution u ∈ C(R;Hs+) of equation (3.1.1) is said to Hs−scatter

forward in time if there exists u+ ∈ Hs+ such that

limt→+∞

‖eit∂2xu+ − u(t)‖Hs = 0.

A global solution u ∈ C(R;Hs+) of equation (3.1.1) is said to Hs−scatter backward in time if there exists

u− ∈ Hs+ such that

limt→−∞

‖eit∂2xu− − u(t)‖Hs = 0.

In the small mass case, equation (3.1.1) is globally well-posed in L2+ and the solution L2−scatters both

forward and backward in time. The proof is similar to Cazenave–Weissler [25, 26], by using Strichartzestimates.

Proposition 3.1.2. There exists ε0 > 0 such that if ‖u0‖L2 ≤ ε0, then there exists a unique functionu ∈ C(R;L2

+) solving equation (3.1.1) and the solution u L2−scatters both forward and backward in time.

There are three conservation laws for (3.1.1) and (3.1.3) : the mass, the momentum and the Hamiltonian

M(u) = ‖u‖2L2 , P (u) = 〈Du, u〉L2 , E(u) =‖∂xu‖2L2

2−‖u‖6L6

6,

where D = −i∂x and u ∈ H1+ for (3.1.1), u ∈ H1(R) for (3.1.3). If u ∈ C(R;H1

+) solves equation (3.1.1),

then the momentum P (u) = ‖|D| 12u‖2L2 and the mass M(u) control H12−norm of the solution, leading

to the global wellposedness of equation (3.1.1). By using Gagliardo–Nirenberg’s interpolation inequality

‖u‖L2m+2 .m ‖|D|12u‖

mm+1

L2 ‖u‖1

m+1

L2 , ∀m ≥ 0, ∀u ∈ H 12 (R), (3.1.5)

one can solve the problem of global wellposedness for all L2−supercritical NLS–Szego equations forall large initial data u0 ∈ H1

+. On the other hand, when U0 ∈ H1(R) such that E(U0) < 0 andU0 ∈ L2(R, x2dx), then the solution U of equation (3.1.3) associated to initial datum U0 blows up


in finite time by the viriel identity (Glassey [61], Cazenave [22, 23], Ogawa–Tsutsumi [115]). We refer toPerelman [118] and Merle–Raphael [106] to see the asymptotic representation of the blow-up dynamicsof equation (3.1.3) in details. Now we state the first result of this paper.

Theorem 3.1.3. For all m ≥ 0, λ = ±1 and u0 ∈ H1+, there exists a unique function u ∈ C(R;H1

+)solving the following L2−supercritical NLS-Szego equation

i∂tu+ ∂2xu = λΠ(|u|2mu), u(0, x) = u0(x), (t, x) ∈ R× R. (3.1.6)

The local well-posedness of equation (3.1.6) is established by the fixed-point theorem and Sobolev esti-mates. In the focusing case λ = −1, since the mass, the momentum and the Hamiltonian are conservedunder the flow of equation (3.1.6), inequality (3.1.5) yields that

supt∈R‖∂xu(t)‖2L2 .m ‖∂xu0‖2L2 + ‖|D| 12u0‖2mL2 ‖u0‖2L2 .

Thus every solution of (3.1.6) is global. Besides the global well-posedness problem, there are still otherdifferences between equation (3.1.1) and equation (3.1.3).

We consider the traveling waves of equation (3.1.1), u(t, x) = eiωtQ(x + ct), for some ω, c ∈ R. u solves(3.1.1) if and only if Q solves the following non local elliptic equation

∂2xQ+ Π(|Q|4Q) = ωQ+ cDQ. (3.1.7)

It suffices to identity equation (3.1.7) to the Euler–Lagrange equation of some functional associated toGagliardo–Nirenberg inequality (3.1.5) in order to obtain the traveling waves. For all m ≥ 2 and γ ≥ 0,we define

I(γ)m (f) :=

‖∂xf‖mL2‖f‖m+2L2 + γ‖|D| 12 f‖2mL2 ‖f‖2L2

‖f‖2m+2L2m+2

, ∀f ∈ H1(R)\0. (3.1.8)

This functional is invariant by space-translation, phase-translation, interior and exterior scaling.

I(γ)m (f) = I(γ)

m (fλ,µ,y,θ), where fλ,µ,y,θ(x) = λeiθf(µx+ y), ∀x, y, θ ∈ R, ∀λ, µ > 0.

We denote its greatest lower bound by J(γ)m = inff∈H1

+\0 I(γ)m (f) and all its minimizers by

G(γ)m = f ∈ H1

+\0 : I(γ)m (f) = J (γ)

m =⋃a,b>0

G(γ)m (a, b), (3.1.9)

where G(γ)m (a, b) = f ∈ G(γ)

m : ‖f‖L2 = a, ‖f‖L2m+2 = b. Then we have

G(γ)m (a, b) = λf(µ·) ∈ H1

+\0 : f ∈ G(γ)m (1, 1), λ = a−

1m b

m+1m and µ = b

2m+2m a−

2m+2m .

A concentration-compactness argument shows that the functional I(γ)m attains its minimum in H1

+\0.We shall follow the idea of profile decomposition of minimizing sequence introduced in Gerard [45] andBahouri–Gerard [6], which is a refinement of the concentration-compactness principle (see Lions [101, 102]and Cazenave–Lions [24] for orbital stability of traveling waves of L2−subcritical NLS equation), in orderto establish the existence of minimizers.


Theorem 3.1.4. For all m ≥ 2 and γ ≥ 0, if (fn)n∈N ∈ H1+ is a minimizing sequence for I

(γ)m such

that ‖fn‖L2 = ‖fn‖L2m+2 = 1 and limn→+∞ I(γ)m (fn) = J

(γ)m , then there exists a profile U ∈ G(γ)

m (1, 1), astrictly increasing function ψ : N→ N and a real-valued sequence (xn)n∈N such that

limn→+∞

‖fψ(n) − U(· − xn)‖H1 = 0. (3.1.10)

Remark 3.1.5. If H1+ is replaced by H1(R), then there exists U ∈ H1(R)\0 such that ‖U‖L2 =

‖U‖L2m+2 = 1, I(γ)m (U) = minf∈H1(R)\0 I

(γ)m (f) and the limit (3.1.10) also holds.

Thus G(γ)m (a, b) is not empty, for all a, b > 0. We refer to Gerard–Lenzmann–Pocovnicu–Raphael [56] to

see the asymptotic dynamics and long time behavior in two different regimes of the two-soliton solutionsof the cubic focusing half-wave equation on R. Similarly, one obtains the existence of ground states oftraveling waves for L2−critical Schrodinger equation on Rd (see Hmidi–Keraani [73]), for cubic Szegoequation (see Pocovnicu [119]) etc. Furthermore, the profile decomposition can be used to prove glo-bal regularity, scattering and a priori bounds for the energy critical Maxwell–Klein–Gordon equation inKrieger–Luhrmann [92].

Given m ≥ 2 and γ ≥ 0, let f ∈ H1+\0 be a minimizer of I

(γ)m , then d

dε

∣∣∣ε=0

log I(γ)m (f + εh) = 0, for all

h ∈ H1+. f solves the following Euler–Lagrange equation

m‖f‖m+2L2 ‖∂xf‖m−2

L2 ∂2xf + 2(m+ 1)J (γ)

m Π(|f |2mf)

=((m+ 2)‖f‖mL2‖∂xf‖mL2 + 2γ‖|D| 12 f‖2mL2 )f + 2γm‖f‖2L2‖|D|12 f‖2m−2

L2 Df.(3.1.11)

From now on, we restrict ourselves to the case m = 2. We want to identify equation (3.1.7) to equation

‖Q‖4L2

3J(γ)2

∂2xQ+ Π(|Q|4Q) =

2‖Q‖2L2‖∂xQ‖2L2 + γ‖|D| 12Q‖4L2

3J(γ)2

Q+2γ‖Q‖2L2‖|D|

12Q‖2L2

3J(γ)2

DQ.

A minimizer Q(γ) ∈ G(γ)2 is called the ground state of functional I

(γ)2 , if ‖Q(γ)‖4L2 = 3J

(γ)2 . If u(t, x) =

eiωtQ(γ)(x+ ct) solves equation (3.1.1) and Q(γ) ∈ G(γ)2 (

4

√3J

(γ)2 , b) for some b, ω > 0 and c, γ ≥ 0, then

c = 0 if and only if γ = 0.

If γ = 0, then we have ‖∂xQ(γ)‖2L2 = ω2

√3J

(0)2 and ‖Q(γ)‖6L6 = 3ω

2

√3J

(0)2 .

If γ > 0, then we have

‖∂xQ(γ)‖2L2 =

√3J

(γ)2

8γ(4γω − c2), ‖|D| 12Q(γ)‖2L2 =

√3J

(γ)2 c

2γ, ‖Q(γ)‖6L6 = 3

√3J

(γ)2 (

ω

2+c2

8γ).

Furthermore, the interpolation inequality ‖|D| 12Q(γ)‖2L2 ≤ ‖Q(γ)‖L2‖∂xQ(γ)‖L2 yields that c2 ≤ 4γ2ωγ+2 .

Even though we do not know how to classify all the ground states of I(γ)2 , for general γ ≥ 0, the

H1−orbital stability with scaling can be established by using theorem 3.1.4 and the conservation lawP (u) = 〈−i∂xu, u〉L2 = ‖|D| 12u‖2L2 .


Theorem 3.1.6. For every ε, b > 0 and γ ≥ 0, there exists δ = δ(b, ε, γ) > 0 such that if

inff∈G(γ)

2 (4√

3J(γ)2 ,b)

‖u0 − f‖H1 < δ,

then we have supt∈R infΨ∈

⋃C(γ)−1≤θ≤C(γ) G

(γ)2 (

4√

3J(γ)2 ,θb)

‖u(t)−Ψ‖H1 < ε, where u is the solution of equa-

tion (3.1.1) with initial datum u(0) = u0 and C(γ) :=

(inff∈H1

+\0‖|D|

13 f‖L2

‖f‖L6

)−16

√J

(γ)2

1+γ .

The problem of uniqueness of ground states of a non-local elliptic equation is difficult to solve (SeeFrank–Lenzmann [40] and Frank–Lenzmann–Silvestre [41] for the fractional Laplacians in R and alsoLenzmann–Sok [99] for a strict rearrangement principle in Fourier space). We refer to subsection 3.4.2

to discuss the classification of ground states of I(γ)2 . For general γ ≥ 0, we only have the H1−orbital

stability with scaling. Since we do not know the uniqueness of ground states of I(γ)2 , the L6−norm of the

ground state Ψ that approaches u(t) is unknown. We can only give a range‖f‖L6

C(γ) ≤ ‖Ψ‖L6 ≤ C(γ)‖f‖L6 ,

where f denotes the ground state that approaches the initial datum u0.

On the other hand, all the ground states can be completely classified in the case γ = 2, by using

Cauchy–Schwarz inequality. The ground state of I(2)2 is unique up to scaling, phase rotation and spatial

translation.

Proposition 3.1.7. In the case γ = m = 2, we have

J(2)2 = min

f∈H1+\0

I(2)2 (f) =

8π2

3, G

(2)2 = x 7→ λeiθ

µx+ y + i∈ H1

+ : ∀λ, µ > 0 ∀θ, y ∈ R. (3.1.12)

If u(t, x) = eiωtQ(x+ ct) is a traveling wave of (3.1.1) and Q ∈ G(2)2 , then we have

3c2 = 8ω, ‖Q‖4L2 = 8π2, ‖Q‖6L6 =3πc2√

2, ‖|D| 12Q‖2L2 =

πc√2, ‖∂xQ‖2L2 =

πc2

2√

2,

thanks to the classification of G(2)2 . In this case, the traveling wave uc(t, x) = e

3c2it8 Qc(x + ct) is

H1−orbitally stable, for every c > 0, where Qc ∈ G(2)2 (

4√

8π2, 6

√3πc2√

2).

Theorem 3.1.8. For every ε, c > 0, there exists δε,c > 0 such that if inff∈G(2)

2 (4√

8π2, 6√

3πc2√2

)‖u0−f‖H1 <

δ, then we have supt∈R inff∈G(2)

2 (4√

8π2, 6√

3πc2√2

)‖u(t)−f‖H1 < ε, where u solves equation (3.1.1) with initial

datum u(0) = u0.

Remark 3.1.9. In the case γ = 2, since all the ground states are completely classified and unique up toscaling, phase rotation and spatial translation by proposition 3.1.7, we obtain the H

12−norm of the ground

state that approaches u(t) by the conservation law P (u) = 〈Du, u〉L2 and formula (3.1.10). Then we knowalso the L6−norm of the very ground state and we have the actual orbital stability without scaling, whichis a refinement of theorem 3.1.6.

Similar results on the classification of ground states of traveling waves by using Cauchy–Schwarz inequa-lity can be found in Foschi [39] for linear Schrodinger equation and linear wave equation on Rd, for d ≥ 1,


Gerard–Grellier [47, 52] for the cubic Szego equation, etc.

In the case γ = 0, we have

I(0)2 (f) :=

‖∂xf‖2L2‖f‖4L2

‖f‖6L6

, ∀f ∈ H1(R)\0.

All of the ground states in H1(R) of I(0)2 have been completely classified in Weinstein [145]. We know

minf∈H1(R)\0 I(0)2 (f) = π2

4 and there exists a unique real-valued, positive, spherically symmetric and

decreasing function R(x) =4√3√

cosh(2x)such that

(I(0)2 )−1(

π2

4) = λeiθR(µ · −y) : λ, µ > 0 θ, y ∈ R.

The traveling wave U(t, x) = eiωtR(x) is an unstable solution of the L2−critical focusing Schodinger

equation (3.1.3) in the following sense : there exists a sequence u(n)0 = (1 + 1

n )R ⊂ H1(R) such that

u(n)0 → R, as n→ +∞, but the corresponding maximal solution u(n) blows up in finite time. We denote

by Q(0) ∈ G(0)2 (

4

√3J

(0)2 , ‖Q(0)‖L6) one of the ground states of I

(0)2 in Hardy space H1

+. Since R /∈ H1+ and

Q+ : x 7→ 1x+i ∈ H

1+, we have π2

4 = I(0)2 (R) < J

(0)2 = I

(0)2 (Q(0)) ≤ I(0)

2 (Q+) = 4π2

3 .

In proposition 3.1.2, we know that the solution of equation (3.1.1) or equation (3.1.3) scatters if theinitial datum has sufficiently small mass. Furthermore, Dodson [31] has proved that if ‖U0‖ < ‖R‖L2 ,then equation (3.1.3) is globally well-posed and the solution L2−scatters both forward and backwardin time. Together with the instability result of traveling waves by Weinstein [145], the scattering mass

threshold of equation (3.1.3) is equal to the mass of ground state R ∈ H1(R) of I(0)2 .

On the other hand, adding the Szego projector in front of the non linear term of the L2−critical focusingSchrodinger equation makes the scattering mass threshold strictly smaller than the mass of ground state

Q(0) ∈ H1+ of I

(0)2 , thanks to the orbital stability theorem 3.1.6. We define E ⊂ R∗+ to be all ε > 0 such

that if ‖u0‖L2 < ε, the corresponding solution of (3.1.1) L2−scatters both forward and backward in time.If an H1−solution of equation (3.1.1) L2−scatters, then it also H1−scatters and its Lr−norm decays,with 2 < r ≤ +∞ (See proposition 3.4.1 and 3.4.2). Thus traveling waves do not L2−scatter and neitherdo the solutions that approach the traveling waves.

Corollary 3.1.10. sup E < ‖Q(0)‖L2 =4

√3J

(0)2 . Precisely, there exists u ∈ C(R;H1

+) solving equation

(3.1.1) such that ‖u(0)‖L2 < ‖Q(0)‖L2 and u does not L2−scatter neither forward nor backward in time.

Remark 3.1.11. The mass of ground state of I(2)2 is strictly larger than the mass of ground state of I

(0)2 .

‖Q(2)‖4L2 = 8π2 > 4π2 ≥ ‖Q(0)‖4L2 .

E(Q(2)) = − πc2

4√

2< 0 = E(Q(0)).

The value of scattering mass threshold of equation (3.1.1) remains as an open problem. The Galileaninvariance is necessary to use interaction Morawetz estimate (see Dodson [31], Killip–Visan [88], Killip–Visan–Zhang [89]). However, the NLS–Szego equation has no Galilean invariance. The next step to study


is to compare the value of the scattering mass threshold of equation (3.1.3) and the scattering massthreshold of the original L2−critical NLS equation.

This paper is organized as follows. In section 3.2, we recall profile decomposition theorem prove theorem3.1.4. In section 3.3, the orbital stability of traveling wave u(t, x) = eiωtQ(x+ ct) is proved at first. Thenwe give the details of the special case γ = 2. In the first appendix, we prove the persistence of regularityof scattering. In the second appendix, we discuss the open problem of uniqueness of ground states of the

functional I(γ)2 , for general γ ≥ 0. It suffices to study the uniqueness of ground states modulo positive

Fourier transformation.

3.2 Profile decomposition

At first, we recall the result of profile decomposition theorem in Gerard [45], which is a refinement ofconcentration-compactness argument of Sobolev embedding introduced in Lions [101, 102]. Every boundedsequence in H1

+ has a subsequence which can be written as a nearly orthogonal sum of a superposition ofsequence of shifted profiles and a sequence tending to zero in Lp(R), for every 2 < p ≤ +∞. It will be usedto find the minimizers of some functionals in calculus of variation and establish the orbital stability ofsome traveling waves. We shall use the version of Hmidi–Keraani [74] and construct the profiles withoutscaling.

Theorem 3.2.1 (Gerard 45, Hmidi–Keraani 74). If (fn)n∈N+is a bounded sequence in H1

+, then there

exists a subsequence of (fn)n∈N+, denoted by (fφ(n))n∈N+

, a sequence of profiles (U (j))j∈N+⊂ H1

+ and

a double-indexed sequence (x(j)n )n,j∈N+

⊂ R such that if j 6= k, then |x(j)n − x(k)

n | → +∞ and for everyl ∈ N+, we have

fφ(n)(x) =

l∑j=1

U (j)(x− x(j)n ) + r(l)

n (x), ∀x ∈ R, (3.2.1)

where lim supn→+∞ ‖r(l)n ‖Lp → 0, as l→ +∞, ∀2 < p ≤ +∞. For every l ∈ N+ and s ∈ [0, 1], we have

∣∣∣‖|D|sfφ(n)‖2L2 −l∑

j=1

‖|D|sU (j)‖2L2 − ‖|D|sr(l)n ‖2L2

∣∣∣→ 0, as n→ +∞. (3.2.2)

Remark 3.2.2. According to Gerard [45] and Hmidi–Keraani [74], we may construct the profiles (U (j))j∈N+⊂

H1+ such that if U (l) = 0 for some l ∈ N+, then U (j) = 0, for every j ≥ l.

Then we shall use this theorem to establish the existence of minimizers in H1+ of I

(γ)m , for every m ≥ 2

and γ ≥ 0.

Proof of theorem 3.1.4. Since supn∈N ‖fn‖H1 < +∞, theorem 3.2.1 gives a subsequence of (fn)n∈N+, de-

noted by (fφ(n))n∈N+, a sequence of profiles (U (j))j∈N+

⊂ H1+ and a double-indexed sequence (x

(j)n )n,j∈N+

⊂R such that |x(j)

n −x(k)n | → +∞, if j 6= k, (3.2.1) and (3.2.2) hold and liml→+∞ lim supn→+∞ ‖r

(l)n ‖L2m+2 →

0.

3.2. PROFILE DECOMPOSITION 75

For all 0 ≤ s ≤ 1, l ∈ N+ and δ > 0, there exists N = N(s, l, δ) ∈ N+ such that

‖|D|sfφ(n)‖2L2 ≥l∑

j=1

‖|D|sU (j)‖2L2 − δ, ∀n > N.

Taking n → +∞, δ → 0 and l → +∞, we have∑lj=1 ‖|D|sU (j)‖2L2 ≤ lim infn→+∞ ‖|D|sfφ(n)‖2L2 , for

every 0 ≤ s ≤ 1. Then, there exists a subsequence of (fφ(n))n∈N, denoted by (fφφ(n))n∈N such that both

sequences (‖|D| 12 fφφ(n)‖L2)n∈N and (‖∂xfφφ(n)‖L2)n∈N converge and we have∑+∞j=1 ‖∂xU (j)‖2L2 ≤ limn→+∞ ‖∂xfφφ(n)‖2L2 ,∑+∞j=1 ‖|D|

12U (j)‖2L2 ≤ limn→+∞ ‖|D|

12 fφφ(n)‖2L2 ,∑+∞

j=1 ‖U (j)‖2L2 ≤ limn→+∞ ‖fφ(n)‖2L2 = 1.

(3.2.3)

Thus 0 ≤ ‖U (j)‖L2 ≤ 1, for every j ∈ N+. We set ψ = φ φ : N→ N. Since m ≥ 2 and

J (γ)m = lim

n→+∞I(γ)m (fψ(n)) = lim

n→+∞‖∂xfψ(n)‖mL2 + γ lim

n→+∞‖|D| 12 fψ(n)‖2mL2 ,

estimates (3.2.3) yields that

J (γ)m ≥(

+∞∑j=1

‖∂xU (j)‖2L2)m2 + γ(

+∞∑j=1

‖|D| 12U (j)‖2L2)m

≥+∞∑j=1

(‖∂xU (j)‖mL2‖U (j)‖m+2L2 + γ‖|D| 12U (j)‖2mL2 ‖U (j)‖2L2)

≥J (γ)m

+∞∑j=1

‖U (j)‖2m+2L2m+2 .

(3.2.4)

We claim that+∞∑j=1

‖U (j)‖2m+2L2m+2 = 1. (3.2.5)

In fact, each profile U (j) ∈ L∞(R). More precisely, we have∑∞j=1 ‖U (j)‖2L∞ .m 1 by Sobolev embedding

H1 → L∞ and estimate (3.2.3). One can easily check that

‖l∑

j=1

U (j)‖2m+2L2m+2 =

l∑j=1

‖U (j)‖2m+2L2m+2 +R(l)

n , ∀l ∈ N+,

where |R(l)n | .m,l

∑1≤j<k≤l

∫R |U

(j)(x−x(j)n )||U (k)(x−x(k)

n )|dx. Since U (j) ∈ L2+, limn→+∞ |x(j)

n −x(k)n | =

+∞, for all 1 ≤ j < k ≤ l, we have |U (j)(· − x(j)n + x

(k)n )| 0 in L2

+, as n→ +∞. So limn→+∞R(l)n = 0.

The profile decomposition theorem (3.2.1) implies that

‖r(l)n ‖L2m+2 ≥

∣∣∣‖fψ(n)‖L2m+2 − ‖l∑

j=1

U (j)‖L2m+2

∣∣∣ =∣∣∣1− (

l∑j=1

‖U (j)‖2m+2L2m+2 +R(l)

n )1

2m+2

∣∣∣.


Taking n → +∞, we have∣∣∣1 − (

∑lj=1 ‖U (j)‖2m+2

L2m+2)1

2m+2

∣∣∣ ≤ lim supn→+∞ ‖r(l)n ‖L2m+2 → 0, as l → +∞

and we obtain (3.2.5).

Combining (3.2.4), we have J(γ)m ≥ J

(γ)m∑+∞j=1 ‖U (j)‖2m+2

L2m+2 = J(γ)m . All inequalities in (3.2.3) and (3.2.4)

are actually equalities. In particular, we have

‖∂xU (1)‖mL2‖U (1)‖2m+2L2 = ‖∂xU (1)‖mL2 , I(γ)

m (U (1))‖U (1)‖2m+2L2m+2 = J (γ)

m ‖U (1)‖2m+2L2m+2 .

If U (1) ≡ 0 a.e. in R, then so is U (j), for every j ≥ 2 by the construction of profiles in remark 3.2.2.It contradicts formula (3.2.5). Thus Gagliardo–Nirenberg inequality yields that ‖∂xU (1)‖L2 > 0 and‖U (1)‖L2 = 1. Since

∑+∞j=1 ‖U (j)‖2L2 ≤ 1, we have U (j) = 0 a.e., for every j ≥ 2. Formula (3.2.5) implies

that ‖U (1)‖L2m+2 = 1 and I(γ)m (U (1)) = J

(γ)m .

Estimate (3.2.3) is also equality, so ‖∂xU (1)‖2L2 = limn→+∞ ‖∂xfψ(n)‖2L2 . We set

ε(l)n,s := ‖|D|sfψ(n)‖2L2 −l∑

j=1

‖|D|sV (j)‖2L2 − ‖|D|sr(l)

φ(n)‖2L2 , ∀n ∈ N, l ∈ N+, 0 ≤ s ≤ 1.

Thus limn→+∞ ε(l)n,s = 0, for all l ∈ N+, 0 ≤ s ≤ 1. Then

‖fψ(n) − U (1)(· − y(1)n )‖2H1 = ‖r(1)

φ(n)‖2H1 = ‖∂xfψ(n)‖2L2 − ‖∂xU (1)‖2L2 − ε(1)

n,0 − ε(1)n,1 → 0,

as n→ +∞.

3.3 Orbital stability

At first, we prove the H1−orbital stability with scaling for traveling wave u(t, x) = eiωtQ(x+ ct) of theL2−critical focusing NLS-Szego equation,

i∂tu+ ∂2xu = −Π(|u|4u), (t, x) ∈ R× R, u(0, ·) = u0,

with ω, c > 0, γ ≥ 0 and Q ∈ G(γ)2 (

4

√3J

(γ)2 , ‖Q‖L6). Then we focus on the case γ = 2 by classifying all

ground states and we refine theorem 3.1.6 as theorem 3.1.8.

3.3.1 Proof of theorem 3.1.6

Demonstration. Fix b > 0 and γ ≥ 0, for every n ∈ N, we choose un0 ∈ H1+ and ϕn ∈ G(γ)

2 (4

√3J

(γ)2 , b)

such that ‖un0 − ϕn‖H1 → 0, as n → +∞. Let un solve (3.1.1) with initial datum un(0) = un0 . We shallprove that

infΨ∈

⋃C(γ)−1≤θ≤C(γ) G

(γ)2 (

4√

3J(γ)2 ,θb)

‖un(tn)−Ψ‖H1 → 0, as n→ +∞, (3.3.1)

3.3. ORBITAL STABILITY 77

up to a subsequence, for every temporal sequence (tn)n∈N ⊂ R. We use the three conservation laws

E(u) =‖∂xu‖2L2

2−‖u‖6L6

6, P (u) = 〈Du, u〉L2 = ‖|D| 12u‖2L2 , M(u) = ‖u‖2L2 .

to construct another conservation law

Kγ(u) := E(u) +γP (u)2

M(u)=‖∂xu‖2L2

2−‖u‖6L6

6+γ‖|D| 12u‖4L2

2‖u‖2L2

=‖u‖6L6

6‖u‖4L2

(3I(γ)2 (u)− ‖u‖4L2). (3.3.2)

Since I(γ)2 (ϕn) = J

(γ)2 and ‖un0 − ϕn‖H1 → 0, as n → +∞, we have limn→+∞ ‖un0‖4L2 = 3J

(γ)2 and

limn→+∞ I(γ)2 (un0 ) = J

(γ)2 . Thus

Kγ(un(tn)) = Kγ(un0 )→ 0, as n→ +∞. (3.3.3)

In order to construct a minimizing sequence of I(γ)2 , we need to prove the following inequalities

C(γ)−1b ≤ lim infn→+∞

‖un(tn)‖L6 ≤ lim supn→+∞

‖un(tn)‖L6 ≤ C(γ)b. (3.3.4)

In fact, we denote by C61 := 1+γ

J(γ)2

C(γ)6, then Sobolev embedding ‖f‖L6 ≤ C1‖|D|13 f‖L2 yields that

‖∂xun(tn)‖2L2

2+γ‖|D| 12un(tn)‖4L2

2‖un(tn)‖2L2

≥(1 + γ)‖|D| 12un0‖4L2

2‖un0‖2L2

≥(1 + γ)‖|D| 13un0‖6L2

2‖un0‖4L2

≥(1 + γ)‖un0‖6L6

2C61‖un0‖4L2

.

Thus ‖un(tn)‖L6 ≥ 6

√3(1+γ)‖un0 ‖6L6

C61‖un0 ‖4L2

− 6Kγ(un(tn))→ bC(γ) , as n→ +∞. Similarly, we have

‖un(tn)‖L6 ≤ C1‖|D|13un(tn)‖L2 ≤ C1‖|D|

12un(tn)‖

23

L2‖un(tn)‖13

L2 = C1‖|D|12un0‖

23

L2‖un0‖13

L2 ,

and ‖un(tn)‖L6 ≤ C1‖∂xun0‖13

L2‖un0‖23

L2 . Thus we have ‖un(tn)‖L6 ≤ C1‖un0‖L66

√I

(γ)2 (un0 )1+γ → C(γ)b, as

n→ +∞. Thus estimates (3.3.4) are proved and we have limn→+∞ I(γ)2 (un(tn)) = J

(γ)2 .

Rescaling vn(x) := λnun(tn, µnx) such that ‖vn‖L2 = ‖vn‖L6 = 1, with λn, µn > 0. Then

√µnλ

−1n = ‖un0‖L2 , 6

√µnλ

−1n = ‖un(tn)‖L6 , lim

n→+∞I

(γ)2 (vn) = J

(γ)2 (3.3.5)

‖∂xvn‖2L2 = I(vn) = I(un(tn)) =‖∂xun(tn)‖2L2

‖un(tn)‖6L6

‖un0‖4L2 → J, as n→ +∞.

Theorem 3.1.4 yields that there exists a profile U ∈ G(γ)2 (1, 1) and a sequence of real numbers (xn)n∈N

such that ‖vn−U(·−xn)‖H1 → 0, as n→ +∞ up to a subsequence, still denoted by (vn)n∈N+. Moreover,

we assume that ‖un(tn)‖L6 → θb, as n→ +∞ in the same subsequence. Then (3.3.4) yields that C−1 ≤θ ≤ C. We denote by λ∞ = limn→+∞ λn and µ∞ = limn→+∞ µn. Then we have

λ∞ > 0, µ∞ > 0,√µ∞λ

−1∞ =

4

√3J

(γ)2 and 6

√µ∞λ

−1∞ = θb. (3.3.6)


Then

limn→∞

‖λnun(tn, µn·)− U(· − xn)‖H1 = 0 ⇐⇒ limn→∞

‖un(tn)− 1

λnU(· − µnxn

µn)‖H1 = 0. (3.3.7)

Since U ∈ H1+, we have limn→∞ ‖ 1

λnU( ·µn )− 1

λ∞U( ·µ∞ )‖H1 = 0. Together with (3.3.7), we have

‖un(tn)−Ψ(· − µnxn)‖H1 → 0, as n→ +∞, (3.3.8)

where Ψ(x) := 1λ∞

U( xµ∞

). Since ‖U‖L2 = ‖U‖L6 = 1, we have Ψ ∈ G(γ)2 (

4

√3J

(γ)2 , θb) by (3.3.6), leading

to (3.3.1) up to a subsequence.

3.3.2 The special case γ = 2

We prove proposition 3.1.7 in order to classify all the ground states of the functional

I(2)2 =

‖∂xf‖2L2‖f‖4L2 + 2‖|D| 12 f‖4L2‖f‖2L2

‖f‖6L6

, ∀f ∈ H1+\0.

as G(2)2 = x 7→ λeiθ

µx+y+i ∈ H1+ : ∀λ, µ > 0 ∀θ, y ∈ R. The idea of using Cauchy–Schwarz inequality to

classify ground states follows from Foschi [39] for linear Schrodinger equation and linear wave equationon Rd, for d ≥ 1, Gerard–Grellier [47, 52] for the cubic Szego equation on the torus S1, Pocovnicu [119]for the cubic Szego equation on the line R.

Proof of proposition 3.1.7. Plancherel formula gives that ‖f‖6L6 = 132π5

∫ξ>0|(f ∗ f ∗ f)(ξ)|2dξ. By using

Cauchy–Schwarz inequality, for every ξ > 0, we have

|(f ∗ f ∗ f)(ξ)|2 =∣∣∣ ∫η1>0,η2>0,η1+η2≤ξ

f(η1)f(η2)f(ξ − η1 − η2)dη1dη2

∣∣∣2≤ξ

2

2|∫η1>0,η2>0,η1+η2≤ξ

∣∣∣f(η1)f(η2)f(ξ − η1 − η2)∣∣∣2dη1dη2.

Thus we have

‖f‖6L6 ≤1

64π5

∫η1>0,η2>0,η3>0

(η1 + η2 + η3)2∣∣∣f(η1)f(η2)f(η3)

∣∣∣2dη1dη2dη3

=3

8π2(‖∂xf‖2L2‖f‖4L2 + 2‖|D| 12 f‖4L2‖f‖2L2).

If I(2)2 (f) = 8π2

3 , then f(η1)f(η2)f(η3) = f(0)2f(η1 + η2 + η3), for all η1, η2, η3 > 0. Since f ∈ H1+\0,

we have f(0) 6= 0. Thus

f(η1)f(η2) = f(η1 + η2)f(0), ∀η1, η2 ≥ 0.

This is true if and only if f(η) = e−ipη f(0), for some p ∈ C such that Imp < 0. Thus we have f(x) = Ax−p ,

for some A ∈ C. We conclude by I(2)2 (f) = I

(2)2 (fλ,µ,θ,y), with fλ,µ,θ,y(x) = λeiθf(µx+ y).

3.4. APPENDICES 79

For every c > 0, we prove the orbital stability of traveling wave u(t, x) = e3c2it

8 Q∗c(x + ct) of equation

(3.1.1), with Q∗c(x) = 24√

2c2

cx+2i . Proposition 3.1.7 yields that G(2)2 (

4√

8π2, 6

√3πc2√

2) = x 7→ 2

4√2c2eiθ

cx+2i+y ∈ H1+ :

∀θ, y ∈ R. If Qc ∈ G(2)2 (

4√

8π2, 6

√3πc2√

2), then we have

‖Qc‖4L2 = 8π2, ‖∂xQc‖2L2 =πc2

2√

2, ‖|D| 12Qc‖2L2 =

πc√2, ‖Qc‖6L6 =

3πc2√2

(3.3.9)

Proof of theorem 3.1.8. Fix c > 0, for every n ∈ N, we choose un0 ∈ H1+ and Qnc ∈ G

(2)2 (

4√

8π2, 6

√3πc2√

2)

such that ‖un0 −Qnc ‖H1 → 0, as n → +∞. Let un solve (3.1.1) with initial datum un(0) = un0 . We shallprove that

infΨ∈G(2)

2 (4√

8π2, 6√

3πc2√2

)

‖un(tn)−Ψ‖H1 → 0, as n→ +∞, (3.3.10)

up to a subsequence, for every temporal sequence (tn)n∈N ⊂ R. We use the same procedure as the proof of

theorem 3.1.6 to obtain that (un(tn))n∈N is a minimizing sequence of I(2)2 . We set vn(x) := λnu

n(tn, µnx)

such that ‖vn‖L2 = ‖vn‖L6 = 1, with λn, µn > 0 and limn→+∞ I(2)2 (vn) = limn→+∞ I

(2)2 (un(tn)) = J

(2)2 .

I(vn) = I(un(tn))→ 1, as n→ +∞.

Theorem 3.1.4 yields that there exists a profile V ∈ G(2)2 (1, 1) and a sequence of real numbers (yn)n∈N+

such that ‖vn−V (·−yn)‖H1 → 0, as n→ +∞ up to a subsequence, still denoted by (vn)n∈N+. Similarly,

we have (3.3.7) for V .

limn→∞

‖λnun(tn, µn·)− V (· − yn)‖H1 = 0 ⇐⇒ limn→∞

‖un(tn)− 1

λnV (· − µnynµn

)‖H1 = 0. (3.3.11)

Since all the ground states are completely classified by proposition 3.1.7, we obtain the values of λ∞ and

µ∞ from (3.3.9) and the conservation law P (u) = ‖|D| 12u‖2L2 . Since λ2n‖|D|

12

1λnV ( ·−µnynµn

)‖2L2 =√

23π,

‖|D| 12un(tn)‖2L2 = ‖|D| 12un0‖2L2 → πc√2

and∣∣∣‖|D| 12un(tn)‖L2 − ‖|D| 12 1

λnV (· − µnynµn

)‖L2

∣∣∣→ 0, as n→ +∞,

we have λ2∞ := limn→+∞ λ2

n = 2√3c

and µ∞ := limn→+∞ µn =√

8π2 limn→+∞ λ2n = 4

√2π√3c

. Together with

(3.3.11), we have‖un(tn)− Ψ(· − µnyn)‖H1 → 0, as n→ +∞,

where Ψ(x) := 1λ∞

V ( xµ∞

). We have Ψ ∈ G(2)2 (

4√

8π2, 6

√3πc2√

2), leading to (3.3.10) up to a subsequence.

3.4 Appendices

In the first appendix, we prove that if a H1−solution of equation (3.1.1) L2−scatters, then it also

H1−scatters. Then we discuss the problem of uniqueness of ground states of the functional I(γ)2 , for

general γ ≥ 0.


3.4.1 Persistence of regularity for scattering

The persistence of regularity for scattering can be established by Strichartz estimates and a bootstrapargument.

Proposition 3.4.1. If u0 ∈ H1+ and there exists u+ ∈ L2

+ such that limt→+∞ ‖u(t) − eit∂2xu+‖L2 = 0,

where u is the unique solution of (3.1.1), then we have u+ = u0 + i∫ +∞

0e−iτ∂

2xΠ(|u(τ)|4u(τ))dτ ∈ H1

+

and limt→+∞ ‖u(t)− eit∂2xu+‖H1 = 0.

Demonstration. We claim that u ∈ L6(0,+∞;L6(R)). In fact (6, 6) is 1-admissible. Set v(t, x) :=

eit∂2xu+(x) and r(t, x) = u(t, x)− v(t, x), then we have v ∈ L6(Rt × Rx) by Strichartz inequality and

i∂tr + ∂2xr = −Π(|u|4u). (3.4.1)

r(t) = ei(t−σ)∂2xr(σ) + i

∫ t

σ

ei(t−τ)∂2xΠ(|u(τ)|4u(τ))dτ. (3.4.2)

Since u ∈ L∞(R;H1+), we have r ∈ L6

loc(R+, L6(R)). Recall that Π is bounded Lp → Lp, for all 1 < p <

+∞. Applying Strichartz inequality to formula (3.4.2), then there exists a constant C∗ > 0 such that

‖r‖L6(T,T ′;L6(R)) ≤C∗(‖r(T )‖L2 + ‖|u|4u‖

L65 (T,T ′;L

65 (R))

)≤C∗

(‖r(T )‖L2 +

5∑k=0

‖v‖kL6(T,T ′;L6(R))‖r‖5−kL6(T,T ′;L6(R))

)

holds for all 0 ≤ T < T ′. Set δ = min1, 12C∗(2+C∗)4 , there exists T > 0 such that ‖v‖L6(T,+∞;L6(R)) ≤ δ

and ‖r(T )‖L2 ≤ δ. We choose T1 := supS ∈ [T,+∞) : ‖r‖L6(T,S;L6(R)) ≤ 12+C∗ . For every T ′ ∈ (T, T1),

we have ‖r‖L6(T,T ′;L6(R)) ≤ C∗(δ + δ5 + (2 + C∗)−5 + 4δ‖r‖L6(T,T ′;L6(R))

). Then we have

‖r‖L6(T,T ′;L6(R)) ≤ 2δC∗ +1

(2 + C∗)4≤ 1

(2 + C∗)3<

1

2 + C∗.

Thus T1 = +∞ and ‖r‖L6(T,+∞;L6(R)) <1

2+C∗ , which yields that u ∈ L6(0,+∞, L6(R)).

Set u+ := u0 + i∫ +∞

0e−iτ∂

2xΠ(|u(τ)|4u(τ))dτ . Since u ∈ L6(Rt × Rx), Strichartz inequality implies that

‖u(t)− eit∂2x u+‖L2 . ‖Π(|u|4u)‖

L65 (t,+∞;L

65 (R))

. ‖u‖5L6(t,+∞;L6(R)) → 0, as t→ +∞. (3.4.3)

Thus u+ = u+. Since the momentum P (u) = 〈−i∂xu, u〉L2 = ‖|D| 12u‖2L2 is conserved, we have

supt∈R‖∂xu(t)‖2L2 . E(u0) + ‖|D| 12u0‖4L2‖u0‖2L2 . (3.4.4)

Thus ∂x Π(|u|4u) ∈ L∞(R;L2+) → L1

loc(R;L2+). The Strichartz estimate yields that ∀T > 0, we have

‖∂xu‖L6(0,T ;L6(R)) . ‖∂xu0‖L2 +

∫ T

0

‖∂xΠ(|u(τ)|4u(τ))‖L2dτ < +∞. (3.4.5)

Since u ∈ L6(Rt × Rx), there exists T0 > 0 such that ‖u‖4L6(T0,+∞;L6(R)) ≤1

2C∗ . For all T > T0, we have

‖∂xu‖L6(T0,T ;L6(R)) ≤C∗(‖∂xu0‖L2 + ‖|∂xu||u|4‖L

65 (T0,T ;L

65 (R))

)

≤C∗(‖∂xu0‖L2 + ‖u‖4L6(T0,+∞;L6(R))‖∂xu‖L6(T0,T ;L6(R)))

3.4. APPENDICES 81

Thus ‖∂xu‖L6(T0,+∞;L6(R)) ≤ 2C∗‖∂xu0‖L2 and ∂xu ∈ L6(0,+∞;L6(R)). Consequently, we use Strichartzestimate to obain

‖∂x(u(t)− eit∂2xu+)‖L2 . ‖u‖4L6(t,+∞;L6(R))‖∂xu‖L6(t,+∞;L6(R)) → 0,

as t→ +∞.

Similarly, we have the following proposition for scattering backward in time.

Proposition 3.4.2. If u0 ∈ H1+ and there exists u− ∈ L2

+ such that limt→−∞ ‖u(t) − eit∂2xu−‖L2 = 0,

where u is the unique solution of (3.1.1), then we have u− = u0 − i∫ 0

−∞ e−iτ∂2xΠ(|u(τ)|4u(τ))dτ ∈ H1

+

and limt→+∞ ‖u(t)− eit∂2xu−‖H1 = 0.

The Lr−norm of a solution of linear Schrodinger equation decays as t→ ±∞, for all 2 < r ≤ +∞.

Lemma 3.4.3. If f ∈ H1(R) and 2 < r ≤ +∞, then ‖eit∂2xf‖Lr → 0, as |t| → +∞.

Demonstration. We set w(t) := eit∂2xf and q = 4r

r−2 , then 2q + 1

r = 12 and w ∈ L∞(R, H1

+)⋂Lq(R;Lr(R))

by Strichartz estimate. Gagliardo–Nirenberg inequality yields that

‖w(t)− w(s)‖Lr ≤ ‖∂xw(t)− ∂xw(s)‖2q

L2‖w(t)− w(s)‖1−2q

L2 ≤ 2‖∂xf‖2q

L2‖w(t)− w(s)‖1−2q

L2 .

Since supt∈R ‖∂tw(t)‖H−1(R) = supt∈R ‖∂2xw(t)‖H−1(R) ≤ ‖f‖H1 , the mapping t → w(t) is Lipschitz

continuous R→ H−1(R). Therefore,

‖w(t)− w(s)‖L2 ≤√‖w(t)− w(s)‖H1‖w(t)− w(s)‖H−1 . ‖f‖H1 |t− s| 12

and ‖w(t) − w(s)‖Lr . ‖f‖H1 |t − s|q−22q . The mapping t → w(t) is uniformly continuous R → Lr(R).

Since w ∈ Lq(R;Lr(R)), we have ‖eit∂2xf‖Lr = ‖w(t)‖Lr → 0, as |t| → +∞.

This lemma yields that a traveling wave u(t, x) = eiωtQ(x) does not H1−scatter neither L2−scatter,

with ω > 0 and Q(0) ∈ G(0)2 (

4

√3J

(0)2 , (

3

√3J

(0)2 ω

2 )16 ). Together with theorem 3.1.6, we can prove corollary

3.1.10.

3.4.2 The open problem of uniqueness of ground states

The problem of classification of ground states of I(γ)2 remains open for general γ, since it is difficult

to solve the non local equation (3.1.7). However, if f is a ground state of I(γ)2 , then so is P (f), where

P (f)(ξ) = |f(ξ)|. Precisely, we have the following proposition.

Proposition 3.4.4. For every m ∈ N⋂

[2,+∞) and γ ≥ 0, if f ∈ G(γ)m , then P (f) ∈ G(γ)

m and there

exist two parameters a, b ∈ R such that f(ξ) = |f(ξ)|ei(aξ+b), for every ξ ≥ 0.


Demonstration. Parseval identity implies that

‖|D|sP (f)‖2L2 =1

2π

∫ +∞

0

|ξ|2s|f(ξ)|2dξ = ‖|D|sf‖2L2 , ∀0 ≤ s ≤ 1.

However, since P (f)(ξ) = |f(ξ)|, for every ξ > 0, we have P (f)m+1 ≥ |fm+1|, for every m ∈ N and

‖P (f)‖2m+2L2m+2 =

1

2π

∫ +∞

0

| P (f)m+1(ξ)|2dξ ≥ 1

2π

∫ +∞

0

|fm+1(ξ)|2dξ = ‖f‖2m+2L2m+2

Thus J(γ)m ≤ I

(γ)m (P (f)) ≤ I

(γ)m (f) = J

(γ)m , for all γ ≥ 0 and m ∈ N. So P (f) ∈ G(γ)

m and all precedent

inequalities become equalities. In particular, P (f)m+1 = |fm+1|. We set h(ξ) = f(ξ), then the Euler–Lagrange equation (3.1.11) reads in Fourier modes as

(Am(f) +Bm,γ(f)ξ + Cm(f)ξ2)h(ξ) = Dm,γ1ξ≥0T (h, h, · · · , h)(ξ), (3.4.6)

where Am(f), Bm,γ(f), Cm(f), Dm,γ > 0 and T : L1(R+)2m+1 → L1(R+) is (2m+ 1)-linear defined as

Tm(h1, h2, · · · , h2m+1)(ξ) =

∫Sm(ξ)

Π2m+1j=1 hj(ηj)dη1 · · · dη2m

with Sm(ξ) = (η1, · · · , η2m+1) ∈ R2m+1+ :

∑m+1j=1 ηj =

∑2m+1j=m+2 ηk + ξ.

We claim that if h(ξ) = 0 for some ξ ∈ R+ then h ≡ 0. In fact, assume by contradiction that h(ζ) 6= 0,

for some ζ ≥ 0. Since P (f) ∈ G(γ)m , we replace h by |h| in equation (3.4.6) in order to get the following

implication :

h(ξ) = 0 =⇒ h(mζ + ξ

m+ 1) = 0.

We construct an iterative sequence ξ0 = ξ and ξn+1 = mζ+ξnm+1 , ∀n ∈ N. Then we have h(ξn) = 0 and

limn→+∞ ξn = ζ. The continuity of h gives that h(ζ) = 0, contradiction. Thus h ≡ 0.

Since h = f and f 6= 0, then h is continuous R→ C∗. Thus there exists a continuous function α : R+ → S1

such that

Πm+1j=1 h(ξj) = α(

m+1∑j=1

ξj)Πm+1j=1 |h(ξj)|.

The lifting theorem yields that there exists a unique continuous function ϕ : R+ → R such that

h(ξ) = |h(ξ)|eiϕ(ξ),

m+1∑j=1

ϕ(ξj) = β(

m+1∑j=1

ξj)

for some continuous function β : R+ → R. We set φ(ξ) = ϕ(ξ)− ϕ(0) then we have

φ(ξ1) + φ(ξ2) = 2φ(ξ1 + ξ2

2), ∀ξ1, ξ2 ≥ 0.

Consequently, ϕ(ξ) = φ(1)ξ + ϕ(0) = (ϕ(1)− ϕ(0))ξ + ϕ(0), for every ξ ≥ 0.

3.4. APPENDICES 83

Thus it suffices to study the uniqueness of ground states modulo the positive Fourier transformation.

We compare theorem 3.1.6 and theorem 3.1.8. Since we do not know the uniqueness of ground states of

the functional I(γ)2 , the conservation law P (u) = ‖|D| 12u‖2L2 can not be completely used to determine the

L6−norm of the final profile that approaches u(t). However, if the ground state is unique up to scaling,phase rotation and spatial translation, then we can determine the L6−norm of the profile that approachesu(t). So we have the actual orbital stability in the case γ = 2.

Chapitre 4

Integrability of the BO equation onthe line


Resume Le but de cette troisieme contribution est de construire des coordonnees action–angle pourl’equation de Benjamin–Ono (BO) sur la droite restreinte a la variete des N -solitons, notee par UN , pourtout entier N strictement positif. Il existe un symplectomorphisme reel analytique de UN vers un cer-tain sous-ensemble ouvert convexe de R2N , pour lequel l’energie de BO ne depend que des N premieresvariables, les variables d’actions. Cette application permet de lineariser l’equation de BO et de decrirecompletement l’orbite a conjugaison reel analytique pres. On peut resoudre cette equation par quadra-ture. Inspire du travail de Gerard–Kappeler [54], on caracterise la variete UN par une representation faconpolynomiale et en terme spectral. Un ingredient important est l’introduction d’une fonction generatriceauxiliaire, qui engendre l’ensemble des hierarchies de BO. Une autre consequence est que la variete UN estle revetement universel de la variete des potentiels a N “gap” pour l’equation de BO sur le tore etudieedans [54].

Mots− clefs : Equation de Benjamin–Ono, Coordonnees d’action–angle, Paire de Lax, Espace deHardy, Transformee inverse spectrale, Multi-soliton, Revetement universel

Abstract This chapter is dedicated to proving the complete integrability of the Benjamin–Ono (BO)equation on the line when restricted to every N -soliton manifold, denoted by UN . We construct gene-ralized action–angle coordinates which establish a real analytic symplectomorphism from UN onto someopen convex subset of R2N and allow to solve the equation by quadrature for any such initial datum. As aconsequence, UN is the universal covering of the manifold of N -gap potentials for the BO equation on thetorus as described by Gerard–Kappeler [54]. The global well-posedness of the BO equation in UN is givenby a polynomial characterization and a spectral characterization of the manifold UN . Besides the spectralanalysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces,the construction of such coordinates also relies on the use of a generating functional, which encodes theentire BO hierarchy.

Keywords Benjamin–Ono equation, Action–angle coordinates, Lax pair, Hardy space, Inverse spectraltransform, Multi-soliton, Universal covering manifold

85

86 CHAPITRE 4. INTEGRABILITY OF THE BO EQUATION ON THE LINE

4.1 Introduction

The Benjamin–Ono (BO) equation on the line reads as

∂tu = H∂2xu− ∂x(u2), (t, x) ∈ R× R, (4.1.1)

where u is real-valued and H = −isign(D) : L2(R)→ L2(R) denotes the Hilbert transform, D = −i∂x,

Hf(ξ) = −isign(ξ)f(ξ), ∀f ∈ L2(R). (4.1.2)

sign(±ξ) = ±1, for all ξ > 0 and sign(0) = 0, f ∈ L2(R) denotes the Fourier–Plancherel transformof f ∈ L2(R). We adopt the convention Lp(R) = Lp(R,C). Its R-subspace consisting of all real-valuedLp-functions is specially emphasized as Lp(R,R) throughout this paper. Equipped with the inner product(f, g) ∈ L2(R)× L2(R) 7→ 〈f, g〉L2 =

∫R f(x)g(x)dx ∈ C, L2(R) is a C-Hilbert space.

Derived by Benjamin [8] and Ono [116], this equation describes the evolution of weakly nonlinear internallong waves in a two-layer fluid. The BO equation is globally well-posed in every Sobolev spaces Hs(R,R),s ≥ 0. (see Tao [137] for s ≥ 1, Burq–Planchon [21] for s > 1

4 , Ionescu–Kenig [80], Molinet–Pilod [108]and Ifrim–Tataru [79] for s ≥ 0, etc.) Recall the scaling and translation invariances of equation (4.1.1) :if u = u(t, x) is a solution, so is uc,y : (t, x) 7→ cu(c2t, c(x− y)). A smooth solution u = u(t, x) is called asolitary wave of (4.1.1) if there exists R ∈ C∞(R) solving the following non local elliptic equation

HR′ +R−R2 = 0, R(x) > 0 (4.1.3)

and u(t, x) = Rc(x − y − ct), where Rc(x) = cR(cx), for some c > 0 and y ∈ R. The unique (up totranslation) solution of equation (4.1.3) is given by the following formula

R(x) =2

1 + x2, ∀x ∈ R, (4.1.4)

in Benjamin [8] and Amick–Toland [4] for the uniqueness statement. Inspired from the complete classifi-cation of solitary waves of the BO equation, we introduce the main object of this paper.

Definition 4.1.1. A function of the form u(x) =∑Nj=1Rcj (x − xj) is called an N -soliton, for some

positive integer N ∈ N+ := Z⋂

(0,+∞), where cj > 0 and xj ∈ R, for every j = 1, 2, · · · , N . LetUN ⊂ L2(R,R) denote the subset consisting of all the N -solitons.

In the point of view of topology and differential manifolds, the subset UN is a simply connected, realanalytic, embedded submanifold of the R-Hilbert space L2(R,R). It has real dimension 2N . The tangentspace to UN at an arbitrary N -soliton is included in an auxiliary space

T := h ∈ L2(R, (1 + x2)dx) : h(R) ⊂ R,∫Rh = 0, (4.1.5)

in which a 2-covector ω ∈ Λ2(T ∗) is well defined by ω(h1, h2) = i2π

∫Rh1(ξ)h2(ξ)

ξ dξ, for every h1, h2 ∈ T ,

by Hardy’s inequality. We define a translation-invariant 2-form ω : u ∈ UN 7→ ω ∈ Λ2(T ∗), endowed with


which UN is a symplectic manifold. The tangent space to UN at u ∈ UN is denoted by Tu(UN ). For everysmooth function f : UN → R, its Hamiltonian vector field Xf ∈ X(UN ) is given by

Xf : u ∈ UN 7→ ∂x∇uf(u) ∈ Tu(UN ),

where ∇uf(u) denotes the Frechet derivative of f , i.e. df(u)(h) = 〈h,∇uf(u)〉L2 , for every h ∈ Tu(UN ).The Poisson bracket of f and another smooth function g : UN → R is defined by

f, g : u ∈ UN 7→ ωu(Xf (u), Xg(u)) = 〈∂x∇uf(u),∇ug(u)〉L2 ∈ R.

Then the BO equation (4.1.1) in the N -soliton manifold (UN , ω) can be written in Hamiltonian form

∂tu = XE(u), where E(u) =1

2〈|D|u, u〉

H−12 ,H

12− 1

3

∫Ru3. (4.1.6)

The Cauchy problem of (4.1.6) is globally well-posed in the manifold UN (see proposition 4.4.9). Inspiredfrom the construction of Birkhoff coordinates of the space-periodic BO equation discovered by Gerard–Kappeler [54], we want to show the complete integrability of (4.1.6) in the Liouville sense.

Let ΩN := (r1, r2, · · · , rN ) ∈ RN : rj < rj+1 < 0, ∀j = 1, 2, · · · , N − 1 denote the subset of actions

and ν =∑Nj=1 drj ∧ dαj denotes the canonical symplectic form on ΩN × RN . The main result of this

paper is stated as follows.

Theorem 1. There exists a real analytic symplectomorphism ΦN : (UN , ω)→ (ΩN × RN , ν) such that

E Φ−1N (r1, r2, · · · , rN ;α1, α2, · · · , αN ) = − 1

2π

N∑j=1

|rj |2. (4.1.7)

Remark 4.1.2. A consequence of theorem 1 is that UN is simply connected. In fact the manifold UNcan be interpreted as the universal covering of the manifold of N -gap potentials for the Benjamin–Onoequation on the torus as described by Gerard–Kappeler in [54]. We refer to section 4.6 for a direct proofof these topological facts, independently of theorem 1.

Remark 4.1.3. Then ΦN : u ∈ UN 7→ (I1(u), I2(u), · · · , IN (u); γ1(u), γ2(u), · · · , γN (u)) ∈ ΩN × RNintroduces the generalized action–angle coordinates of the BO equation in the N -soliton manifold, i.e.

Ik, E(u) = 0, γk, E(u) =Ik(u)

π, ∀u ∈ UN . (4.1.8)

Theorem 1 gives a complete description of the orbit structure of the flow of equation (4.1.6) up to realbi-analytic conjugacy. Let u : t ∈ R 7→ u(t) ∈ UN denote the solution of equation (4.1.6), rk(t) = Ik u(t)denotes action coordinates and αk(t) = γk u(t) denotes the generalized angle coordinates, then we have

rk(t) = rk(0), αk(t) = αk(0)− rk(0)t

π, ∀k = 1, 2, · · · , N. (4.1.9)

We refer to definition 4.5.1 and theorem 4.5.2 for a precise description of ΦN .

In order to establish the link between the action–angle coordinates and the translation–scaling parametersof an N -soliton, we introduce the inverse spectral matrix associated to ΦN , denoted by

M : u ∈ UN 7→ (Mkj(u))1≤j,k≤N ∈ CN×N , Mkj(u) =

2πiIk(u)−Ij(u)

√Ik(u)Ij(u) , if j 6= k,

γj(u) + πiIj(u) , if j = k,

(4.1.10)

where Ik, γk : U → R is given by remark 4.1.3. Then UN has the following polynomial characterization.


Proposition 4.1.4. A real-valued function u ∈ UN if and only if there exists a monic polynomial

Qu ∈ C[X] of degree N , whose roots are contained in the lower half-plane C− and u = −2ImQ′uQu

. Precisely,

Qu is unique and is the characteristic polynomial of the matrix M(u) ∈ CN×N defined by (4.1.10).

An N -soliton is expressed by u(x) =∑Nj=1Rcj (x − xj) if and only if its translation–scaling parameters

xj − c−1j i1≤j≤N ⊂ CN− are the roots of the characteristic polynomial Qu(X) = det(X −M(u)), whose

coefficients are expressed in terms of the action–angle coordinates (Ij(u), γj(u))1≤j≤N ∈ ΩN × RN .Proposition 4.1.4 is restated with more details in proposition 4.4.1, formula (4.5.11) and theorem 4.4.8which gives a spectral characterization of UN . If u : t ∈ R 7→ u(t) ∈ UN solves the BO equation (4.1.1),then we have the following explicit formula

u(t, x) = 2Im〈(M(u0)− (x+ t

πV(u0)))−1

X(u0), Y (u0)〉CN , (t, x) ∈ R× R, (4.1.11)

where the inner product of CN is 〈X,Y 〉CN = XTY , for every u ∈ UN , the matrix V(u) ∈ CN×N and thevectors X(u), Y (u) ∈ CN are defined by

√2πX(u)T = (

√|I1(u)|,

√|I2(u)|, · · · ,

√|IN (u)|),

√2π−1Y (u)T = (

√|I1(u)|−1,

√|I2(u)|−1, · · · ,

√|IN (u)|−1),

V(u) =

I1(u)I2(u)

. . .IN (u)

.

4.1.1 Notation

Before outlining the construction of action–angle coordinates, we introduce some notations used in thispaper. The indicator function of a subset A ⊂ X is denoted by 1A, i.e. 1A(x) = 1 if x ∈ A and1A(x) = 0 if x ∈ X\A. Recall that H : L2(R) → L2(R) denotes the Hilbert transform given by (4.1.2).Set IdL2(R)(f) = f , for every f ∈ L2(R). Let Π : L2(R)→ L2(R) denote the Szego projector, defined by

Π :=IdL2(R) + iH

2⇐⇒ Πf(ξ) = 1[0,+∞)(ξ)f(ξ), ∀ξ ∈ R, ∀f ∈ L2(R). (4.1.12)

If O is an open subset of C, we denote by Hol(O) all holomorphic functions on O. Let the upper half-plane and the lower half-plane be denoted by C+ = z ∈ C : Imz > 0 and C− = z ∈ C : Imz < 0respectively. For every p ∈ (0,+∞], we denote by Lp+ to be the Hardy space of holomorphic functions onC+ such that Lp+ = g ∈ Hol(C+) : ‖g‖Lp+ < +∞, where

‖g‖Lp+ = supy>0

(∫R|g(x+ iy)|pdx

) 1p

, if p ∈ (0,+∞), (4.1.13)

and ‖g‖L∞+ = supz∈C+|g(z)|. A function g ∈ L∞+ is called an inner function if |g| = 1 on R. When p = 2,

the Paley–Wiener theorem yields the identification between L2+ and Π[L2(R)] :

L2+ = F−1[L2(0,+∞)] = f ∈ L2(R) : suppf ⊂ [0,+∞) = Π(L2(R)),

where F : f ∈ L2(R) 7→ f ∈ L2(R) denotes the Fourier–Plancherel transform. Similarly, we setL2− = (IdL2(R) − Π)(L2(R)). Let the filtered Sobolev spaces be denoted as Hs

+ := L2+

⋂Hs(R) and

Hs− := L2

−⋂Hs(R), for every s ≥ 0.


The domain of definition of an unbounded operator A on some Hilbert space E is denoted by D(A) ⊂ E .Given another operator B on D(B) ⊂ E such that A(D(A)) ⊂ D(B) and B(D(B)) ⊂ D(A), their Liebracket is an operator defined on D(A)

⋂D(B) ⊂ E , which is given by

[A,B] := AB − BA. (4.1.14)

If the operator A is self-adjoint, let σ(A) denote its spectrum, σpp(A) denotes the set of its eigenva-

lues and σcont(A) denotes its continuous spectrum. Then σcont(A)⋃σpp(A) = σ(A) ⊂ R. Given two

C-Hilbert spaces E1 and E2, let B(E1, E2) denote the C-Banach space of all bounded C-linear transforma-tions E1 → E2, equipped with the uniform norm.

Given a smooth manifold M of real dimension N , let C∞(M) denote all smooth functions f : M → Rand the set of all smooth vector fields is denoted by X(M). The tangent (resp. cotangent) space to M atp ∈M is denoted by Tp(M) (resp. T ∗p (M)). Given k ∈ N, the R-vector space of smooth k-forms on M is

denoted by Ωk(M). Given a R-vector space V, we denote by Λk(V∗) the vector space of all k-covectorson V. Given a smooth covariant tensor field A on M and X ∈ X(M), the Lie derivative of A with respectto X is denoted by LX(A), which is also a smooth tensor field on M. If N is another smooth manifold,F : N→M is a smooth map and A is a smooth covariant k-tensor field on M, the pullback of A by Fis denoted by F∗A, which is a smooth k-tensor field on N defined by ∀p ∈ N, ∀j = 1, 2, · · · , k,

(F∗A)p(v1, v2, · · · , vk) = AF(p) (dF(p)(v1),dF(p)(v2), · · · ,dF(p)(vk)) , ∀vj ∈ Tp(N). (4.1.15)

Given a positive integer N , let C≤N−1[X] denote the C-vector space of all polynomials with complexcoefficients whose degree is no greater than N − 1 and CN [X] = C≤N [X]\C≤N−1[X] consists of allpolynomials of degree exactly N . R+ = [0,+∞) and R∗+ = (0,+∞). D(z, r) ⊂ C denotes the open discof radius r > 0, whose center is z ∈ C.

4.1.2 Organization of this paper

The construction of action–angle coordinates for the BO equation (4.1.6) mainly relies on the Lax pairformulation ∂tLu = [Bu, Lu], discovered by Nakamura [110] and Bock–Kruskal [12]. Section 4.2 is dedica-ted to the spectral analysis of the Lax operator Lu : h ∈ H1

+ 7→ −i∂xh−Π(uh) ∈ L2+ given by definition

4.2.1 for general symbol u ∈ L2(R,R), where Π denotes the Szego projector given in (4.1.12) and theHardy space L2

+ is defined in (4.1.13). Lu is an unbounded self-adjoint operator on L2+ that is bounded

from below, it has essential spectrum σess(Lu) = [0,+∞). If x 7→ xu(x) ∈ L2(R) in addition, everyeigenvalue is negative and simple, thanks to an identity firstly found by Wu [146]. Then we introduce agenerating function which encodes the entire BO hierarchy,

Hλ(u) = 〈(Lu + λ)−1Πu,Πu〉L2 , if λ ∈ C\σ(−Lu), (4.1.16)

in definition 4.2.9. It provides a sequence of conservation laws controlling every Sobolev norms.

In section 4.3, we study the shift semigroup (S(η)∗)η≥0 acting on the Hardy space L2+, where S(η)f = eηf

and eη(x) = eiηx. Then a weak version of Beurling–Lax theorem can be obtained by solving a linear dif-ferential equation with constant coefficients. Every N -dimensional subspace of L2

+ that is invariant under

its infinitesimal generator G = i ddη

∣∣η=0+S(η)∗ is of the form

C≤N−1[X]

Q , for some monic polynomial Q

whose roots are contained in the lower half-plane C−.


In section 4.4, the real analytic structure and symplectic structure of the N -soliton subset UN are es-tablished at first. Then we continue the spectral analysis of the Lax operator Lu, ∀u ∈ UN . Lu has Nsimple eigenvalues λu1 < λu2 < · · · < λuN < 0 and the Hardy space L2

+ splits as

L2+ = Hcont(Lu)

⊕Hpp(Lu), Hcont(Lu) = Hac(Lu) = ΘuL

2+, Hpp(Lu) =

C≤N−1[X]

Qu. (4.1.17)

where Qu denotes the characteristic polynomial of u given by proposition 4.1.4 and Θu = QuQu

is an in-

ner function on the upper half-plane C+. Proposition 4.1.4 is proved by identifying M(u) in (4.1.10)as the matrix of the restriction G|Hpp(Lu) associated to the spectral basis ϕu1 , ϕu2 , · · · , ϕuN, whereϕuj ∈ Ker(λuj − Lu) such that ‖ϕuj ‖L2 = 1 and

∫R uϕ

uj > 0. The generating function Hλ in (4.1.16)

can be identified as the Borel–Cauchy transform of the spectral measure of Lu associated to the vectorΠu, which yields the invariance of UN under the BO flow in H∞(R,R). Hence (4.1.6) is a globally well-posed Hamiltonian system on UN .

Section 4.5 is dedicated to completing the proof of theorem 1. The generalized angle-variables are the realparts of the diagonal elements of the matrix M(u), i.e. γj : u ∈ UN 7→ Re〈Gϕuj , ϕuj 〉L2 ∈ R and the action-

variables are Ij : u ∈ UN 7→ 2πλuj ∈ R. Thanks to the Lax pair formulation dL(u)(XHλ(u)) = [Bλu , Lu],

where L : u ∈ UN 7→ Lu ∈ B(H1+, L

2+) is R-affine and Bλu is some skew-adjoint operator on L2

+, we havethe following formulas of Poisson brackets,

2πλj , γk = 1j=k, γj , γk = 0 on UN , 1 ≤ j, k ≤ N. (4.1.18)

which implies that ΦN : u ∈ UN 7→ (I1(u), I2(u), · · · , IN (u); γ1(u), γ2(u), · · · , γN (u)) ∈ ΩN × RN isa real analytic immersion. The diffeomorphism property of ΦN is given by Hadamard’s global inverse

theorem. The inverse spectral formula Πu = iQ′uQu

with Qu(X) = det(X − G|Hpp(Lu)), which is restated

as formula (4.5.11), implies the explicit formula (4.1.11) of all multi-soliton solutions of the BO equa-tion (4.1.1) and (4.5.11) provides an alternative proof of the injectivity of ΦN . Finally, we show thatΦN : (UN , ω) → (ΩN × RN , ν) is a symplectomorphism by restricting the 2-form ω − Φ∗Nν to a special

Lagrangian submanifold ΛN :=⋂Nj=1 γ

−1j (0) ⊂ UN .

In appendix 4.6, we establish the simple connectedness of UN and a covering map from UN to the manifoldof N -gap potentials from their constructions without using the integrability theorems.

4.1.3 Related work

The BO equation has been extensively studied for nearly sixty years in the domain of partial differentialequations. We refer to Saut [128] for an excellent account of these results. Besides the global well-posednessproblem, various properties of its multi-soliton solutions has been investigated in details. Matsuno [104]has found the explicit expression of multi-soliton solutions of (4.1.1) by following the bilinear methodof Hirota [75]. The multi-phase solutions (periodic multi-solitons) have been constructed by Satsuma–Ishimori [126] at first. We point out the work of Amick–Toland [4] on the characterization of 1-solitonsolutions which can also be revisited by theorem 1 and proposition 4.1.4. In Dobrokhotov–Krichever [30],the multi-phase solutions are constructed by finite zone integration and they have also established aninversion formula for multi-phase solutions. Compared to their work, we give a geometric description ofthe inverse spectral transform by proving the real bi-analyticity and the symplectomorphism property ofthe action–angle map. Furthermore, the inverse spectral formula

Πu(x) = iQ′u(x)

Qu(x), Qu(x) = det(x−G|Hpp(Lu)) = det(x−M(u)), ∀x ∈ R. (4.1.19)

4.2. THE LAX OPERATOR 91

provides a spectral connection between the Lax operator Lu and the infinitesimal generator G. The ideaof introducing generating function Hλ has also been used for the quantum BO equation in Nazarov–Sklyanin [111]. Their method has also been developed by Moll [109] for the classical BO equation. Theasymptotic stability of soliton solutions and of solutions starting with sums of widely separated solitonprofiles is obtained by Kenig–Martel [85].

Concerning the investigation of integrability for the BO equation on R besides the discovery of Lax pairformulation, we mention the pioneering work of Ablowitz–Fokas [1], Coifman–Wickerhauser [28], Kaup–Matsuno [84] and Wu [146, 147] for the inverse scattering transform. In the space-periodic regime, theBO equation on the torus T admits global Birkhoff coordinates on L2

r,0(T) := v ∈ L2(T,R) :∫T v = 0

in Gerard–Kappeler [54]. We refer to Gerard–Kappeler–Topalov [55] to see that the Birkhoff coordinatesof the BO equation on the torus can be extended to a larger Sobolev space Hs

r,0(T) := v ∈ Hs(T,R) :∫T v = 0, for every − 1

2 < s < 0. We point out that both Korteweg–de Vries equation on T (seeKappeler–Poschel [81]) and the defocusing cubic Schrodinger equation on T (see Grebert–Kappeler [65])admit global Birkhoff coordinates. The theory of finite-dimensional Hamiltonian system is transferredto the BO, KdV and dNLS equation on T through the submanifolds of corresponding finite-gap poten-tials, which are introduced to solve the periodic KdV initial problem. We refer to Matveev [105] for details.

Moreover, the cubic Szego equation both on T (see Gerard–Grellier [47, 49, 51, 52]) and on R (see Po-covnicu [119, 120]) admit global (generalized) action–angle coordinates on all finite-rank generic rationalfunction manifolds, denoted respectively byM(N)Tgen andM(N)Rgen. Moreover, the cubic Szego equationboth on T and on R have inverse spectral formulas which permit the Szego flows to be expressed explicitlyin terms of time-variables and initial data without using action–angle coordinates. The shift semigroup(S(η)∗)η≥0 and its infinitesimal generator G are also used in Pocovnicu [120] to establish the integrabilityof the cubic Szego equation on the line.

The BO equation admits an infinite hierarchy of conservation laws controlling every Hs-norm (seeAblowitz–Fokas [1], Coifman–Wickerhauser [28] in the case 2s ∈ N and Talbut [136] in the case− 1

2 < s < 0and conservation law controlling Besov norms etc.), so does the KdV equation and the NLS equation (seeKillip–Visan–Zhang [90], Koch–Tataru [91], Faddeev–Takhtajan [33], Gerard [46] and Sun [131] etc.)

Throughout this chapter, the main results of each section are stated at the beginning. Theirproofs are left inside the corresponding subsections.

4.2 The Lax operator

This section is dedicated to studying the Lax operator Lu in the Lax pair formulation of the BOequation (4.1.1), discovered by Nakamura [110] and Bock–Kruskal [12]. Then we describe the locationand revisit the simplicity of eigenvalues of Lu. At last, we introduce a generating functional Hλ whichencodes the entire BO hierarchy. The equation ∂tu = ∂x∇uHλ(u) also enjoys a Lax pair structure withthe same Lax operator Lu.

Definition 4.2.1. Given u ∈ L2(R,R), its associated Lax operator Lu is an unbounded operator on L2+,

given by Lu := D− Tu, where D : h ∈ H1+ 7→ −i∂xh ∈ L2

+ and Tu denotes the Toeplitz operator of symbolu, defined by Tu : h ∈ H1

+ 7→ Π(uh) ∈ L2+, where the Szego projector Π : L2(R)→ L2

+ is given by (4.1.12).If u ∈ H1(R,R) in addition, we define Bu := i(T|D|u − T 2

u) ∈ B(H1+, L

2+).


Both D and Tu are densely defined symmetric operators on L2+ and ‖Tu(h)‖L2 ≤ ‖u‖L2‖h‖L∞ , for every

h ∈ H1+ and u ∈ L2(R,R). Moreover, the Fourier–Plancherel transform implies that D is a self-adjoint

operator on L2+, whose domain of definition is H1

+.

Proposition 4.2.2. If u ∈ L2(R,R), then Lu is an unbounded self-adjoint operator on L2+, whose

domain of definition is D(Lu) = H1+. Moreover, Lu is bounded from below. The essential spectrum of Lu

is σess(Lu) = σess(D) = [0,+∞) and its pure point spectrum satisfies σpp(Lu) ⊂ [−C2

4 ‖u‖2L2 ,+∞), where

C = inff∈H1+\0

‖|D|14 f‖L2

‖f‖L4denotes the Sobolev constant.

Thanks to an identity firstly found by Wu [146] in the negative eigenvalue case, we show the simplicityof the pure point spectrum σpp(Lu), if u ∈ L2(R, (1 + x2)dx) is real-valued.

Proposition 4.2.3. Assume that u ∈ L2(R;R) and x 7→ xu(x) ∈ L2(R). For every λ ∈ R and ϕ ∈Ker(λ− Lu), we have uϕ ∈ C1(R)

⋂H1(R) and the following identity holds,∣∣∣ ∫

Ruϕ∣∣∣2 = −2πλ

∫R|ϕ|2. (4.2.1)

Thus σpp(Lu) ⊂ (−∞, 0) and for every λ ∈ σpp(Lu), we have

Ker(λ− Lu) ⊂ ϕ ∈ H1+ : ϕ|R+

∈ C1(R+)⋂H1(R+) and ξ 7→ ξ[ϕ(ξ) + ∂ξϕ(ξ)] ∈ L2(R+). (4.2.2)

Corollary 4.2.4. Assume that u ∈ L2(R;R) and x 7→ xu(x) ∈ L2(R). Then every eigenvalue of Lu is

simple. If u ∈ L∞(R) in addition, then σpp(Lu) is a finite subset of [−C2‖u‖2

L2

4 , 0).

Demonstration. Fix λ ∈ σpp(Lu) and set Vλ = Ker(λ−Lu), then dimC(Vλ) ≥ 1. We define a linear formA : Vλ → C such that

A(ϕ) :=

∫Ruϕ

Then identity (4.2.1) yields that Ker(A) = 0. Thus V ∼= V/Ker(A) ∼= Im(A) → C. So we havedimC(Vλ) = 1. When u ∈ L∞(R) in addition, the finiteness of σpp(Lu)

⋂(−∞, 0) is given by Theorem

1.2 of Wu [146].

We recall some known results of global well-posedness of the BO equation on the line.

Proposition 4.2.5. For every s ≥ 0, the Frechet space C(R, Hs(R)) is endowed with the topologyof uniform convergence on every compact subset of R. There exists a unique continuous mapping u0 ∈Hs(R) 7→ u ∈ C(R, Hs(R)) such that u solves the BO equation (4.1.1) with initial datum u(0) = u0.

Demonstration. See Tao [137], Burq–Planchon [21], Ionescu–Kenig [80], Molinet–Pilod [108], Ifrim–Tataru[79] etc.

Proposition 4.2.6. For every n ∈ N, if u0 ∈ Hn2 (R,R), let u : t ∈ R 7→ u(t) ∈ H

n2 (R,R) solves

equation (4.1.1) with initial datum u(0) = u0, then C(‖u0‖H n2

) := supt∈R ‖u(t)‖Hn2< +∞.

Demonstration. See Ablowitz–Fokas [1], Coifman–Wickerhauser [28].

When u ∈ H2(R,R), the Toeplitz operators T|D|u and Tu are bounded both on L2+ and on H1

+. So Bu isa bounded skew-adjoint operator both on L2

+ and on H1+.


Proposition 4.2.7. Let u : t ∈ R 7→ u(t) ∈ H2(R,R) denote the unique solution of equation (4.1.1),then

∂tLu(t) = [Bu(t), Lu(t)] ∈ B(H1+, L

2+), ∀t ∈ R. (4.2.3)

Let U : t 7→ U(t) ∈ B(L2+) := B(L2

+, L2+) denote the unique solution of the following equation

U ′(t) = Bu(t)U(t), U(0) = IdL2+, (4.2.4)

if u : t ∈ R 7→ u(t) ∈ H2(R,R) denote the unique solution of equation (4.1.1). The system (4.2.4) isglobally well-posed in B(L2

+), thanks to proposition 4.2.6, the following estimate

‖Bu(h)‖L2 . (‖u‖H2 + ‖u‖2H1)‖h‖L2 , ∀h ∈ L2+, ∀u ∈ H2(R,R).

and a classical Cauchy theorem (see for instance lemma 7.2 of Sun [131]). Since B∗u = −Bu, the operatorU(t) is unitary for every t ∈ R. Thus, the Lax pair formulation (4.2.3) of the BO equation (4.1.1) isequivalent to the unitary equivalence between Lu(t) and Lu(0),

Lu(t) = U(t)Lu(0)U(t)∗ ∈ B(H1+, L

2+). (4.2.5)

On the one hand, the spectrum of Lu is invariant under the BO flow. In particular, we have σpp(Lu(t)) =σpp(Lu(0)). On the other hand, there exists a sequence of conservation laws controlling every Sobolev

norms Hn2 (R), n ≥ 0. Furthermore, the Lax operator in the Lax pair formulation is not unique. If

f ∈ L∞(R) and p is a polynomial with complex coefficients, then

f(Lu(t)) = U(t)f(Lu(0))U(t)∗ ∈ B(L2+), p(Lu(t)) = U(t)p(Lu(0))U(t)∗ ∈ B(HN

+ , L2+), (4.2.6)

where N is the degree of the polynomial p.

Proposition 4.2.8. Given n ∈ N, let u : t ∈ R 7→ u(t) ∈ H n2 (R,R) denote the solution of equation

(4.1.1), we setEn(u) := 〈LnuΠu,Πu〉

H−n2 ,H

n2. (4.2.7)

Then En(u(t)) = En(u(0)), for every t ∈ R. In particular, E1 = E on H12 (R,R), where the energy

functional E is given by (4.1.6).

Definition 4.2.9. Given u ∈ L2(R,R) and λ ∈ C\σ(−Lu), the C-linear transformation λ + Lu isinvertible in B(H1

+, L2+) and the generating function is defined by Hλ(u) = 〈(Lu + λ)−1Πu,Πu〉L2 . The

subset X := (λ, u) ∈ R× L2(R,R) : 4λ > C2‖u‖2L2 is open in the R-Banach space R× L2(R,R), where

the Sobolev constant is given by C = inff∈H1+\0

‖|D|14 f‖L2

‖f‖L4and we have σ(Lu) ⊂ [−C

2‖u‖2L2

4 ,+∞) by

proposition 4.2.2.

The map (λ, u) ∈ X 7→ Hλ(u) = 〈(Lu + λ)−1Πu,Πu〉L2 ∈ R is real analytic.

Proposition 4.2.10. Let u : t ∈ R 7→ u(t) ∈ H∞(R,R) denote the solution of the BO equation (4.1.1)and we choose λ ∈ C\σ(−Lu(0)), then Hλ(u(t)) = Hλ(u(0)), for every t ∈ R.

Given (λ, u) ∈ X , there exists a neighbourhood of u in L2(R,R), denoted by Vu such that the restrictionHλ : v ∈ Vu 7→ Hλ(v) ∈ R is real analytic. The Frechet derivative of Hλ at u is computed as follows,

dHλ(u)(h) = 〈wλ,Πh〉L2 + 〈wλ,Πh〉L2 + 〈Thwλ, wλ〉L2 = 〈h,wλ + wλ + |wλ|2〉L2 , ∀h ∈ L2(R,R).


where wλ ∈ H1+ is given by wλ ≡ wλ(u) ≡ wλ(x, u) = [(Lu + λ)−1 Π]u(x), for every x ∈ R. Then

∇uHλ(u) = |wλ(u)|2 + wλ(u) + wλ(u). (4.2.8)

Given (λ, u0) ∈ X fixed, the pseudo-Hamiltonian equation associated to Hλ is defined by

∂tu = ∂x∇uHλ(u) = ∂x(|wλ(u)|2 + wλ(u) + wλ(u)

), u(0) = u0. (4.2.9)

There exists an open subset Vu0of L2(R,R) such that v ∈ Vu0

7→ ∂x(|wλ(v)|2 + wλ(v) + wλ(v)

)∈

L2(R,R) is real analytic and u0 ∈ Vu0. Hence (4.2.9) admits a local solution by Cauchy–Lipschitz theorem.

Remark 4.2.11. The word ’pseudo-Hamiltonian’ is used here because no symplectic form has been definedon L2(R,R) until now. In section 4.4, we show that ∂x∇f(u) is exactly the Hamiltonian vector field ofthe smooth function f : UN → R with respect to the symplectic form ω on the N -soliton manifold UNdefined in (4.4.2).

Proposition 4.2.12. Given (λ, u0) ∈ X fixed, there exists ε > 0 such that (λ, u(t)) ∈ X , for everyt ∈ (−ε, ε), where u : t ∈ (−ε,+ε) 7→ u(t) ∈ L2(R,R) denotes the local solution of (4.2.9) with initialdatum u(0) = u0. We have

∂tLu(t) = [Bλu(t), Lu(t)], where Bλv := i(Twλ(v)Twλ(v) + Twλ(v) + Twλ(v)), if (λ, v) ∈ X . (4.2.10)

i.e. (Lu, Bλu) is a Lax pair of equation (4.2.9).

Remark 4.2.13. The Toeplitz operators Twλ(v) and Twλ(v) are bounded both on L2+ and on H1

+, so is

the skew-adjoint operator Bλv , if (λ, v) ∈ X .

For every u ∈ H∞(R,R) and ε ∈ (0, 4C2‖u‖2

L2), we set Hε(u) := 1

εH 1ε(u) and Bε,u := 1

εB1εu . Recall that

En(u) = 〈LnuΠu,Πu〉L2 , we have the following Taylor expansion

Hε(u) =

M∑k=0

(−ε)nEn(u)− (−ε)M 〈(Lu + 1ε )−1Πu, LMu Πu〉L2 , ∀M ∈ N. (4.2.11)

Proposition 4.2.12 then leads to a Lax pair formulation for the equations corresponding to the conservationlaws in the BO hierarchy,

∂tLu = [dn

dεn

∣∣∣ε=0

Bε,u, Lu],

where now u evolves according to the pseudo-Hamiltonian flow of En = (−1)n dn

dεn

∣∣ε=0Hε. In the case

n = 1, we have E1 = E and Bu = ddε

∣∣ε=0

Bε,u.

This section is organized as follows. In subsection 4.2.1, we recall some basic facts concerning unitarilyequivalent self-adjoint operators on different Hilbert spaces. The subsection 4.2.2 is dedicated to theproofs of proposition 4.2.2 and 4.2.3. Proposition 4.2.8 and 4.2.10 that concern the conservation laws areproved in subsection 4.2.3. Proposition 4.2.7 and proposition 4.2.12 that indicate the Lax pair structuresare proved in subsection 4.2.4.


4.2.1 Unitary equivalence

Generally, if E1 and E2 are two Hilbert spaces, let A be a self-adjoint operator defined on D(A) ⊂ E1 andB be a self-adjoint operator defined on D(B) ⊂ E2. Both A and B have spectral decompositions

E1 = Hac(A)⊕

Hsc(A)⊕

Hpp(A), E2 = Hac(B)⊕

Hsc(B)⊕

Hpp(B). (4.2.12)

If A and B are unitarily equivalent i.e. there exists a unitary operator U : E1 → E2 such that

B = UAU∗, D(B) = UD(A), (4.2.13)

then we have the following identification result.

Proposition 4.2.14. The operators A and B have the same spectrum and UHxx(A) = Hxx(B), forevery xx ∈ ac, sc,pp. Moreover, for every bounded borel function f : R→ C, f(A) is a bounded operatoron E1, f(B) is a bounded operator on E2, we have f(B) = Uf(A)U∗.

Demonstration. If f is a bounded Borel function, ψ ∈ E1, consider the spectral measure of A associatedto the vector ψ ∈ E1, denoted by µAψ . Similarly, we denote by µBUψ the spectral measure of B associatedto the vector Uψ ∈ E2. Clearly, we have

supp(µAψ ) ⊂ σ(A) ⊂ R, supp(µBUψ) ⊂ σ(B) ⊂ R.

For every λ ∈ C\σ(A) = C\σ(B), formula (4.2.13) implies that U(λ − A)−1U∗ = (λ − B)−1. So theBorel–Cauchy transforms of these two spectral measures are the same.∫

R

dµAψ (ξ)

λ− ξ= 〈(λ−A)−1ψ,ψ〉E1 = 〈(λ− B)−1Uψ,Uψ〉E2 =

∫R

dµBUψ(ξ)

λ− ξ.

Both of these two spectral measures have finite total variations : µAψ (R) = µBUψ(R) = ‖ψ‖2E1 . Sinceevery finite Borel measure is uniquely determined by its Borel–Cauchy transform (see Theorem 3.21 ofTeschl [139] page 108), we have µAψ = µBUψ. So the restriction U|Hxx(A) : Hxx(A) → Hxx(B) is a linearisomorphism, for every xx ∈ ac, sc,pp. Finally, we use the definition of the spectral measures to obtain

〈f(A)ψ,ψ〉E1 =

∫Rf(ξ)dµAψ (ξ) =

∫Rf(ξ)dµBUψ(ξ) = 〈f(B)Uψ,Uψ〉E2

We may assume that f is real-valued, so that f(A) is self-adjoint. The polarization identity implies that〈f(A)ψ, φ〉E1 = 〈f(B)Uψ,Uφ〉E2 , for every ψ, φ ∈ E1. So we obtain f(B) = Uf(A)U∗ in the case f isreal-valued bounded Borel function. In the general case, it suffices to use f = Ref + iImf .

4.2.2 Spectral analysis I

In this subsection, we study the essential spectrum and discrete spectrum of the Lax operator Lu byproving proposition 4.2.2 and 4.2.3. The spectral analysis of Lu such that u is a multi-soliton in definition4.1.1, will be continued in subsection 4.4.2.

Proof of proposition 4.2.2. For every h ∈ L2+, let µD

h denote the spectral measure of D associated to h,then

〈f(D)h, h〉L2 =

∫ +∞

0

f(ξ)|h(ξ)|2

2πdξ =⇒ dµD

h (ξ) =1[0,+∞)(ξ)|h(ξ)|2

2πdξ.


Thus we have σ(D) = σess(D) = σac(D) = [0,+∞). If u ∈ L2(R,R), we claim that Pu := Tu (D + i)−1

is a Hilbert–Schmidt operator on L2+.

Recall that R∗+ = (0,+∞). In fact, let F : h ∈ L2+ 7→ h√

2π∈ L2(R∗+) denotes the renormalized Fourier–

Plancherel transform, then Au := F Pu F−1 is an operator on L2(R∗+). Then we have

Aug(ξ) =

∫ +∞

0

Ku(ξ, η)g(η)dη, Ku(ξ, η) :=u(ξ − η)

2π(η + i), ∀ξ, η ∈ R∗+.

Hence its Hilbert–Schmidt norm ‖Au‖HS(L2(R∗+)) ≤ ‖K‖L2(R∗+×R∗+) ≤‖u‖L2

2 . Since Pu is unitarily equiva-

lent to Au, we have ‖Pu‖2HS(L2+)

=∑λ∈σ(Pu) λ

2 =∑λ∈σ(Au) λ

2 = ‖Au‖2HS(L2(R∗+)) ≤‖u‖2

L2

4 .

Then the symmetric operator Tu is relatively compact with respect to D and Weyl’s essential spectrumtheorem (Theorem XIII.14 of Reed–Simon [122]) yields that σess(Lu) = σess(D) and Lu is self-adjoint withD(Lu) = D(D) = H1

+. An alternative proof of the self-adjointness of Lu can be given by Kato–Rellichtheorem (Theorem X.12 of Reed–Simon [121]) and the following estimate, for every f ∈ H1

+,

2π‖f‖L∞ ≤ ‖f‖L1 ≤ ‖f‖L2

√A+ ‖∂xf‖L2

√A−1 ≤ 2

(‖f‖L2‖∂xf‖L2

) 12

, A =

√‖∂xf‖L2

‖f‖L2.

So ‖Tu(f)‖L2 ≤ ‖u‖L2‖f‖L∞ ≤ 2π‖∂xf‖L2 +

‖u‖2L2

4 ‖f‖L2 .

Moreover, |〈Tuf, f〉L2 | = |∫R u|f |

2| ≤ ‖u‖L2‖f‖2L4 ≤ C‖u‖L2‖f‖L2‖|D| 12 f‖L2 holds by Sobolev embed-

ding ‖f‖L4 ≤ C‖|D| 14 f‖L2 , for every f ∈ H1+. Then Lu is bounded from below, precisely

〈Luf, f〉L2 = ‖|D| 12 f‖2L2 − 〈Tuf, f〉L2 ≥ −C2‖u‖2

L2‖f‖2L2

4 .

When λ < −C2‖u‖2

L2

4 , the map Lu − λ : H1+ → L2

+ is injective. Hence σpp(Lu) ⊂ [−C2

4 ‖u‖2L2 ,+∞).

Before the proof of proposition 4.2.3, we recall a lemma concerning the regularity of convolutions.

Lemma 4.2.15. For every p ∈ (1,+∞) and m,n ∈ N, we have

Wm,p(R) ∗Wn, pp−1 (R) → Cm+n(R)

⋂Wm+n,+∞(R). (4.2.14)

For every f ∈Wm,p(R) ∗Wn, pp−1 (R), we have lim|x|→+∞ ∂αx f(x) = 0, for every α = 0, 1, · · · ,m+ n.

Demonstration. In the case m = n = 0, it suffices use Holder’s inequality and the density argument ofthe Schwartz class S (R) ⊂ Wm,p(R). In the case m = 0 and n = 1, recall that a continuous functionwhose weak-derivative is continuous is of class C1 and 〈f, ϕ〉D(R)′,D(R) = f ∗ ϕ(0), we use the densityargument of the test function class D(R) ⊂ Lp(R). We conclude by induction on n ≥ 1 and m ∈ N.

Remark 4.2.16. Identity (4.2.1) was firstly found by Wu [146] in the case λ < 0. We show that (4.2.1)still holds in the case λ ≥ 0. Hence the operator Lu has no eigenvalues in [0,+∞).


Proof of proposition 4.2.3. We choose u ∈ L2(R; (1+x2)dx) such that u(R) ⊂ R, λ ∈ R and ϕ ∈ D(Lu) =H1

+ such that Lu(ϕ) = λϕ. Applying the Fourier–Plancherel transform, we obtain

uϕ(ξ)1ξ≥0 = (ξ − λ)ϕ(ξ) =: gλ(ξ). (4.2.15)

Since u ∈ H1(R) and ϕ ∈ L2(R), their convolution uϕ = 12π u ∗ ϕ ∈ C

1(R)⋂C0(R), where C0(R) denotes

the uniform closure of Cc(R) with respect to the L∞(R)-norm, by lemma 4.2.15. Recall R+ = [0,+∞).

We claim that if λ < 0, then ϕ ∈ C1(R+);

if λ ≥ 0, then ϕ ∈ C(R+)⋂C1(R+\λ).

In fact, if λ ≥ 0, we have gλ(λ) = 0. Otherwise, λ would be a singular point of ϕ that prevents ϕ frombeing a L2 function on R+, because ξ → 1

ξ−λ /∈ L2(R+). By using the fact gλ ∈ C1(R+) (gλ is right

differentiable at ξ = 0 and the derivative g′λ is right continuous at ξ = 0), we have

ϕ(ξ) =gλ(ξ)− gλ(λ)

ξ − λ→

g′λ(λ), if λ > 0;

g′λ(0+), if λ = 0;

when ξ → λ. So ϕ ∈ C(R+) and limξ→+∞ ϕ(ξ) = 0. Then we derive formula (4.2.15) with respect to ξ toget the following

− ixu ∗ ϕ(ξ) = g′λ(ξ) = (uϕ)′(ξ) = ϕ(ξ) + (ξ − λ)(ϕ)′(ξ), ∀ξ ∈ [0,+∞)\λ. (4.2.16)

Thus we have

d

dξ[(ξ − λ)|ϕ(ξ)|2] = |ϕ(ξ)|2 + 2Re[((ξ − λ)(ϕ)′(ξ))ϕ(ξ)] = 2Re[(uϕ)′(ξ)ϕ(ξ)]− |ϕ(ξ)|2. (4.2.17)

When λ < 0, it suffices to integrate equation (4.2.17) on [0,+∞) and use the Plancherel formula∫ +∞

0

(uϕ)′(ξ)ϕ(ξ)dξ = −2πi

∫Rxu(x)|ϕ(x)|2dx.

We also use the fact (ξ − λ)|ϕ(ξ)|2 = uϕ(ξ)ϕ(ξ)→ 0, as ξ → +∞. Thus,

λ|ϕ(0)|2 =

∫ +∞

0

d

dξ[(ξ − λ)|ϕ(ξ)|2]dξ = 4πIm

∫Rxu(x)|ϕ(x)|2dx−

∫ +∞

0

|ϕ(ξ)|2dξ = −2π‖ϕ‖2L2(R).

When λ > 0, there may be some problem of derivability of ϕ at ξ = λ. We replace the integral∫ +∞

0by

two integrals∫ λ−ε

0and

∫ +∞λ+ε

, for some ε ∈ (0, λ). Set

I(ε) :=λ|ϕ(0)|2 − ε|ϕ(λ− ε)|2 − ε|ϕ(λ+ ε)|2

=2Re

(∫ +∞

0

(uϕ)′(ξ)ϕ(ξ)dξ −∫ λ+ε

λ−ε(uϕ)′(ξ)ϕ(ξ)dξ

)−∫ +∞

0

|ϕ(ξ)|2dξ +

∫ λ+ε

λ−ε|ϕ(ξ)|2dξ

Thanks to the continuity of ϕ on R+, we have λ|ϕ(0)|2 = limε→0+ I(ε) = −2π‖ϕ‖2L2(R).

When λ = 0, we use the same idea and integrate (4.2.17) over interval [ε,+∞), for some ε > 0. Then


J (ε) := −ε|ϕ(ε)|2 = 2Re

∫ +∞

ε

(uϕ)′(ξ)ϕ(ξ)dξ −∫ +∞

ε

|ϕ(ξ)|2dξ → 0,

as ε→ 0. So we always have

− 2π‖ϕ‖2L2(R) = λ|ϕ(0)|2, if ϕ ∈ Ker(λ− Lu). (4.2.18)

As a consequence Lu has only negative eigenvalues, if the real-valued function u ∈ L2(R, (1 + x2)dx).Finally we use uϕ(0) = −λϕ(0) to get identity (4.2.1). If λ ∈ σpp(Lu) and ϕ ∈ Ker(λ−Lu)\0, we wantto prove that

ξ 7→ (1 + |ξ|)∂ξϕ(ξ) ∈ L2(0,+∞). (4.2.19)

In fact, since ϕ ∈ H1+ → L∞(R) and u ∈ L2(R, (1+x2)dx), we have uϕ = u∗ϕ

2π ∈ H1(R). Formula (4.2.15)

yields that ξ 7→ (|λ| + ξ)ϕ(ξ) ∈ L2(R) and we have ϕ ∈ L1(R). The hypothesis u ∈ L2(R, x2dx) impliesthat the convolution term xu ∗ ϕ ∈ L2(R). Since λ < 0, we obtain (4.2.19) by using formula (4.2.16).

4.2.3 Conservation laws

Proposition 4.2.8 and 4.2.10 are proved in this subsection. We begin with the following proposition.

Proposition 4.2.17. If u : t ∈ R 7→ u(t) ∈ H2(R,R) denotes the unique solution of the BO equation(4.1.1), then we have

∂tΠu(t) = Bu(t)(Πu(t)) + iL2u(t)(Πu(t)) ∈ L2

+. (4.2.20)

Demonstration. For every u ∈ H2(R,R), Bu is a bounded operator on both L2+ and H1

+, Πu ∈ D(Lu) =

H1+. We have u(−ξ) = u(ξ), u = Πu+Πu and |D|u = DΠu−DΠu. Since DΠu ∈ L2

−, we have Π(ΠuDΠu) =

Π(uDΠu). Thus the following two formulas hold,

Bu(Πu) = i(T|D|u − T 2u)(Πu) = i(Πu)(DΠu)− iΠ(uDΠu)− iT 2

u(Πu) = Πu∂xΠu−Π(u∂xΠu)− iT 2u(Πu),

iL2u(Πu) = iD2Πu− iTu(DΠu)− iD Tu(Πu) + iT 2

u(Πu) = −i∂2xΠu− Tu(∂xΠu)− ∂x[Tu(Πu)] + iT 2

u(Πu).

Then we add them together to get the following

Bu(Πu) + iL2u(Πu) = −i∂2

xΠu− 2Π[Πu∂xΠu+ Πu∂xΠu+ Πu∂xΠu]

Finally we replace u by u(t), where u : t ∈ R 7→ u(t) ∈ H2(R,R) solves equation (4.1.1) to obtain(4.2.20).

Proof of proposition 4.2.8. It suffices to prove (4.2.7) in the case u0 ∈ H∞(R,R). Then we use the densityargument and the continuity of the flow map

u0 ∈ Hs(R) 7→ u ∈ C([−T, T ];Hs(R)) with T > 0, s ≥ 0,

in proposition 4.2.5. We choose u = u(t) ∈ H∞(R,R) =⋂s≥0H

s(R,R), so the functions LnuΠu, ∂tΠuand ∂t(L

nu)Πu = [Bu, L

nu]Πu are in H∞(R,C). Thus

∂tEn(u) = 2Re〈LnuΠu, ∂tΠu〉L2 + 〈∂t(Lnu)Πu,Πu〉L2 .


Since Bu + iL2u is skew-adjoint, we use formula (4.2.20) to get the following

2Re〈LnuΠu, ∂tΠu〉L2 = 〈[Lnu, Bu + iL2u]Πu,Πu〉L2 = 〈[Lnu, Bu]Πu,Πu〉L2 .

Since (Lnu, Bu) is also a Lax pair of the Benjamin–Ono equation (4.1.1), we have

∂tEn(u) = 〈([Lnu, Bu] + ∂t(Lnu))Πu,Πu〉L2 = 0.

In the case n = 1, we assume that u ∈ H1(R,R). Since u = Πu+Πu, |D|u = DΠu−DΠu and∫R(Πu)3 = 0,

we have 〈|D|u, u〉L2 = 2〈DΠu,Πu〉L2 and∫R u

3 = 3∫R(Πu+ Πu)|Πu|2 = 3

∫R u|Πu|

2. In the general case

u ∈ H 12 (R,R), we use the density argument.

Proof of proposition 4.2.10. Let u : t ∈ R 7→ u(t) ∈ H∞(R,R) solve equation (4.1.1). Since σ(−Lu(t)) =σ(−Lu(0)) by proposition 4.2.14, the operator λ+ Lu(t) ∈ B(H1

+, L2+) is invertible and we have

∂tHλ(u) = 2Re〈(Lu + λ)−1Πu, ∂tΠu〉L2 − 〈(Lu + λ)−1∂tLu(Lu + λ)−1Πu,Πu〉L2 . (4.2.21)

Formula (4.2.20) yields that

2Re〈(Lu + λ)−1Πu, ∂tΠu〉L2 = 〈[(Lu + λ)−1, Bu + iL2u]Πu,Πu〉L2 = 〈[(Lu + λ)−1, Bu]Πu,Πu〉L2 ,

〈[(Lu + λ)−1, Bu]Πu,Πu〉L2 = 〈(Lu + λ)−1[Bu, Lu + λ](Lu + λ)−1Πu,Πu〉L2 .

Then ∂tLu = [Bu, Lu] yields that ∂tHλ(u(t)) = 0.

4.2.4 Lax pair formulation

In this subsection, we prove proposition 4.2.12 and 4.2.7. The Hankel operators whose symbols are inL2(R)

⋃L∞(R) will be used to calculate the commutators of Toeplitz operators. We notice that the Hankel

operators are C-anti-linear and the Toeplitz operators are C-linear. For every symbol v ∈ L2(R)⋃L∞(R),

we define its associated Hankel operator to be Hv(h) = Thv = Π(vh), for every h ∈ H1+. If v ∈ L∞(R),

then Hv : L2+ → L2

+ is a bounded operator. If v ∈ L2(R), then Hv may be an unbounded operator onL2

+ whose domain of definition contains H1+. For every b ∈ H1(R), we have ‖Tb(h)‖H1 + ‖Hb(h)‖H1 .

‖b‖H1‖h‖H1 , for every h ∈ H1+, so both Tb and Hb are bounded on L2

+ and on H1+.

Lemma 4.2.18. For every v, w ∈ L2+

⋂L∞(R) and u ∈ L2(R), we have

[Tv, Tw] = −Hv Hw ∈ B(L2+). (4.2.22)

If w ∈ H1+ in addition, then we have Tu(w) ∈ L2

+ and

HTuw = Tw HΠu +Hw Tu = Tu Hw +HΠu Tw ∈ B(H1+, L

2+). (4.2.23)

Demonstration. For every v, w ∈ L2+

⋂L∞(R) and h ∈ L2

+, we have wh = Π(wh) + Π(wh) ∈ L2+. Thus,

[Tv, Tw]h = Π(vΠ(wh)− wΠ(vh)) = Π(vwh− vΠ(wh)− vwh) = −Π(vΠ(wh)) = −Hv Hw(h) ∈ L2+.

Given u ∈ L2(R) and w ∈ H1+, for every h ∈ H1

+, we have wh = Π(wh) + Π(wh) ∈ H1(R) and

Hw(h), Tw(h) ∈ H1+. So Π(uΠ(wh)) = Π(Π(wh)Πu) = HΠu Tw(h) ∈ L2

+ and we have

HTuw(h) = Π(Π(uw)h) = Π(uwh) = Π(uΠ(wh) + uΠ(wh)) = (Tu Hw +HΠu Tw)(h) ∈ L2+.


Similarly, h ∈ H1+ yields that uh = Π(uh) + Π(uh) ∈ L2(R) and Π(uh) = Π(hΠu) = HΠu(h) ∈ L2

+. Thus,

HTuw(h) = Π(wuh) = Π(wΠ(uh) + wΠ(uh)) = (Tw HΠu +Hw Tu)(h) ∈ L2+.

Lemma 4.2.19. Given (λ, u) ∈ X given in definition 4.2.9, set wλ(u) = (Lu + λ)−1 Π(u) ∈ H1

+, then

[D− Tu, Twλ(u)Twλ(u) + Twλ(u) + Twλ(u)] = TD[|wλ(u)|2+wλ(u)+wλ(u)] ∈ B(H1+, L

2+). (4.2.24)

Demonstration. We use abbreviation wλ := wλ(u) ∈ H1+, then wλ ∈ H1

−. If f+, g+ ∈ H1+ and f−, g− ∈

H1−, then we have [Tf+ , Tg+ ] = [Tf− , Tg− ] = 0, because for every h ∈ L2

+, we have

Tf+ [Tg+(h)] = f+g+h = Tg+ [Tf+(h)], ∀h ∈ L2+.

and Tf− [Tg−(h)] = Π(f−Π(g−h)) = Π(f−g−h) = Π(g−Π(f−h)) = Tg− [Tf−(h)]. Since Πu ∈ L2+ and

Πu ∈ L2−, we use Leibnitz’s rule and formula (4.2.22) to obtain that

[D− Tu, Twλ + Twλ ] =TDwλ + TDwλ − [Tu, Twλ ]− [Tu, Twλ ]

=TDwλ + TDwλ − [TΠu, Twλ ]− [TΠu, Twλ ]

=TDwλ + TDwλ −HwλHΠu +HΠuHwλ .

(4.2.25)

Similarly, formula (4.2.22) implies that

[Tu, TwλTwλ ] =[Tu, Twλ ]Twλ + Twλ [Tu, Twλ ]

=[TΠu, Twλ ]Twλ + Twλ [TΠu, Twλ ]

=HwλHΠuTwλ − TwλHΠuHwλ .

(4.2.26)

For every h ∈ H1+, since wλ,Dwλ ∈ L2

−, we have

[D, TwλTwλ ]h =[D, Twλ ]Twλh+ Twλ [D, Twλ ]h

=TDwλ(Twλh) + Twλ(TDwλh)

=Π[DwλΠ(wλh) + wλΠ(Dwλh)] = Π[(wλDwλ + wλDwλ)h] ∈ L2+.

So [D, TwλTwλ ] = TD|wλ|2 ∈ B(H1+, L

2+). We use formula (4.2.22) and Leibnitz’s Rule to obtain that

[D, TwλTwλ ] = [D, TwλTwλ ]− [D, H2wλ

] = TD|wλ|2 −HDwλHwλ +HwλHDwλ (4.2.27)

Recall that wλ = (λ+ Lu)−1Πu, then we have

Dwλ = Tu(wλ)− λwλ + Πu. (4.2.28)

The formula (4.2.23) and (4.2.28) yield that

HDwλ − TwλHΠu = HTuwλ − λHwλ +HΠu − TwλHΠu = HwλTu − λHwλ +HΠu (4.2.29)

and

HDwλ −HΠuTwλ = HTuwλ − λHwλ +HΠu −HΠuTwλ = TuHwλ − λHwλ +HΠu. (4.2.30)


We use formulas (4.2.26), (4.2.27), (4.2.29) and (4.2.30) to get the following formula

[D− Tu, TwλTwλ ]

=TD|wλ|2 − (HDwλ − TwλHΠu)Hwλ +Hwλ(HDwλ −HΠuTwλ)

=TD|wλ|2 − (HwλTuHwλ − λH2wλ

+HΠuHwλ) + (HwλTuHwλ − λH2wλ

+HwλHΠu)

=TD|wλ|2 −HΠuHwλ +HwλHΠu

(4.2.31)

At last, we combine formulas (4.2.25) and (4.2.31) to obtain formula (4.2.24).

End of the proof of proposition 4.2.12. Since L : u ∈ L2(R,R) 7→ Lu = D− Tu ∈ B(H1+, L

2+) is R-affine,

for every u : t 7→ u(t) ∈ L2(R,R) solves equation (4.2.9), we have

d

dt(L u)(t) = −T∂tu(t) = −iTD(wλ(u(t))wλ(u(t))+wλ(u(t))+wλ(u(t))).

Thus the Lax equation (4.2.10) is equivalent to identity (4.2.24) in lemma 4.2.19.

The proof of proposition 4.2.7 can be found in Gerard–Kappeler [54], Wu [146] etc. In order to make thispaper self contained, we recall it here.

Proof of proposition 4.2.7. Since the Lax map L : u ∈ H2(R,R) 7→ D− Tu ∈ B(H1+, L

2+) is R-affine,

d

dt(L u)(t) = −T∂tu(t) = −TH∂2

xu(t)−∂x(u(t)2).

It suffices to prove [Bu, Lu] + TH∂2xu−∂x(u2) = 0 for every u ∈ H2(R,R).

In fact, u is real-valued, we have u(−ξ) = u(ξ), u = Πu + Πu and |D|u = DΠu − DΠu. Since both Tuand Bu are bounded operators L2

+ → L2+ and bounded operators H1

+ → H1+ , their Lie Bracket [Bu, Lu]

is given by

[Bu, Lu]f =−Π(f∂x|D|u) + iΠ[uΠ(f |D|u)− |D|uΠ(uf)] + Π[∂xuΠ(uf) + uΠ(f∂xu)]

=−Π(fH∂2xu) + I1 + I2 ∈ L2

+,(4.2.32)

for every f ∈ H1+, where the terms I1 and I2 are given by

I1 :=iΠ[uΠ(f |D|u)− |D|uΠ(uf)]

=Π[fΠu∂xΠu+ fΠu∂xΠu]−ΠuΠ(f∂xΠu)−Π(fΠu)∂xΠu+ Π[Π(fΠu)∂xΠu−ΠuΠ(f∂xΠu)],

I2 :=Π[∂xuΠ(uf) + uΠ(f∂xu)] = Π(fΠu)∂xΠu+ ΠuΠ(f∂xΠu) + Π(ΠuΠ(f∂xΠu))

+ 2fΠu∂xΠu+ Π[fΠu∂xΠu+ fΠu∂xΠu+ Π(fΠu)∂xΠu].

If h1 ∈ H1− and h2 ∈ L2

−, then h1h2 ∈ L2−. Since ∂xΠu ∈ L2

−, we have Π[Π(fΠu)∂xΠu] = Π[fΠu∂xΠu].Thus

I1 + I2 = 2fΠu∂xΠu+ 2Π[fΠu∂xΠu+ fΠu∂xΠu+ Π(fΠu)∂xΠu] = Π[f∂x(u2)] ∈ H1+. (4.2.33)

Formulas (4.2.32) and (4.2.33) yield that [Bu, Lu]f = Π[f(∂x(u2)− H∂2xu)]. Thus equation (4.2.3) holds

along the evolution of equation (4.1.1).


Remark 4.2.20. As indicated in Gerard–Kappeler [54], there are many choices of the operator Bu. Wecan replace Bu by any operator of the form Bu + Pu such that Pu is a skew-adjoint operator commutingwith Lu. For instance, we set Cu := Bu + iL2

u and we obtain Cu = iD2 − 2iDTu + 2iTDΠu. So (Lu, Cu)is also a Lax pair of the BO equation (4.1.1). The advantage of the operator Bu = i(T|D|u − T 2

u) is thatBu : L2

+ → L2+ is bounded if u is sufficiently regular. For instance, u ∈ H2(R,R).

4.3 The action of the shift semigroup

In this section, we introduce the semigroup of shift operators (S(η)∗)η≥0 acting on the Hardy space L2+

and classify all finite-dimensional translation-invariant subspaces of L2+.

For every η ≥ 0, we define the operator S(η) : L2+ → L2

+ such that S(η)f = eηf , where eη(x) = eiηx. Itsadjoint is given by S(η)∗ = Te−η . We have

S(η)∗ Lu S(η) = Lu + ηIdL2+, ∀η ≥ 0.

Since ‖S(η)∗‖B(L2+) = ‖S(η)‖B(L2

+) = 1, (S(η)∗)η≥0 is a contraction semi-group. Let −iG be its infinite-

simal generator, i.e. Gf = i ddη

∣∣∣η=0+

S(η)∗f ∈ L2+, ∀f ∈ D(G), where

D(G) :=f ∈ L2+ : f|R+

∈ H1(0,+∞), (4.3.1)

because limε→0 ‖ψ−τεψε −∂xψ‖L2(0,+∞) = 0, where τεψ(x) = ψ(x−ε) and ψ ∈ H1(0,+∞). Every functionf ∈ D(G) has bounded Holder continuous Fourier transform by Morrey’s inequality and Sobolev extension

operator yields the existence of f(0+) := limξ→0+ f(ξ). The operator G is densely defined and closed.The Fourier transform of Gf is given by

Gf(ξ) = i∂ξ f(ξ), ∀f ∈ D(G), ∀ξ > 0. (4.3.2)

In accordance with the Hille–Yosida theorem, we have

(−∞, 0) ⊂ ρ(iG), ‖(G− λi)−1‖B(L2+) ≤ λ−1, ∀λ > 0. (4.3.3)

Lemma 4.3.1. For every b ∈ L2(R)⋂L∞(R), we have Tb(D(G)) ⊂ D(G) and the following identity

[G,Tb]ϕ =iϕ(0+)

2πΠb (4.3.4)

holds for every ϕ ∈ D(G).

Demonstration. For every η > 0 and ϕ ∈ D(G), both S(η)∗ and Tb are bounded operators, so we have

([S(η)∗

η , Tb]ϕ)∧

(ξ) =b ∗ ϕ(ξ + η)− b ∗ [1R+

(τ−ηϕ)](ξ)

2πη=

1

2πη

∫ ξ+η

ξ

b(ζ)ϕ(ξ + η − ζ)dζ, ∀ξ > 0,

4.3. THE ACTION OF THE SHIFT SEMIGROUP 103

where τ−ηϕ(x) = ϕ(x+ η), for every x ∈ R. Then we change the variable ζ = ξ + tη, for 0 ≤ t ≤ 1,([S(η)∗−Id

L2+

η , Tb]ϕ

)∧(ξ) =

1

2π

∫ 1

0

b(ξ + tη)ϕ((1− t)η)dt = aη b(ξ) + φη(ξ), ∀ξ > 0, (4.3.5)

where aη := 12π

∫ 1

0ϕ((1− t)η)dt ∈ C and φη ∈ L2

+ such that

φη(ξ) :=1

2π

∫ 1

0

[b(ξ + tη)− b(ξ)]ϕ((1− t)η)dt, ∀ξ > 0.

Since ϕ|R+ ∈ H1(0,+∞), ϕ is bounded and limη→0+ ϕ(η) = ϕ(0+), Lebesgue’s dominated convergence

theorem yields that limη→0+ aη = ϕ(0+)2π . Since b ∈ L2(R), we have limε→0 ‖τεb − b‖L2 = 0. By using

Cauchy–Schwarz inequality and Fubini’s theorem, we have

‖φη‖2L2 . ‖ϕ‖2L∞∫ 1

0

∫ +∞

0

|b(ξ + tη)− b(ξ)|2dξdt = ‖ϕ‖2L∞∫ 1

0

‖τ−tη b− b‖2L2dt→ 0,

when η → 0+, by Lebesgue’s dominated convergence theorem. Thus (4.3.5) implies that

[S(η)∗−Id

L2+

η , Tb]ϕ = aηΠb+ φη →ϕ(0+)

2πΠb, in L2

+, when η → 0+.

Since ϕ ∈ D(G) and Tb is bounded, we have 1ηTb[(S(η)∗ − IdL2

+)ϕ]→ (TbG)ϕ in L2

+, consequently

1η (S(η)∗ − IdL2

+)(Tbϕ)→ (TbG)ϕ+

ϕ(0+)

2πin L2

+, when η → 0+.

So Tbϕ ∈ D(G) and (4.3.4) holds.

The following scalar representation theorem of Lax [96] allows to classify all translation-invariant sub-spaces of the Hardy space L2

+, which plays the same role as Beurling’s theorem in the case of Hardy spaceon the circle (see Theorem 17.21 of Rudin [124]).

Theorem 4.3.2 (Beurling–Lax). Every nonempty closed subspace of L2+ that is invariant under the

semigroup of shift operators (S(η))η≥0 is of the form ΘL2+, where Θ is a holomorphic function in the

upper-half plane C+ = z ∈ C : Imz > 0. We have |Θ(z)| ≤ 1, for all z ∈ C+ and |Θ(x)| = 1, ∀x ∈ R.Moreover, Θ is uniquely determined up to multiplication by a complex constant of absolute value 1.

The following lemma classifies all finite-dimensional subspaces that are invariant under the semi-group(S(η)∗)η≥0, which is a weak version of theorem 4.3.2.

Lemma 4.3.3. Let M be a subspace of L2+ of finite dimension N = dimCM ≥ 1 and G(M) ⊂ M .

Then there exists a unique monic polynomial Q ∈ CN [X] such that Q−1(0) ⊂ C− and M =C≤N−1[X]

Q ,

where C≤N−1[X] denotes all the polynomials whose degrees are at most N − 1. Q is the characteristicpolynomial of the operator G|M .


Demonstration. We set M = f ∈ L2(0,+∞) : f ∈ M, then dimC M = N . Since Gf = i∂ξ f on R\0,the restriction G|M is unitarily equivalent to i∂ξ|M by the renormalized Fourier–Plancherel transforma-

tion. So the characteristic polynomial Q ∈ CN [X] of i∂ξ|M is well defined, let β1, β2, · · · , βn ⊂ Cdenote the distinct roots of Q and mj denote the multiplicity of βj , we have

∑nj=1mj = N and

Q(z) = det(z − i∂ξ|M ) =

n∏j=1

(z − βj)mj = zd +

N−1∑k=0

ckzk, ck ∈ C.

The Cayley–Hamilton theorem implies that Q(i∂ξ) = 0 on the subspace M . If ψ ∈ M ⊂ L2(0,+∞), thenψ is a weak-solution of the following differential equation

i−NQ(−D)ψ = ∂Nξ ψ +

N−1∑k=0

ik−Nck∂kξψ = 0 on (0,+∞), ψ ≡ 0 on (−∞, 0). (4.3.6)

The differential operator Q(−D) is elliptic is on the open interval (0,+∞) in the following sense : thesymbol of the principal part of Q(−D), denoted by aQ : (x, ξ) ∈ (0,+∞)× R 7→ (−ξ)N , does not vanishexcept for ξ = 0. Theorem 8.12 of Rudin [125] yields that ψ is a smooth function. The solution space

Sol(4.3.6) = SpanCfj,l0≤l≤mj−1,1≤j≤n, fj,l(ξ) = ξle−iβjξ1R+. (4.3.7)

has complex dimension∑nj=1mj = N so we have Sol(4.3.6) = M ⊂ L2

+ and Imβj = Re(iβj) > 0 and

Q−1(0) ⊂ C−. At last, we have M = SpanCfj,l0≤l≤mj−1,1≤j≤n =C≤N−1[X]

Q , where

fj,l(x) =l!

2π[(−i)(x− βj)]l+1, ∀x ∈ R. (4.3.8)

The uniqueness is obtained by identifying all the roots.

4.4 The manifold of multi-solitons

This section is dedicated to a geometric description of the multi-soliton subsets in definition 4.1.1. Wegive at first a polynomial characterization then a spectral characterization for the real analytic symplecticmanifold of N -solitons in order to prove the global well-posedness of the BO equation with N -solitonsolutions (4.1.6).

Recall that every N -soliton has the form u(x) =∑Nj=1Rη−1

j(x−xj) =

∑Nj=1

2ηj(x−xj)2+η2

jwith xj ∈ R and

ηj > 0, then we have the following polynomial characterization of the N -solitons.

Proposition 4.4.1. The N -soliton subset UN ⊂ H∞(R,R)⋂L2(R, x2dx) and UN

⋂UM = ∅, for every

M 6= N . Moreover, each of the following three properties implies the others :

(a). u ∈ UN .(b). There exists a unique monic polynomial Qu ∈ CN [X] whose roots are contained in the lower half-

plane C− such that Πu = iQ′uQu

.

(c). There exists Q ∈ CN [X] such that Q−1(0) ⊂ C− and Πu = iQ′

Q .

4.4. THE MANIFOLD OF MULTI-SOLITONS 105

Demonstration. We only prove the uniqueness in (a)⇒ (b). If Πu = iQ′uQu

= iP′

P , then we have(PQu

)′≡ 0

on R. Since P and Qu are monic polynomials, we have P = Qu. The other assertions are consequencesof u = Πu+ Πu.

Definition 4.4.2. For every u ∈ UN , the unique monic polynomial Qu ∈ CN [X] given by proposition4.4.1 is called the characteristic polynomial of u. Its roots are denoted by zj = xj − iηj ∈ C−, for1 ≤ j ≤ N (not necessarily all distinct). The unordered N -uplet cl(z1, z2, · · · , zN ) ∈ CN−/SN is calledthe translation–scaling parameters of u, where CN−/SN denotes the orbit space of the action (4.6.3) ofsymmetric group SN on CN− .

The real analytic structure of UN is given in the next proposition.

Proposition 4.4.3. Equipped with the subspace topology of L2(R,R), the subset UN is a connected, realanalytic, embedded submanifold of the R-Hilbert space L2(R,R) and dimR UN = 2N . For every u ∈ UN ,its translation–scaling parameters are denoted by cl(x1 − iη1, x2 − iη2, · · · , xN − iηN ) for some xj ∈ Rand ηj > 0, then the tangent space to UN at u is given by

Tu(UN ) =

N⊕j=1

(Rfuj⊕

Rguj ), where fuj (x) =2[(x−xj)2−η2

j ]

[(x−xj)2+η2j ]2, guj (x) =

4ηj(x−xj)[(x−xj)2+η2

j ]2. (4.4.1)

Every tangent space Tu(UN ) is contained in the auxiliary space T defined by (4.1.5) in which the global2-covector ω ∈ Λ2(T ∗) is well defined. Recall that the nondegenerate 2-form ω on UN is given by

ωu(h1, h2) = ω(h1, h2) =i

2π

∫R

h1(ξ)h2(ξ)

ξdξ, ∀h1, h2 ∈ Tu(UN ). (4.4.2)

It provides the symplectic structure of the manifold UN .

Proposition 4.4.4. The nondegenerate real analytic 2-form ω is closed on UN . Endowed with thesymplectic form ω, the real analytic manifold (UN , ω) is a symplectic manifold.

For every smooth real-valued function f : UN → R, let Xf ∈ X(UN ) denote its Hamiltonian vector field,defined as follows : for every u ∈ UN and h ∈ Tu(UN ),

df(u)(h) = 〈h,∇uf(u)〉L2 =i

2π

∫R

h(ξ)

ξiξ(∇uf(u))∧(ξ)dξ = ωu(h,Xf (u)).

Then we haveXf (u) = ∂x∇uf(u) ∈ Tu(UN ), ∀u ∈ UN . (4.4.3)

Remark 4.4.5. There are several ways to prove the simple connectedness of UN . Firstly, it is irrelevantto the proof of proposition 4.5.16. In subsection 4.5.4, we show that the real analytic manifold UN isdiffeomorphic to some open convex subset of R2N , hence UN is homotopy equivalent to a one-point space.On the other hand, the simple connectedness of the Kahler manifold Π(UN ) can be directly obtained fromits construction (see proposition 4.6.5).

Then, we return back to spectral analysis in order to establish a spectral characterization of the manifold

UN . For every monic polynomial Q ∈ CN [X] with roots in C−, we set Θ = ΘQ := QQ ∈ Hol(C+), where

Q(x) :=

N−1∑j=0

ajxj + xN , if Q(x) =

N−1∑j=0

ajxj + xN .


Then Θ is an inner function on the upper half-plane C+, because |Θ| ≤ 1 on C+ and |Θ| = 1 on R.Recall the shift operator S(η) : L2

+ → L2+ defined in section 4.3, we have S(η)[Θh] = Θ[S(η)h], for every

h ∈ L2+, so ΘL2

+ is a closed subspace of L2+ that is invariant by the semigroup (S(η))η≥0 (see also the

Beurling–Lax theorem 4.3.2 of the complete classification of the translation-invariant subspaces of theHardy space L2

+). We define KΘ to be the orthogonal complement of ΘL2+, thus

L2+ = ΘL2

+

⊕KΘ, S(η)∗(KΘ) ⊂ KΘ and G(D(G)

⋂KΘ) ⊂ KΘ. (4.4.4)

where the infinitesimal generator G is defined in (4.3.2). Recall that the C-vector space C≤N−1[X] consists

of all polynomials with complex coefficients of degree at most N − 1. SoC≤N−1[X]

Q is an N -dimensional

subspace of L2+.

The Lax map L : u ∈ L2(R,R) 7→ Lu = D − Tu ∈ B(H1+, L

2+) is R-affine. Defined on D(Lu) = H1

+, theunbounded self-adjoint operator Lu has the following spectral decomposition

L2+ = Hac(Lu)

⊕Hsc(Lu)

⊕Hpp(Lu). (4.4.5)

The following proposition gives an identification of these subspaces in the spectral decomposition (4.4.5).

Proposition 4.4.6. If u ∈ UN , then Lu has exactly N simple negative eigenvalues. Let Qu denote the

characteristic polynomial of the N -soliton u given in definition 4.4.2 and Θu := ΘQu = QuQu

denote theassociated inner function. Then we have the following identification,

Hac(Lu) = ΘuL2+, Hsc(Lu) = 0, Hpp(Lu) = KΘu =

C≤N−1[X]

Qu. (4.4.6)

For every u ∈ UN , we have the following spectral decomposition of Lu :

σ(Lu) = σac(Lu)⋃σsc(Lu)

⋃σpp(Lu), where σac(Lu) = [0,+∞), σsc(Lu) = ∅ (4.4.7)

and σpp(Lu) = λu1 , λu2 , · · · , λuN consists of all eigenvalues of Lu. Proposition 4.2.2 yields that Lu is

bounded from below and −C2

4 ‖u‖2L2 ≤ λu1 < · · · < λuN < 0, where C = inff∈H1

+\0‖|D|

14 f‖L2

‖f‖L4denotes the

Sobolev constant. Hence the min-max principle (Theorem XIII.1 of Reed–Simon [122]) yields that

λun = supdimC F=n−1

I(F,Lu), I(F,Lu) = inf〈Luh, h〉L2 : h ∈ H1+

⋂F⊥, ‖h‖L2 = 1 (4.4.8)

where, the above supremum, F describes all subspaces of L2+ of complex dimension n, 1 ≤ n ≤ N . When

n ≥ N + 1, supdimC F=n I(F,Lu) = inf σess(Lu) = 0. Proposition 4.2.3 and corollary 4.2.4 yield that thereexist eigenfunctions ϕj : u ∈ UN 7→ ϕuj ∈Hpp(Lu) such that

Ker(λuj − Lu) = Cϕuj , ‖ϕuj ‖L2 = 1, 〈ϕuj , u〉L2 =

∫Ruϕuj =

√2π|λuj |, (4.4.9)

for every j = 1, 2, · · · , N . Then ϕu1 , ϕu2 , · · · , ϕuN is an orthonormal basis of the subspace Hpp(Lu). Wehave the following result.

Proposition 4.4.7. For every j = 1, 2, · · · , N , the j th eigenvalue λj : u ∈ UN 7→ λuj ∈ R is realanalytic.


We refer to proposition 4.4.14 and formula (4.4.4) to see that the subspace Hpp(Lu) ⊂ D(G) is invariantby G. The matrix representation of G|Hpp(Lu) with respect to the orthonormal basis ϕu1 , ϕu2 , · · · , ϕuN isgiven in proposition 4.5.4. Then the following theorem gives the spectral characterization for N -solitons.

Theorem 4.4.8. A function u ∈ UN if and only if u ∈ L2(R, (1+x2)dx) is real-valued, dimC Hpp(Lu) =N and Πu ∈Hpp(Lu). Moreover, we have the following inversion formula

Πu(x) = id

dx det(x−G|Hpp(Lu))

det(x−G|Hpp(Lu)), ∀x ∈ R. (4.4.10)

Then Qu in definition 4.4.2 is the characteristic polynomial of G|Hpp(Lu). The translation–scaling para-meters of u can be identified as the spectrum of G|Hpp(Lu). Finally the invariance of UN under the BOflow is obtained by its spectral characterization, so we have the global well-posedness of the BO equationin the N -soliton manifold (4.1.6).

Proposition 4.4.9. If u0 ∈ UN , we denote by u : t ∈ R 7→ u(t) ∈ H∞(R,R) the solution of the BOequation (4.1.1) with initial datum u(0) = u0. Then u(t) ∈ UN , for every t ∈ R.

This section is organized as follows. The real analytic structure and the symplectic structure are given insubsection 4.4.1. Then the spectral decomposition of the Lax operator Lu and the real analyticity of itseigenvalues are given in subsection 4.4.2, for every u ∈ UN . The characterization theorem 4.4.8 is provedin subsection 4.4.3. Finally, we show the stability of UN under the BO flow in subsection 4.4.4.

4.4.1 Differential structure

The construction of real analytic structure and symplectic structure of UN is divided into three steps.Firstly, we describe the complex structure of Π(UN ). Then the Hermitian metric H for the complex ma-nifold Π(UN ) is introduced in (4.4.15) and we establish a real analytic diffeomorphism between UN andΠ(UN ). The third step is to prove dω = 0 on UN . Since ω = −Π∗(ImH), (Π(UN ),H) is a Kahler manifold.

Step I. The Viete map V : (β1, β2, · · · , βN ) ∈ CN 7→ (a0, a1, · · · , aN−1) ∈ CN is defined as follows

N∏j=1

(X − βj) =

N−1∑k=0

akXk +XN . (4.4.11)

Both addition and multiplication of two complex numbers are open continuous maps C2 → C, the Vietemap V : CN → CN is an open quotient map. So V(CN− ) is an open connected subset of CN (see also

proposition 4.6.5). With the subspace topology and the Hermitian form HCN (X,Y ) = 〈X,Y 〉CN = XTY ,the subset (V(CN− ),HCN ) is a connected Kahler manifold of complex dimension N .

Lemma 4.4.10. Equipped with the subspace topology of L2+, the subset Π(UN ) is a connected topological

manifold of complex dimension N and it has a unique complex analytic structure making it into anembedded submanifold of the C-Hilbert space L2

+. For every u ∈ UN , its translation–scaling parametersare denoted by cl(x1 − iη1, x2 − iη2, · · · , xN − iηN ), for some xj ∈ R and ηj > 0, then the tangent spaceto Π(UN ) at Πu is given by

TΠu(Π(UN )) =

N⊕j=1

Chuj , where huj (x) =1

(x− xj + ηji)2. (4.4.12)


Demonstration. We define ΓN : a = (a0, a1, · · · , aN−1) ∈ V(CN− ) 7→ Πu = iQ′

Q ∈ Π(UN ) ⊂ L2+ such that

Q(X) =

N−1∑k=0

akXk +XN .

The surjectivity of ΓN is given by the definition of UN . Since the monic polynomial Q is uniquelydetermined by u ∈ UN , the map ΓN is injective. For every h = (h0, h1, · · · , hN−1) ∈ CN , we have

dΓN (a0, a1, · · · , aN−1)h = iQH ′ −Q′H

Q2, where H(X) =

N−1∑k=0

hkXk.

If dΓN (a0, a1, · · · , aN−1)h = 0, then (HQ )′ ≡ 0. Since degH ≤ degQ − 1, we have H = 0. Thus

ΓN : V(CN− )→ L2+ is a complex analytic immersion. We claim that ΓN is a topological embedding.

In fact we set a(n) = (a(n)0 , a

(n)1 , · · · , a(n)

N−1) ∈ V(CN− ) such that

∂xQnQn

→ ∂xQ

Qin L2

+, as n→ +∞, where Qn(x) =

N−1∑j=0

a(n)j xj + xN , ∀x ∈ R.

Since a(n) ∈ V(CN− ), we have a(n)0 = Qn(0) 6= 0. For every x ∈ R, we have

Qn(x)

Qn(0)= exp(

∫ x

0

∂yQn(y)

Qn(y)dy)→ exp(

∫ x

0

∂yQ(y)

Q(y)dy) =

Q(x)

Q(0), as n→ +∞. (4.4.13)

Every coefficient of QnQn(0) converges to the corresponding coefficient of Q(x)

Q(0) . Since Qn, Q are monic, we

have limn→+∞1

Qn(0) = 1Q(0) and limn→+∞ a(n) = a. Then Γ−1

N : Π(UN ) ⊂ L2+ → V(CN− ) is continuous.

Since ΓN is a complex analytic embedding, with the subspace topology of L2+, there exists a unique

complex analytic structure making Π(UN ) = ΓN V(CN− ) into an embedded complex analytic submanifold

of L2+. The map ΓN : V(CN− ) → Π(UN ) is biholomorphic. Set u(x) =

∑Nj=1

2ηj(x−xj)2+η2

jfor some xj =

xj(u) ∈ R and ηj = ηj(u) > 0. Then every h ∈ TΠu(Π(UN )) is identified as the velocity of the smoothcurve c : t ∈ (−1, 1)→ Π(UN ) such that c(0) = Πu at t = 0. If we choose

c(t, x) =

N∑j=1

i

x− xj(t) + ηj(t)iwhere xj(t) ∈ R, ηj(t) > 0.

Then we have xj(0) = xj , ηj(0) = ηj and

h(x) = ∂t∣∣t=0

c(t, x) =

N∑j=1

η′j(0) + ix′j(0)

(x− xj + ηji)2. (4.4.14)

We have huj = Πfuj = −iΠguj and (huj )∧(ξ) = −2π1ξ≥0ξe−(ixj(u)+ηj(u))ξ. For every h ∈ TΠu(Π(UN )), we

have ξ 7→ ξ−1h(ξ) ∈ L2(R) (see also Hardy’s inequality (4.4.18)).

Step II. Given u ∈ UN , the Hermitian metric HΠu is defined as follows

HΠu(h1, h2) =

∫ +∞

0

h1(ξ)h2(ξ)

πξdξ, ∀h1, h2 ∈ TΠu(Π(UN )). (4.4.15)


The sesquilinear form HΠu is positive definite because HΠu(h, h) =∫ +∞

0|h(ξ)|2πξ dξ > 0, if h 6= 0. Hence

the smooth symmetric covariant 2-tensor field ReH is positive definite on Π(UN ), so (Π(UN ),ReH) is aRiemannian manifold of real dimension 2N .

We consider the R-linear isomorphism between the Hilbert spaces

Π : u ∈ L2(R,R) 7→ Πu ∈ L2+, f ∈ L2

+ 7→ 2Ref ∈ L2(R,R).

Then Π 2Re = IdL2+

and 2Re Π = IdL2(R,R) and ‖u‖L2 =√

2‖Πu‖L2 . Then UN = 2Re Π(UN ) is a

real analytic manifold of real dimension 2N . Furthermore we have fuj = 2Rehuj , guj = 2iRehuj and

2Re : TΠu(Π(UN ))→ Tu(UN ) (4.4.16)

is an R-linear isomorphism. Since H is Hermitian, the 2-form ω = −Π∗(ImH) is nondegenerate on UN .

Step III. We set E := L2(R,R)⋂L2(R, x2dx), Ec := u ∈ E :

∫R u = c, for every c ∈ R. Then

UN ⊂ E2πN , Tu(UN ) ⊂ T := E0, ∀u ∈ UN .

The nondegenerate 2-form ω can be extended to a 2-covector of the subspace T . Recall that

ω(h1, h2) =i

2π

∫R

h1(ξ)h2(ξ)

ξdξ, ∀h1, h2 ∈ T . (4.4.17)

If h ∈ T , then we have h(0) = 0 and h ∈ H1(R). Hence the Hardy’s inequality (see Brezis [16], Bahouri–Chemin–Danchin [5] etc.) yields that∫

R

|h(ξ)|2

|ξ|2dξ ≤ 4‖∂ξh‖2L2 =⇒ ξ 7→ h(ξ)

ξ∈ L2(R), (4.4.18)

so the 2-covector ω ∈ Λ2(T ∗) is well defined and ωu(h1, h2) = ω(h1, h2). For every smooth vector fieldX ∈ X(UN ), let Xyω ∈ Ω1(UN ) denote the interior multiplication by X, i.e. (Xyω)(Y ) = ω(X,Y ), forevery Y ∈ X(UN ). We shall prove that dω = 0 on UN by using Cartan’s formula :

LXω = Xy(dω) + d(Xyω). (4.4.19)

Proof of proposition 4.4.4. For any smooth vector field X ∈ X(UN ), let φ denote the smooth maximalflow of X. If t is sufficiently close to 0, then φt : u ∈ UN 7→ φ(t, u) ∈ UN is a local diffeomorphism by thefundamental theorem on flows (see Theorem 9.12 of Lee [98]). For every u ∈ UN , h1, h2 ∈ Tu(UN ), wecompute the Lie derivative of ω with respect to X,

(LXω)u(h1, h2) = limt→0

ωφt(u)(dφt(u)h1,dφt(u)h2)− ωu(h1, h2)

t

= limt→0

ω

(dφt(u)h1 − h1

t,dφt(u)h2

)+ limt→0

ω

(h1,

dφt(u)h2 − h2

t

).

Since limt→0dφt(u)hj−hj

t = dX(u)hj ∈ Tu(UN ), for every j = 1, 2, we have

(LXω)u(h1, h2) = ω(dX(u)h1, h2) + ω(h1,dX(u)h2) = (h1ω(X,h2)) (u)− (h2ω(X,h1)) (u).


We choose (V, xi) a smooth local chart for UN such that u ∈ V and the tangent vector hk has the

coordinate expression hk =∑2Nj=1 h

(j)k

∂∂xj

∣∣u, for some h

(j)k ∈ R, j = 1, 2 · · · , 2N and k = 1, 2. The tangent

vector hk can be identified as some locally constant vector field Yk ∈ X(UN ) defined by

Yk : v ∈ V 7→2N∑j=1

h(j)k

∂

∂xj

∣∣∣v∈ Tv(UN ), Yk : u 7→ (Yk)u = hk, k = 1, 2.

Then the vector field [Y1, Y2] vanishes in the open subset V . The exterior derivative of the 1-form β = Xyωis computed as dβ(Y1, Y2) = Y1 (β(Y2))− Y2 (β(Y1)) + β([Y1, Y2]). Thus

d(Xyω)u(h1, h2) = h1ωu(Xu, h2)− h2ωu(Xu, h1) + ωu(Xu, [Y1, Y2]u) = (LXω)u(h1, h2).

Then Cartan’s formula (4.4.19) yields that Xy(dω) = 0. Since X ∈ X(UN ) is arbitrary, we have dω = 0.As a consequence, the real analytic 2-form ω : u ∈ UN 7→ ω ∈ Λ2(T ∗) is a symplectic form.

Since ImH = (−2Re)∗ω, where −2Re : Π(UN ) → UN is a real analytic diffeomorphism, the associated2-form ImH is closed. So (Π(UN ),H) is a Kahler manifold. The simple connectedness of Π(UN ) is provedin subsection 4.6.1.

4.4.2 Spectral analysis II

We continue to study the spectrum of the Lax operator Lu introduced in definition 4.2.1. The generalcases u ∈ L2(R,R) and u ∈ L2(R, (1 + x2)dx) have been studied in subsection 4.2.2. We restrict ourstudy to the case u ∈ UN in this subsection. Let Q = Qu denote the characteristic polynomial of u and

Θ := QQ , KΘ = (ΘL2

+)⊥. Since Lu is an unbounded self-adjoint operator of L2+, we have the following

L2+ = ΘL2

+

⊕KΘ = Hac(Lu)

⊕Hsc(Lu)

⊕Hpp(Lu).

We shall at first identify those subspaces by proving proposition 4.4.6 and formula (4.4.7). Then we turnto study the real analyticity of each eigenvalue λj : u ∈ UN 7→ λuj ∈ R.

Proof of proposition 4.4.6. The first step is to prove KΘ =C≤N−1[X]

Q . In fact, for every h ∈ L2+ and

f = PQ ∈

C≤N−1[X]

Q , for some P ∈ C≤N−1[X], we have

〈f,Θh〉L2 =

∫R

P (x)Θ(x)h(x)

Q(x)dx =

∫R

P (x)h(x)

Q(x)dx = 〈P

Q, h〉L2 .

Since Q(x) =∏Nj=1(x− αj) with Im(αj) > 0, the meromorphic function P

Qhas poles in C+, so P

Q∈ L2

−.

Thus 〈f,Θh〉L2 = 〈PQ, h〉L2 = 0. Thus

C≤N−1[X]

Q ⊂ (ΘL2+)⊥ = KΘ.

Conversely, if f ∈ KΘ, then 〈Θ−1f, h〉L2 = 〈f,Θh〉L2 = 0, for every h ∈ L2+. Thus g := Q

Qf ∈ L2

−. It

suffices to prove that P := Qf = Qg ∈ C[X]. In fact,

Qf = Q(i∂ξ)f and supp(f) ⊂ [0,+∞) =⇒ supp(Qf) ⊂ [0,+∞).


Similarly, supp((Qg)∧) ⊂ (−∞, 0]. Thus supp(P ) ⊂ 0 and P is a polynomial. Since f = PQ ∈ L

2(R),

we have degP ≤ N − 1. So KΘ ⊂C≤N−1[X]

Q .

The second step is to prove Lu(ΘL2+) ⊂ ΘL2

+. Precisely, we have

Lu(Θh) = ΘDh, ∀h ∈ L2+. (4.4.20)

SinceC≤N−1[X]

Q ⊂ L2+, Θ = Q

Q and DΘΘ = DQ

Q− DQ

Q = iQ′

Q − iQ′

Q= Πu+ Πu = u on R, we have

Lu(Θh) = (D− Tu)(Θh) = ΘDh+ h(

DΘ− iQ′

Q Θ + iQ′

Q

)= ΘDh+ hΘ

(DΘΘ − i

Q′

Q + Q′

Q

)= ΘDh.

Recall that Lu = L∗u, so we have Lu(KΘ) ⊂ KΘ. Since dimCKΘ = N , corollary 4.2.4 yields that theHermitian matrix Lu|KΘ

has exactly N distinct eigenvalues. Hence KΘ ⊂Hpp(Lu).

On the other hand, we set UΘ : L2+ → ΘL2

+ such that UΘh = Θh. Thus ‖UΘ‖B(L2+,ΘL

2+) = 1 and

U−1Θ = U∗Θ : g ∈ ΘL2

+ 7→ Θ−1g ∈ L2+.

So UΘ : L2+ → ΘL2

+ is a unitary operator. UΘ(H1+) = ΘH1

+ = H1+

⋂ΘL2

+. Formula (4.4.20) yields that

U∗ΘLu|ΘL2+UΘ = D, UΘ[D(D)] = ΘH1

+ = H1+

⋂ΘL2

+ = D(Lu|ΘL2+

).

For every bounded Borel function f : R → C, we have f(Lu)UΘ = UΘf(D) by proposition 4.2.14. Wedenote by µψ = µLuψ the spectral measure of Lu associated to ψ ∈ L2

+, then ∀h ∈ L2+, we have∫

Rf(ξ)dµΘh(ξ) = 〈f(Lu)UΘh, UΘh〉L2 = 〈Θf(D)h,Θh〉L2 = 〈f(D)h, h〉L2 =

1

2π

∫ +∞

0

f(ξ)|h(ξ)|2dξ.

So dµΘh(ξ) =1R+|h(ξ)|2

2π dξ. The spectral measure µΘh is absolutely continuous with respect to the Le-besgue measure on R. Thus ΘL2

+ ⊂Hac(Lu) ⊂Hcont(Lu) = (Hpp(Lu))⊥ ⊂ ΘL2+ and (4.4.6) is obtained.

We have supp(µΘh) ⊂ [0,+∞), for every h ∈ L2+. ∀ξ ≥ 0, there exists h ∈ L2

+ such that h(ξ) 6= 0. So wehave σess(Lu) = σcont(Lu) = σac(Lu) = [0,+∞).

Before proving the real analyticity of each eigenvalue, we show its continuity at first.

Lemma 4.4.11. For every j = 1, 2, · · · , N , the j th eigenvalue λj : u ∈ UN 7→ λuj ∈ R is Lipschitzcontinuous on every compact subset of UN .

Demonstration. For every f ∈ H1(R), the Sobolev embedding ‖f‖L4 ≤ C‖|D| 14 f‖L2 yields that ∀u, v ∈UN , ∣∣〈Luh, h〉L2 − 〈Lvh, h〉L2

∣∣ ≤ ‖u− v‖L2‖h‖2L4 ≤ C‖u− v‖L2‖|D| 12h‖L2‖h‖L2 , ∀h ∈ H1+. (4.4.21)

Given j = 1, 2, · · · , N and a subspace F ⊂ L2+ with complex dimension j − 1, we choose

h ∈ F⊥⋂ j⊕

k=1

Ker(λuk − Lu) ⊂ H1+, ‖h‖L2 = 1, h =

j∑k=1

hkϕuk .


Then 〈Luh, h〉L2 =∑jk=1 |hk|2λuk ≤ λuj < 0, because λuk < λuk+1. We have the following estimate

‖D| 12h‖2L2 = 〈Dh, h〉L2 = 〈Luh, h〉L2 + 〈uh, h〉L2 ≤ λuj +‖u‖L2‖h‖2L4 ≤ C‖u‖L2‖|D| 12h‖L2‖h‖L2 . (4.4.22)

So estimates (4.4.21) and (4.4.22) yield that 〈Lvh, h〉L2 ≤ λuj + C2‖u‖L2‖u− v‖L2 . Since F is arbitrary,the max–min formula (4.4.8) implies that

|λuj − λvj | ≤ C2(‖u‖L2 + ‖v‖L2)‖u− v‖L2 .

Every compact subset K ⊂ UN is bounded in L2(R,R). Hence u ∈ K 7→ λuj ∈ R is Lipschitz continuous.

Proof of proposition 4.4.7. For every u ∈ UN , the Lax operator Lu has N negative simple eigenvalues,denoted by λu1 < λu2 < · · · < λuN < 0. Let Pju denotes the Riesz projector of the eigenvalue λuj and

D(z, ε) = η ∈ C : |η − z| < ε, C (z, ε) = ∂D(z, ε) = η ∈ C : |η − z| = ε, ∀z ∈ C, ε > 0.

Then there exists ε0 > 0 such that the family of closed discs D(λuj , ε0)1≤j≤N⋃D(0, ε0) is mutually

disjoint and for every j, k = 1, 2 · · · , N and any closed path Γuj (piecewise C1 closed curve) in D(λuj , ε0)with respect to which the eigenvalue λuj has winding number 1, we have

Pju =1

2πi

∮Γuj

(ζ − Lu)−1dζ, Pju Pju = Pju, Pjuϕuk = 1j=kϕuk . (4.4.23)

by Theorem XII.5 of Reed–Simon [122]. We choose Γuj to be the counterclockwise-oriented circle C (λuj , ε)

in (4.4.23) for some ε ∈ (0, ε0). We claim that ImPju = Ker(λuj − Lu) = Cϕuj .

It suffices to show that Pju|Hac(Lu) = 0. In fact the operator Pju = gλuj (Lu) is self-adjoint by Theorem

VIII.6 of Reed–Simon [123], where the real-valued bounded Borel function gλ : R→ R is given by

gλ(x) :=1

2πi

∮C (λ,ε)

(ζ − x)−1dζ = 1(λ−ε,λ+ε)(x), a.e. on R,

for every λ ∈ R. Since Pju(Hpp(Lu)) ⊂ Cϕuj ⊂Hpp(Lu), we have Pju(Hac(Lu)) ⊂Hac(Lu). Let µψ = µLuψdenote the spectral measure of Lu associated to the function ψ ∈Hac(Lu), whose support is included in[0,+∞) by formula (4.4.7), we have

〈Pjuψ,ψ〉L2 =1

2πi

∮C (λuj ,ε)

〈(ζ − Lu)−1ψ,ψ〉L2dζ =1

2πi

∫ +∞

0

(∮C (λuj ,ε)

(ζ − ξ)−1dζ

)dµψ(ξ) = 0.

Set ψ = Pjuψ ∈Hac(Lu), then ‖ψ‖2L2 = 〈Pjuψ, ψ〉L2 = 0. So the claim is obtained.

For every fixed j = 1, 2, · · ·N , we have λuj = Tr(Lu Pju). Since every eigenvalue λk : v ∈ UN 7→ λvk ∈ R iscontinuous, there exists an open subset V ⊂ UN containing u such that supv∈V sup1≤k≤N |λvk − λuk | <

ε03 .

We set ε = 2ε03 , then λvj ∈ D(λuj , ε)\D(λuk , ε0), for every v ∈ V and k 6= j. For example, in the next picture,

the dashed circles denote respectively C (λuj , ε0) and C (λuk , ε0) ; the smaller circles denote respectivelyC (λuj , ε) and C (λuk , ε) with j < k. The segments inside small circles denote the possible positions of λvjand λvk.


λuj

λvj λvk

λuk

0

Then σ(Lv)⋂D(λuj , ε0) = λvj and C (λuj , ε) is a closed path in D(λuj , ε0) with respect to which λvj has

winding number 1. Thus,

Pjv =1

2πi

∮C (λuj ,ε)

(ζ − Lv)−1dζ, λvj = Tr(Lv Pjv), ∀v ∈ V. (4.4.24)

Since v ∈ V 7→ Lv ∈ B(H1+, L

2+) is R-affine and i : A ∈ BI(H1

+, L2+) 7→ A−1 ∈ B(L2

+, H1+) is complex

analytic, where BI(H1+, L

2+) ⊂ B(H1

+, L2+) denotes the open subset of all bijective bounded C-linear

transformations H1+ → L2

+, we have the real analyticity of the following map

(ζ, v) ∈(D(λuj ,

3

4ε0)\D(λuj ,

1

2ε0)

)× V 7→ (ζ − Lv)−1 ∈ B(L2

+, H1+). (4.4.25)

Hence the maps Pj : v ∈ V 7→ Pjv ∈ B(L2+, H

1+) and λj : v ∈ V 7→ Tr(Lv Pjv) ∈ R are both real analytic

by composing (4.4.24) and (4.4.25).

Recall that Hpp(Lu) =C≤N−1[X]

Qu, where Qu denotes the characteristic polynomial of u ∈ UN whose zeros

are contained in C−, so Hpp(Lu) ⊂ D(G) is given by (4.3.7). We have the following consequence.

Corollary 4.4.12. For every j = 1, 2, · · · , N , the map fj : u ∈ UN 7→ 〈Gϕuj , ϕuj 〉L2 ∈ C is real analytic.

Demonstration. For every u, v ∈ UN , we have Pjvϕuj = 〈ϕuj , ϕvj 〉L2ϕvj . Since the Riesz projector Pj : v ∈UN 7→ Pjv ∈ B(L2

+, H1+) is real analytic in the proof of proposition 4.4.7 and ‖Pjuϕuj ‖L2 = 1, there exists

a neighbourhood of u, denoted by V, such that ‖Pjvϕuj ‖L2 > 12 for every v ∈ V and Pj : v ∈ V 7→ Pjv ∈

B(L2+, H

1+) can be expressed by power series. Then

ϕvj =Pjvϕuj

〈ϕuj , ϕvj 〉L2

, fj(v) =〈G Pjv(ϕuj ),Pjv(ϕuj )〉L2

‖Pjv(ϕuj )‖2L2

.

Hence the restriction fj : v ∈ V 7→ ‖Pjv(ϕuj )‖−2L2 〈G Pjv(ϕuj ),Pjv(ϕuj )〉L2 ∈ C is real analytic.


4.4.3 Characterization theorem

The characterization theorem 4.4.8 is proved in this subsection. The direct sense is given by proposition4.4.1 and proposition 4.4.6. Before proving the converse sense of theorem 4.4.8, we need the followinglemmas to prove the invariance of Hpp(Lu) under G, if u ∈ L2(R, (1+x2)dx) is real-valued, Πu ∈Hpp(Lu)and dimC Hpp(Lu) = N ≥ 1. The following lemma gives another version of formula of commutators (seealso lemma 4.3.1).

Lemma 4.4.13. For u ∈ L2(R, (1 + x2)dx), ϕ ∈ Ker(λ − Lu) for some λ ∈ σpp(Lu), then we haveϕ, Tuϕ,Luϕ ∈ D(G) and

[G,Tu]ϕ =iϕ(0+)

2πΠu, [G,Lu]ϕ = iϕ− iϕ(0+)

2πΠu. (4.4.26)

where Θ = Θu = QuQu

with Qu the characteristic polynomial of u.

Demonstration. In proposition 4.2.3, we have shown that uϕ ∈ H1(R), so (Tuϕ)∧ = uϕ1R+∈ H1(0,+∞)

and Tuϕ ∈ D(G). We recall the regularity of eigenfunctions (4.2.2)

Ker(λ− Lu) ⊂ ϕ ∈ H1+ : ϕ|R+

∈ C1(R+)⋂H1(R+) and ξ 7→ ξ[ϕ(ξ) + ∂ξϕ(ξ)] ∈ L2(R+). (4.4.27)

So Gϕ ∈ H1+ = D(Lu) = D(Tu). Moreover, we have ϕ is right-continuous at ξ = 0+ and ϕ ∈ C1(0,+∞).

The weak-derivative of ϕ is denoted by ∂wξ ϕ, δ0 denotes the Dirac measure with support 0, then

∂wξ ϕ = 1R∗+d

dξϕ+ ϕ(0+)δ0, ∂ξ(u ∗ ϕ) = ∂wξ (u ∗ ϕ) = u ∗ ∂wξ ϕ (4.4.28)

by lemma 4.2.15. Since ϕ = 1R∗+ ϕ a.e. in R and u ∈ H1(R), we have u ∗ Gϕ(ξ) = u ∗ [1R∗+Gϕ](ξ), for

every ξ > 0 and ([G,Tu]ϕ)∧(ξ) = i2π∂ξ(u ∗ ϕ)(ξ) − i

2π u ∗ [1R∗+ddξ f ](ξ) = i

2π ϕ(0+)u(ξ). Together with

(4.5.9), the first formula of (4.4.26) is obtained. Since Lu = D − Tu, we claim that Dϕ ∈ D(G). In fact,

∂ξ(Dϕ)∧(ξ) = ϕ(ξ) + ξ∂ξϕ(ξ), ∀ξ > 0. Thus (4.4.27) implies that Dϕ ∈ H1(0,+∞). Then

([G,D]ϕ)∧

(ξ) = i∂ξ(ξϕ)(ξ)− ξ · i∂ξϕ(ξ) = iϕ(ξ), ∀ξ > 0. (4.4.29)

So we have [∂x, G] = IdL2+

. The second formula of (4.4.26) holds.

Proposition 4.4.14. If u ∈ L2(R, (1 + x2)dx) is real-valued, dimC Hpp(Lu) = N ≥ 1 and Πu ∈Hpp(Lu), then we have Hpp(Lu) ⊂ D(G) and G(Hpp(Lu)) ⊂Hpp(Lu).

Demonstration. There exists an orthonormal basis of Hpp(Lu), denoted by ψ1, ψ2, · · · , ψN, such that

Luψj = λjψj , where σpp(Lu) = λ1, λ2, · · · , λN ⊂ (−∞, 0), λj < λj+1.

Since (4.4.27) implies that Hpp(Lu) ⊂ G−1(H1+)⋂

D(G), formula (4.4.26) gives that

fj := [Lu, G]ψj = −iψj +iψj(0

+)

2πΠu ∈Hpp(Lu), ∀j = 1, 2, · · · , N.

So we have 〈fj , ψj〉L2 = 〈Gψj , Luψj〉L2 − 〈GLuψj , ψj〉L2 = λ(〈Gψj , ψj〉L2 − 〈Gψj , ψj〉L2) = 0.


For every j = 1, 2, · · · , N , we set gj :=∑

1≤k≤N,k 6=j〈fj ,ψk〉L2

λk−λj ψk. Since fj =∑

1≤k≤N,k 6=j〈fj , ψk〉L2ψk,

we have (Lu − λj)gj = fj = (Lu − λj)Gψj . Then Gψj − gj ∈ Ker(Lu − λj) = Cψj and

Gψj ∈ gj + Cψj ⊂Hpp(Lu).

We conclude by Hpp(Lu) = SpanCψ1, ψ2, · · · , ψN. (see also formulas (4.4.4) and (4.4.6))

Now, we perform the proof of converse sense of theorem 4.4.8 give the explicit formula of Qu.

End of the proof of theorem 4.4.8. ⇐ : Proposition 4.4.14 yields that G(Hpp(Lu)) ⊂ Hpp(Lu). Let Q

denote the characteristic polynomial of the operator G|Hpp(Lu), then we have Hpp(Lu) =C≤N−1[X]

Q by

lemma 4.3.3. So Πu = P0

Q , for some P0 ∈ C[X] such that deg P0 ≤ N − 1. It remains to show that

P0 = iQ′. Since Hpp(Lu) is invariant under Lu, for every P ∈ C≤N−1[X], we have

Lu(P

Q) = (D− TP0

Q− TP0

Q

)(P

Q) =

DP

Q−Π(

P0P

QQ) +

(iQ′ − P0)P

Q2∈ C≤N−1[X]

Q.

Partial-fraction decomposition implies that Π(P0PQQ

) ∈ C≤N−1[X]

Q . So (iQ′−P0)PQ ∈ C≤N−1[X] for every

P ∈ C≤N−1[X]. Choose P = 1, since deg(iQ′ − P0) ≤ N − 1, we have P0 = iQ′, so u ∈ UN . SinceQ ∈ CN [X] is monic and Q−1(0) ⊂ C−, we have Qu(x) = Q(x) = det(x−G|Hpp(Lu)).

4.4.4 The stability under the Benjamin–Ono flow

Finally we prove proposition 4.4.9 in this subsection. Two lemmas will be proved at first in order toobtain the invariance of the property x 7→ xu(x) ∈ L2(R) under the BO flow.

Lemma 4.4.15. If u0 ∈ H2(R,R)⋂L2(R, x2dx), let u = u(t, x) solves the BO equation (4.1.1) with

initial datum u(0) = u0, then u(t) ∈ L2(R, x2dx), for every t ∈ R.

Remark 4.4.16. This result can be strengthened by replacing the assumption u0 ∈ H2(R,R) by a weaker

assumption u0 ∈ H32 +(R,R) =

⋃s> 3

2Hs(R,R), because one can construct the conservation law of BO

equation controlling the Hs-norm for every s > − 12 by using the method of perturbation of determinants.

We refer to Talbut [136] to see details and Killip–Visan–Zhang [90] for the KdV and the NLS cases(see also Koch–Tataru [91]). Lemma 4.4.15 is a special case of the result in Fonseca–Ponce [37] andFonseca–Linares–Ponce [38].

It suffices to use lemma 4.4.15 to prove proposition 4.4.9. Before proving lemma 4.4.15, we need somecommutator estimates used in Gerard–Lenzmann–Pocovnicu–Raphael [56], we recall it here.

Lemma 4.4.17. For a general locally Lipschitz function χ : R → R such that ∂xχ, ∂3xχ, ∂

5xχ ∈ L1(R),

then we have the following commutator estimates

‖[|D|, χ]g‖L2 + ‖[∂x, χ]g‖L2 . (‖∂xχ‖L1‖∂3xχ‖L1)

12 ‖g‖L2 , ∀g ∈ L2(R),

‖|D|[∂x, χ]g‖L2 . (‖∂xχ‖L1‖∂3xχ‖L1)

12 ‖∂xg‖L2 + (‖∂xχ‖L1‖∂5

xχ‖L1)12 ‖g‖L2 , ∀g ∈ H1(R).

(4.4.30)


Demonstration. We use∣∣|ξ| − |η|∣∣ ≤ |ξ − η| to estimate the Fourier modes of [|D|, χ]g.

2π∣∣∣ ([|D|, χ]g)

∧(ξ)∣∣∣ ≤ ∫

η∈R

∣∣|ξ| − |η|∣∣|χ(ξ − η)||g(η)|dη ≤∫η∈R|ξ − η||χ(ξ − η)||g(η)|dη = |∂xχ| ∗ |g|(ξ).

Then Young’s convolution inequality yields that ‖[|D|, χ]g‖L2 . ‖∂xχ| ∗ |g|‖L2 . ‖∂xχ‖L1‖g‖L2 . In order

to estimate ‖∂xχ‖L1 , we divide the integral as two parts. Wet set R1 = ‖∂xχ‖− 1

2

L1 ‖∂3xχ‖

12

L1 , so

‖∂xχ‖L1 ≤ ‖∂xχ‖L∞∫|ξ|≤R1

dξ +

∫|ξ|>R1

‖∂3xχ‖L∞|ξ|2

dξ . ‖∂xχ‖L1R1 +‖∂3xχ‖L1

R1= (‖∂xχ‖L1‖∂3

xχ‖L1)12 .

Similarly, we have ‖[∂x, χ]g‖L2 . ‖∂xχ‖L1‖g‖L2 . (‖∂xχ‖L1‖∂3xχ‖L1)

12 . Thus (4.4.30) is obtained.

2π∣∣∣ (|D|[∂x, χ]g)

∧(ξ)∣∣∣ ≤|ξ|∫

η∈R|ξ − η||χ(ξ − η)||g(η)|dη

≤∫η∈R|ξ − η|2

∣∣|χ(ξ − η)||g(η)|dη +

∫η∈R|ξ − η||χ(ξ − η)||η||g(η)|dη

=|∂2xχ| ∗ |g|(ξ) + |∂xχ| ∗ |∂xg|(ξ)

So we have ‖|D|[∂x, χ]g‖L2 . ‖|∂2xχ| ∗ |g|‖L2 +‖|∂xχ| ∗ |∂xg|‖L2 . ‖∂2

xχ‖L1‖g‖L2 +‖∂xχ‖L1‖∂xg‖L2 . Then

we use the same idea to estimate ‖∂2xχ‖L1 , we set R2 := ‖∂xχ‖

− 14

L1 ‖∂5xχ‖

14

L1 . Thus,

‖∂2xχ‖L1 ≤ ‖∂xχ‖L∞

∫|ξ|≤R1

|ξ|dξ+

∫|ξ|>R1

‖∂5xχ‖L∞|ξ|3

dξ . ‖∂xχ‖L1R22 +‖∂5xχ‖L1

R22

= (‖∂xχ‖L1‖∂5xχ‖L1)

12 .

Finally, we add them together to get the second estimate in (4.4.30).

Now we prove the invariance of the property x 7→ xu(x) ∈ L2(R) is invariant under the BO flow.

Proof of lemma 4.4.15. We choose a cut-off function χ ∈ C∞c (R) such that χ decreases in [0,+∞), χ iseven and

0 ≤ χ ≤ 1, χ ≡ 1 on [−1, 1], supp(χ) ⊂ [−2, 2]. (4.4.31)

If u0 ∈ H2(R)⋂L2(R, x2dx), we claim that there exists a constant C = C(‖u(0)‖H1) such that

I(R, t) :=

∫Rχ2( xR )|x|2|u(t, x)|2dx ≤ Ce|t|(

∫R|x|2|u(0, x)|2dx+ 1), ∀t ∈ R, ∀R > 1, (4.4.32)

if u solves the BO equation ∂tu = H∂2xu− ∂x(u2) = |D|∂xu− 2u∂xu.

In fact, we define ρ(x) := xχ(x). For every R > 0, we set ρR(x) := Rρ( xR ) = xχ( xR ). Thus

∂tI(R, t) = 2Re〈ρ2R∂tu(t), u(t)〉L2 = 2Re〈ρ2

R|D|∂xu(t)− 2ρ2Ru(t)∂xu(t), u(t)〉L2 = J1(u(t)) + J2(u(t)),

where for every u ∈ H2(R), we define

J1(u) := −4Re〈ρ2Ru∂xu, u〉L2 =⇒ |J1(u)| ≤ 4‖∂xu‖L∞‖ρRu‖2L2 . ‖u‖H2‖ρRu‖2L2 (4.4.33)


andJ2(u) := 2Re〈ρ2

R|D|∂xu, u〉L2 = 〈[ρ2R, |D|∂x]u, u〉L2 ,

because |D|∂x = −(|D|∂x)∗ is an unbounded skew-adjoint operator on L2(R), whose domain of definitionis H2(R), u 7→ ρRu is a bounded self-adjoint operator on Hs(R), for every s ≥ 0. Since

[ρ2R, |D|∂x] = ρR[ρR, |D|∂x] + [ρR, |D|∂x]ρR, [ρR, |D|∂x] = [ρR, |D|∂x]∗ = [ρR, |D|]∂x + |D|[ρR, ∂x],

we have

J2(u) =〈ρR[ρR, |D|∂x]u+ [ρR, |D|∂x]ρRu, u〉L2

=2Re〈[ρR, |D|∂x]u, ρRu〉L2

=2Re〈[ρR, |D|]∂xu, ρRu〉L2 + 2Re〈|D|[ρR, ∂x]u, ρRu〉L2 .

(4.4.34)

Since ‖∂xρR‖L1 = R‖∂xρ‖L1 , ‖∂3xρR‖L1 = R−1‖∂xρ‖L1 and ‖∂5

xρR‖L1 = R−3‖∂xρ‖L1 , the commutatorestimates (4.4.30) yield that if u ∈ H2(R), then

|J2(u)| ≤2‖ρRu‖2L2 + ‖[ρR, |D|]∂xu‖2L2 + ‖|D|[ρR, ∂x]u‖2L2

.‖ρRu‖2L2 + ‖∂xρR‖L1‖∂3xρR‖L1‖∂xu‖2L2 + ‖∂xρR‖L1‖∂5

xρR‖L1‖u‖2L2

.‖ρRu‖2L2 + ‖∂xρ‖L1‖∂3xρ‖L1‖∂xu‖2L2 +R−2‖∂xρ‖L1‖∂5

xρ‖L1‖u‖2L2

.‖ρRu‖2L2 + ‖u‖2H1

(4.4.35)

for every R ≥ 1. Proposition 4.2.6 and 4.2.8 yield that there exists a conservation law of (4.1.1) controllingH2-norm of the solution. Let u : t ∈ R 7→ u(t) ∈ H2(R) denote the solution of the BO equation (4.1.1).Then supt∈R ‖u(t)‖H2 .‖u0‖H2

1. Since I(R, t) = ‖ρRu(t)‖2L2 , estimates (4.4.33) and (4.4.35) imply that

|∂tI(R, t)| ≤ C(I(R, t) + 1), t ∈ R,

for some constant C = C(‖u0‖H2). Thus (4.4.32) is obtained by Gronwall’s inequality. Let R → +∞, weconclude by using Lebesgue’s monotone convergence theorem.

Since the generating function λ ∈ C\σ(−Lu) 7→ Hλ(u) ∈ C is the Borel–Cauchy transform of the spectralmeasure of Lu, the invariance of the N−soliton manifold UN under BO flow is obtained by using theinverse spectral transform.

End of the proof of proposition 4.4.9. If u0 ∈ UN ⊂ H∞(R,R)⋂L2(R, x2dx), let u = u(t, x) be the

unique solution of the BO equation (4.1.1) with initial datum u(0) = u0, then u(t) ∈ H∞(R,R)⋂L2(R, x2dx)

by proposition 4.2.5 and lemma 4.4.15. Recall the generating function Hλ : u ∈ L2(R,R)→ R defined as

Hλ(u) = 〈(λ+ Lu)−1Πu,Πu〉L2 =

∫R

dmu(ξ)

ξ + λ, mu := µLuΠu, ∀λ ∈ C\σ(−Lu), (4.4.36)

where µLuψ denotes the spectral measure of Lu associated to the function ψ ∈ L2+. So the holomorphic

function λ ∈ C\σ(−Lu) 7→ Hλu is the Borel–Cauchy transform of the positive Borel measure mu. Werecall that the total variation mu(R) = ‖Πu‖2L2 is a conservation law of the BO equation (4.1.1) byproposition 4.2.8 and formula (4.2.20). Every finite Borel measure is uniquely determined by its Borel–Cauchy transform (see Theorem 3.21 of Teschl [139] page 108), precisely for every a ≤ b real numbers,we use Stieltjes inversion formula to obtain that

1

2mu((a, b)) +

1

2mu([a, b]) = − 1

πlimε→0+

∫ b

a

ImHx+iε(u)dx.


For every t ∈ R, proposition 4.2.10 yields that Hλ[u(t)] = Hλ[u(0)], ∀λ ∈ C\σ(−Lu(0)) = C\σ(−Lu(t)).Since u(0) ∈ UN , we have Π[u(0)] ∈Hpp(Lu(0)) by proposition 4.4.6 and there exist c1, c2, · · · , cN ∈ R\0such that

µLu(t)

Π[u(t)] = mu(t) = mu(0) = µLu(0)

Π[u(0)] =

N∑j=1

cjδλu(0)j

.

The spectral measure µLu(t)

Π[u(t)] is purely point, so Π[u(t)] ∈Hpp(Lu(t)) for every t ∈ R. The Lax pair struc-

ture yields the unitary equivalence between Lu(t) and Lu(0). So dimC Hpp(Lu(t)) = dimC Hpp(Lu(0)) = Nis given by proposition 4.2.14. We conclude by theorem 4.4.8.

4.5 The generalized action–angle coordinates

In this section, we construct the (generalized) action–angle coordinates ΦN in theorem 1 of the BOequation (4.1.6) with solutions in the real analytic symplectic manifold (UN , ω) of real dimension 2Ngiven in proposition 4.4.3. The goal of this section is to establish the diffeomorphism property and thesymplectomorphism property of ΦN .

Recall that the BO equation with N -soliton solutions is identified as a globally well-posed Hamiltoniansystem reading as

∂tu(t) = XE(u(t)), u(t) ∈ UN , (4.5.1)

whose energy functional E(u) = 〈LuΠu,Πu〉L2 is well defined on UN and the Hamiltonian vector fieldXE : u ∈ UN 7→ XE(u) = ∂x(|D|u − u2) ∈ Tu(UN ) coincides with the definition (4.4.3). The Poissonbracket of two smooth functions f, g : UN → R is given by

f, g : u ∈ UN 7→ ωu(Xf (u), Xg(u)) = 〈∂x∇uf(u),∇ug(u)〉L2 ∈ R. (4.5.2)

Given u ∈ UN , proposition 4.4.6 yields that there exist λu1 < λu2 < · · · < λuN < 0 and ϕuj ∈ Ker(λuj −Lu) ⊂D(G) such that ‖ϕuj ‖L2 = 1 and 〈u, ϕuj 〉L2 =

√2π|λuj |, thanks to the spectral analysis in subsection 4.4.2.

Definition 4.5.1. For every j = 1, 2, · · · , N , the map Ij : u ∈ UN 7→ 2πλuj ∈ R is called the j th action.The map γj : u ∈ UN 7→ Re〈Gϕuj , ϕuj 〉L2 ∈ R is called the j th (generalized) angle.

Set ΩN := (r1, r2, · · · , rN ) ∈ RN : r1 < r2 < · · · < rN < 0 ⊂ RN , the canonical symplectic form on

R2N = (r1, r2, · · · , rN ;α1, α2, · · · , αN ) : ∀rj , αj ∈ R is given by ν =∑Nj=1 drj ∧ dαj . Endowed with

the subspace topology and the embedded real analytic structure of R2N , the submanifold (ΩN × RN , ν)is a symplectic manifold of real dimension 2N . The action–angle map is defined by

ΦN : u ∈ UN 7→ (I1(u), I2(u), · · · , IN (u); γ1(u), γ2(u), · · · , γN (u)) ∈ ΩN × RN . (4.5.3)

Theorem 1 is restated here.

Theorem 4.5.2. The map ΦN has following properties :

(a). The map ΦN : UN → ΩN × RN is a real analytic diffeomorphism.(b). The pullback of ν by ΦN is ω, i.e. Φ∗Nν = ω.

(c). We have E Φ−1N : (r1, r2, · · · , rN ;α1, α2, · · · , αN ) ∈ ΩN × RN 7→ − 1

2π

∑Nj=1 |rj |2 ∈ (−∞, 0).

4.5. THE GENERALIZED ACTION–ANGLE COORDINATES 119

Remark 4.5.3. The real analyticity of ΦN : UN → ΩN ×RN is given by proposition 4.4.7 and corollary4.4.12. The symplectomorphism property (b) is equivalent to the following Poisson bracket characterization(see proposition 4.5.24)

Ij , Ik = 0, Ij , γk = 1j=k, γj , γk = 0 on UN , ∀j, k = 1, 2, · · · , N. (4.5.4)

The family (XI1 , XI2 , · · · , XIN ;Xγ1, Xγ2

, · · · , XγN ) is linearly independent in X(UN ) and we have

dΦN (u) : XIk(u) 7→ ∂

∂αk

∣∣∣ΦN (u)

, dΦN (u) : Xγk(u) 7→ − ∂

∂rk

∣∣∣ΦN (u)

.

The assertion (c) is obtained by a direct calculus : Πu =∑Nj=1〈Πu, ϕuj 〉L2ϕuj , formula (4.4.9) yields that

E(u) = 〈Lu(Πu),Πu〉L2 =

N∑j=1

|〈Πu, ϕuj 〉L2 |2λuj = −N∑j=1

Ij(u)2

2π.

Thus theorem 4.5.2 introduces (generalized) action–angle coordinates of the BO equation (4.5.1) in thesense of (4.1.8), i.e. Ij , E(u) = 0 and γj , E(u) = 2λuj , for every u ∈ UN .

This section is organized as follows. The matrix associated to G|Hpp(Lu) is expressed in terms of ac-tions and angles in subsection 4.5.1. Then the injectivity of ΦN is given by inversion formulas in sub-section 4.5.2. In subsection 4.5.3, the Poisson brackets of actions and angles are used to show thelocal diffeomorphism property of ΦN . The surjectivity of ΦN is obtained by Hadamard’s global in-verse theorem in subsection 4.5.4. Finally, we use subsection 4.5.5 and subsection 4.5.6 to prove thatΦN : (UN , ω)→ (ΩN × RN , ν) preserves the symplectic structure.

4.5.1 The associated matrix

We continue to study the infinitesimal generator G defined in (4.3.2) when restricted to the invariantsubspace Hpp(Lu) with complex dimension N . Let M(u) = (Mkj(u))1≤k,j≤N denote the matrix associa-ted to the operator G|Hpp(Lu) with respect to the basis ϕu1 , ϕu2 , · · · , ϕuN. Then we state a general linearalgebra lemma that describes the location of eigenvalues of the matrix M(u).

Proposition 4.5.4. For every u ∈ UN , the coefficients of matrix M(u) = (Mkj(u))1≤k,j≤N are givenby

Mkj(u) = 〈Gϕuj , ϕuk〉L2 =

i

λuk−λuj

√|λuk ||λuj |

, if j 6= k,

γj(u)− i2|λuj |

, if j = k.(4.5.5)

Demonstration. Since Lu is a self-adjoint operator on L2+ and Hpp(Lu) ⊂ D(G), we have

(λuj − λuk)Mkj(u) = 〈GLuϕuj , ϕuk〉L2 − 〈Gϕuj , Luϕuk〉L2 = 〈[G,Lu]ϕuj , ϕuk〉L2 .

Since formulas (4.2.15) and (4.4.9) imply that −λuj ϕuj (0) = uϕuj (0) =√

2π|λuj |, we use (4.4.26) to obtain

(λuj − λuk)Mkj(u) = 〈iϕuj −i

2πϕuj (0+)Πu, ϕuk〉L2 = − i

2πϕuj (0+)uϕuk(0) = −i

√|λuk ||λuj |

.


In the case k = j, we use Plancherel formula and integration by parts to calculate

〈G∗f, g〉L2 = 〈f,Gg〉L2 = − i2π

∫ +∞

0

f(ξ)∂ξ g(ξ)dξ = i2π

[f(0+)g(0+) +

∫ +∞

0

∂ξ f(ξ)g(ξ)dξ

]Thus we have 〈G∗f, g〉L2 = 〈Gf, g〉L2 + i

2π f(0+)g(0+), for every f, g ∈Hpp(Lu). Then

ImMjj(u) =1

2i(〈Gϕuj , ϕuj 〉L2 − 〈G∗ϕuj , ϕuj 〉L2) = −

|ϕuj (0)|2

4π= − 1

2|λuj |.

We conclude by γj(u) = Refj(u) = 〈Gϕuj , ϕuj 〉L2 defined in corollary 4.4.12.

Then we state a linear algebra lemma that describe the location of spectrum of all matrices of the formdefined as (4.5.5).

Lemma 4.5.5. For every N ∈ N+, we choose N negative numbers λ1 < λ2 < · · · < λN < 0 and N realnumbers γ1, γ2, · · · , γN ∈ R. The matrix M = (Mkj)1≤k,j≤N ∈ CN×N is defined as

Mkj =

iλk−λj

√|λk||λj | , if k 6= j,

γj − i2|λj | , if k = j.

(4.5.6)

Then ImM = M−M∗2i is negative semi-definite and σpp(M) ⊂ C−. Furthermore, the map

(λ1, λ2, · · · , λN ; γ1, γ2, · · · , γN ) 7→ M = (Mkj)1≤k,j≤N

defined as (4.5.6) is real analytic on ΩN × RN .

Demonstration. The vector Vλ ∈ RN is defined as V Tλ := ((2|λ1|)−12 , (2|λ2|)−

12 , · · · , (2|λN |)−

12 ). So we

have

ImM =

(− 1

2√|λj ||λk|

)1≤k,j≤N

= −Vλ · V Tλ .

Recall that 〈X,Y 〉CN := XT · Y , thus 〈(ImM)X,X〉CN = −|〈X,Vλ〉CN |2 ≤ 0. So ImM is a negativesemi-definite matrix. If µ ∈ σpp(M) and V ∈ Ker(µ−M)\0, it suffices to show that Imµ < 0.

− |〈V, Vλ〉CN |2 = 〈(ImM)V, V 〉CN = Imµ‖V ‖2CN , where ‖V ‖2CN = 〈V, V 〉CN > 0. (4.5.7)

So we have Imµ ≤ 0. Assume that µ ∈ R, then formula (4.5.7) yields that V ⊥ Vλ. Moreover, we have(M−M∗)V = −2i〈V, Vλ〉CNVλ = 0. We set Dλ ∈ CN×N to be the diagonal matrix whose diagonal

elements are λ1, λ2, · · · , λN , i.e. Dλ =

λ1

λ2

. . .λN

. Then we have the following formula

[M, Dλ] = i(IN + 2DλVλVTλ ). (4.5.8)

So [M, Dλ]V = iV by (4.5.8). Recall that M∗V =MV = µV . Finally,

i‖V ‖2CN = 〈[M, Dλ]V, V 〉CN = 〈(M− µ)DλV, V 〉CN = 〈DλV, (M∗ − µ)V 〉CN = 0

contradicts the fact that V 6= 0. Consequently, we have µ ∈ C−.


Corollary 4.5.6. For every u ∈ UN , let M(u) = (Mkj(u))1≤k,j≤N ∈ CN×N denote the matrix defined

by formula (4.5.5), then ImM(u) = M(u)−M(u)∗

2i is negative semi-definite and σpp(M(u)) ⊂ C−.

Remark 4.5.7. The fact σpp(M(u)) ⊂ C− can also be given by using the inversion formula (4.4.10) andproposition 4.4.1. The characteristic polynomial Qu(x) = det(x−M(u)) has zeros in C−.

4.5.2 Inverse spectral formulas

The injectivity of ΦN is proved in this subsection by using inverse spectral formulas. The followinglemma describes the relation between the Fourier transform of an eigenfunction ϕ ∈ Hpp(Lu) and the

inner function associated to u defined by Θu = QuQu

with Qu(x) = det(x−M(u)).

Lemma 4.5.8. For every monic polynomial Q ∈ CN [X] such that Q−1(0) ⊂ C−, the associated inner

function is defined by Θ = QQ . The following identity holds for every ϕ ∈ C≤N−1[X]

Q ,

ϕ(ξ) = 〈S(ξ)∗ϕ, 1−Θ〉L2 . (4.5.9)

In particular, ϕ(0+) = 〈ϕ, 1−Θ〉L2 .

Demonstration. Since ϕ = PQ , for some P ∈ C≤N−1[X] and Q−1(0) ⊂ C−, recall that Q(x) =

∏nj=1(x−

zj)mj with Imzj < 0, z1, z2, · · · , zN are all distinct and

∑nj=1mj = N . Formulas (4.3.7) and (4.3.8) imply

that

fj,l(x) =l!

2π[(−i)(x− zj)]l+1=⇒ fj,l(ξ) = ξle−izjξ1R+

(ξ).

Since ϕ ∈ SpanCfj,l1≤j≤mj ,1≤j≤n, partial-fractional decomposition implies that ϕ ∈ C1(R∗+), and the

right limit ϕ(0+) = limξ→0+ ϕ(ξ) exists. Recall that Θ = QQ , so we have Θϕ = Q

QPQ = P

Q∈ L2

−. Since

Θ(x) = 1 + 2i∑Nj=1

Imzjx−zj +O( 1

x2 ), when x→ +∞, we have 1−Θ ∈ L2+. Then

ϕ(ξ) =

∫Rϕ(y)(1−Θ(y))e−iyξdy = 〈ϕ, S(ξ)(1−Θ)〉L2 = 〈S(ξ)∗ϕ, 1−Θ〉L2 , ∀ξ ≥ 0.

Proposition 4.5.9. For every u ∈ UN , we set Qu ∈ CN [X] to be the characteristic polynomial of u and

we define the associated inner function as Θu = QuQu

. Then the following inversion formula holds,

f(z) =1

2πi〈(G− z)−1f, 1−Θu〉L2 , f ∈Hpp(Lu), ∀z ∈ C+. (4.5.10)

Demonstration. If f ∈Hpp(Lu) =C≤N−1[X]

Qu, then formula (4.5.9) yields that

f(ξ) = 〈S(ξ)∗f, 1−Θu〉L2 = 〈e−iξGf, 1−Θu〉L2 .

Since ImG := G−G∗2i is a negative semi-definite operator on Hpp(Lu) by proposition 4.5.4 and lemma

4.5.5, the operator Re(i(z−G))|Hpp(Lu) = (ImG− Imz)|Hpp(Lu) is negative definite, for every z ∈ C+. So

f(z) =1

2π

∫ +∞

0

〈eiξ(z−G)f, 1−Θu〉L2dξ =1

2πi〈(G− z)−1f, 1−Θu〉L2 .


Recall that 〈Πu, ϕuj 〉L2 =√

2π|λuj | and 〈1−Θ, ϕuj 〉L2 =√

2π|λuj |

, for every j = 1, 2, · · · , N , by (4.2.15) and

(4.4.9). Since Πu ∈ Hol(z ∈ C : Imz > −ε), for some ε > 0, we have the following inversion formula

Πu(x) = 12πi 〈(G− x)−1Πu, 1−Θ〉L2 = −i〈(M(u)− x)−1X(u), Y (u)〉CN , ∀x ∈ R, (4.5.11)

where the two vectors X(u), Y (u) ∈ RN are defined as

X(u)T = (√|λu1 |,

√|λu2 |, · · · ,

√|λuN |), Y (u)T = (

√|λu1 |−1,

√|λu2 |−1, · · · ,

√|λuN |−1), (4.5.12)

and M(u) is the N × N matrix of the infinitesimal generator G associated to the orthonormal basisϕu1 , ϕu1 , · · · , ϕuN, defined in (4.5.4). A consequence of the inverse spectral formula (4.5.11) is the explicitformula of the BO flow with N -soliton solutions as described by formula (4.1.11).

Corollary 4.5.10. The map ΦN : UN → ΩN × RN is injective.

Demonstration. If ΦN (u) = ΦN (v) for some u, v ∈ UN , then λuj = λvj and γj(u) = γj(v), for every j. So

M(u) = M(v), X(u) = X(v), Y (u) = Y (v).

Then the inversion formula (4.5.11) gives that Πu = Πv. Thus, u = 2ReΠu = 2ReΠv = v.

At last we show the equivalence between the inversion formulas (4.4.10) and (4.5.11).

Revisiting formula (4.4.10). For every k, j = 1, 2, · · · , N , let Kukj(x) denote the (N − 1)× (N − 1) sub-

matrix obtained by deleting the k th column and j th row of the matrix M(u) − x, for every x ∈ R. Sothe inversion formula (4.5.11) and the Cramer’s rule imply that

iΠu(x) =∑

1≤k,j≤N

(−1)k+j det(Kukj(x))

det(M(u)− x)

√λukλuj

=

∑Nj=1 det(Ku

jj(x)) +R

det(M(u)− x), (4.5.13)

where R :=∑

1≤k 6=j≤N (−1)k+j det(Kukj(x))

√λukλuj

. The coefficients of the matrix M(u)− x satisfies that

(M(u)− x)kj = Mkj(u) = iλuk−λ

uj

√λukλuj, if 1 ≤ j 6= k ≤ N,

by formula (4.5.5). Using expansion by minors, we have

iR =∑

1≤k,j≤N

(−1)k+j(λuk − λuj )(M(u)− x)kj det(Kukj(x)) = (

N∑k=1

λuk −N∑j=1

λuj ) det(M(u)− x) = 0.

Finally, let Q denote the characteristic polynomial of the operator G|Hpp(Lu), so

Q(x) = det(x−G|Hpp(Lu)) = det(x−M(u)), Q′(x) = (−1)NN∑j=1

det(Kujj(x)).


4.5.3 Poisson brackets

In this subsection, the Poisson bracket defined in (4.5.2) is generalized in order to obtain the first twoformulas of (4.5.4). It can be defined between a smooth function from UN to an arbitrary Banach spaceand another smooth function from UN to R.

The N-soliton subset (UN , ω) is a real analytic symplectic manifold of real dimension 2N , where

ωu(h1, h2) =i

2π

∫R

h1(ξ)h2(ξ)

ξdξ, ∀h1, h2 ∈ Tu(UN ), ∀u ∈ UN .

For every smooth function f : UN → R, its Hamiltonian vector field Xf ∈ X(UN ) is given by (4.4.3).Recall that Xf (u) = ∂x∇uf(u) and df(u)(h) = ω(h,Xf (u)), ∀h ∈ Tu(UN ). For any Banach space E andany smooth map F : u ∈ UN 7→ F (u) ∈ E , we define the Poisson bracket of f and F as follows

f, F : u ∈ UN 7→ f, F(u) := dF (u)(Xf (u)) ∈ TF (u)(E) = E . (4.5.14)

If E = R, then the definition in formula (4.5.14) coincide with (4.5.2) and we recall it here,

f, F(u) = dF (u)(Xf (u)) = ωu(Xf (u), XF (u)). (4.5.15)

For every λ ∈ C\σ(−Lu), the generating function Hλ(u) = 〈(Lu + λ)−1Πu,Πu〉L2 is well defined. Since

Πu =∑Nj=1〈Πu, ϕuj 〉L2ϕuj , we have

Hλ(u) =

N∑j=1

|〈Πu, ϕuj 〉L2 |2

λ+ λuj= −

N∑j=1

2πλujλ+ λuj

. (4.5.16)

The analytical continuation allow to extend the generating function λ 7→ Hλ(u) to the domain C\σpp(−Lu),

and it has simple poles at every λ = −λuj . Proposition 4.2.2 yields that −C2

4 ‖u‖2L2 ≤ λu1 < · · · < λuN < 0,

where C = inff∈H1+\0

‖|D|14 f‖L2

‖f‖L4denotes the Sobolev constant. So we introduce

Y = (λ, u) ∈ R× UN : 4λ > C2‖u‖2L2 = X⋂

(R× UN ) , (4.5.17)

where X is given by definition 4.2.9. Then the subset Y is open in R×UN and the map H : (λ, u) ∈ Y 7→−∑Nj=1

2πλujλ+λuj

∈ R is real analytic by proposition 4.4.7. Recall that the Frechet derivative (4.2.8) is given

by

dHλ(u)(h) = 〈wλ,Πh〉L2 + 〈wλ,Πh〉L2 + 〈Thwλ, wλ〉L2 = 〈h,wλ + wλ + |wλ|2〉L2 , ∀h ∈ Tu(UN ).

where wλ ∈ H1+ is given by wλ ≡ wλ(u) ≡ wλ(x, u) = [(Lu + λ)−1 Π]u(x), for every x ∈ R. Thus

XHλ(u) = ∂x∇uHλ(u) = ∂x(|wλ(u)|2 + wλ(u) + wλ(u)), ∀(λ, u) ∈ Y. (4.5.18)

by (4.4.3). The Lax map L : u ∈ UN 7→ Lu = D − Tu ∈ B(H1+, L

2+) is R-affine, hence real analytic. The

following proposition restates the Lax pair structure of the Hamiltonian equation associated to Hλ. Eventhough the stability of UN under the Hamiltonian flow of Hλ remains as an open problem, the Poissonbracket defined in (4.5.14) provides an algebraic method to obtain the first two formulas of (4.5.4).


Proposition 4.5.11. Given (λ, u) ∈ Y defined by (4.5.17), we have Hλ, L(u) = [Buλ , Lu] and

Hλ, λj(u) = 0, Hλ, γj(u) = Re〈[G,Bλu ]ϕuj , ϕuj 〉L2 = − λ

(λ+ λuj )2, (4.5.19)

for every j = 1, 2, · · · , N , where Buλ = i(Twλ(u)Twλ(u) + Twλ(u) + Twλ(u)).

Demonstration. Since L : u ∈ L2(R,R) 7→ Lu = D− Tu ∈ B(H1+, L

2+), for every u ∈ L2

+, we have

dL(u)(h) = −Th, ∀h ∈ L2+.

If (λ, u) ∈ Y, then the C-linear transformation Lu + λ ∈ B(H1+, L

2+) is bijective. So formula (4.5.18)

yields that Hλ, L(u) = dL(u)(XHλ(u)) = −TD(|wλ(u)|2+wλ(u)+wλ(u)). Then identity (4.2.24) yields theLax equation for the Hamiltonian flow of the generating function Hλ, i.e.

Hλ, L(u) = [Buλ , Lu] ∈ B(H1+, L

2+). (4.5.20)

Consider the map Lϕj : u ∈ UN 7→ Luϕuj = λujϕ

uj ∈ H1

+, for every (λ, u) ∈ Y, we have

Hλ, L(u)ϕuj + Lu (Hλ, ϕj(u)) = λuj Hλ, ϕj(u) + Hλ, λj(u)ϕuj

with Hλ, ϕj(u) ∈ H1+ and Hλ, λj(u) ∈ R. Then (4.5.20) yields that

(λuj − Lu)(Bλuϕ

uj − Hλ, ϕj(u)

)= Hλ, λj(u)ϕuj .

Since ϕuj ∈ Ker(λuj − Lu) and ‖ϕuj ‖L2 = 1 by the definition in (4.4.9), we have

Hλ, λj(u) = 〈(λuj − Lu)(Bλuϕ

uj − Hλ, ϕj(u)

), ϕuj 〉L2 = 0.

Let N2 : ϕ ∈ L2 7→ ‖ϕ‖2L2 , then we have N2 ϕj ≡ 1 on UN . Then we have

0 = d(N2 ϕj)(u) = 2Re〈ϕuj , Hλ, λj(u)〉L2 . (4.5.21)

So there exists r ∈ R such that Bλuϕuj − Hλ, ϕj(u) = irϕuj because Ker(λuj − Lu) = Cϕuj by corollary

4.2.4 and formula (4.5.21). Recall that Bλu is skew-adjoint and γj = Re〈Gϕuj , ϕuj 〉L2 , we have

Hλ, γj(u) = Re(〈GHλ, ϕj(u), ϕuj 〉L2 + 〈Gϕuj , Hλ, ϕj(u)〉L2

)= Re〈[G,Bλu ]ϕuj , ϕ

uj 〉L2 .

Furthermore, for every (λ, u) ∈ Y, formula (4.3.4) implies that [G,Twλ(u)] = 0 and

[G,Bλu ]f = i[G,Twλ(u)](Twλ(u)(f) + f) = − 12π [(wλ(u)f)∧(0+) + f(0+)]wλ(u), ∀f ∈ D(G). (4.5.22)

Since (wλ(u)ϕuj )∧(0+) = 〈ϕuj , wλ(u)〉L2 = (λ+ λuj )−1〈u, ϕuj 〉L2and 〈u, ϕuj 〉L2

= −λuj ϕuj (0+), we replace f

by ϕuj in formula (4.5.22) to obtain the following

〈[G,Bλu ]ϕuj , ϕuj 〉L2 =

〈u, ϕuj 〉L2

2π(

1

λuj− 1

λ+ λuj)〈wλ(u), ϕuj 〉L2 = − λ

(λ+ λuj )2, ∀(λ, u) ∈ Y.


Remark 4.5.12. Recall that Hε = 1εH 1

εand Bε,u := 1

εB1εu for every (ε−1, u) ∈ Y. In general, the identity

En, γj(u) = Re〈[G, dn

dεn

∣∣ε=0

Bε,u]ϕuj , ϕuj 〉L2 , 1 ≤ j ≤ N

holds for every conservation law En = (−1)n dn

dεn

∣∣ε=0Hε in the BO hierarchy.

Corollary 4.5.13. For every j, k = 1, 2, · · · , N , we have

2πλj , γk(u) = 1j=k, λk, λj(u) = 0, ∀u ∈ UN . (4.5.23)

Demonstration. Given u ∈ UN , for every λ >C2‖u‖2

L2

4 then (λ, u) ∈ Y, then (4.5.16) and (4.5.19) implythat

− λ

(λ+ λuj )2= Hλ, γj(u) = 2π

N∑k=1

λ

λ+ λuk, γj(u) = −2πλ

N∑k=1

λk, γj(u)

(λ+ λuk)2,

and 0 = Hλ, λj(u) = 2πλ∑Nk=1

λk,λj(u)(λ+λuk )2 , for every j = 1, 2, · · · , N . The uniqueness of analytic

continuation yields that the following formula holds for every z ∈ C\R,

− z

(z + λuj )2= −2πz

N∑k=1

λk, γj(u)

(z + λuk)2,

N∑k=1

λk, λj(u)

(z + λuk)2= 0.

Recall that the actions Ij : u ∈ UN 7→ 2πλuj and the generalized angles γj : u ∈ UN 7→ Re〈Gϕuj , ϕuj 〉L2 areboth real analytic functions by proposition 4.4.7 and corollary 4.4.12.

Proposition 4.5.14. For every u ∈ UN , the family of differentials

dI1(u),dI2(u), · · · dIN (u); dγ1(u),dγ2(u), · · · dγN (u)

is linearly independent in the cotangent space T ∗u (UN ).

Demonstration. For every a1, a2, · · · , aN , b1, b2, · · · , bN ∈ R such that N∑j=1

ajdIj(u) + bjdγj(u)

(h) = 0, ∀h ∈ Tu(UN ). (4.5.24)

Formula of Poisson brackets (4.5.23) yields that for every j, k = 1, 2, · · · , N , we have

dIj(u)(XIk(u)) = Ik, Ij(u) = 0, dγj(u)(XIk(u)) = Ik, γj(u) = 1j=k

We replace h by XIk(u) in (4.5.24) to obtain that bk = 0, ∀k = 1, 2, · · · , N . Then set h = Xγk(u)

−ak =

N∑j=1

ajγk, Ij(u) =

N∑j=1

ajdIj(u)

(Xγk(u)) = 0, ∀k = 1, 2, · · · , N.


As a consequence, ΦN : UN → ΩN × RN is a local diffeomorphism. Moreover, since all the actions(Ij)1≤j≤N are in evolution by (4.5.23) and the differentials (dIj(u))1≤j≤N are linearly independent forevery u ∈ UN , for every r = (r1, r2, · · · , rN ) ∈ ΩN , the level set

Lr =

N⋂j=1

I−1j (rj), where r = (r1, r2, · · · , rN )

is a smooth Lagrangian submanifold of UN and Lr is invariant under the Hamiltonian flow of Ij , for everyj = 1, 2, · · · , N , by the Liouville–Arnold theorem (see Theorem 5.5.21 of Katok–Hasselblatt [83], seealso Fiorani–Giachetta–Sardanashvily [35] and Fiorani–Sardanashvily [36] for the non-compact invariantmanifold case).

4.5.4 The diffeomorphism property

This subsection is dedicated to proving the real bi-analyticity of ΦN : UN → ΩN × RN . It remains toshow the surjectivity. Its proof is based on Hadamard’s global inverse theorem 4.5.18.

Lemma 4.5.15. The map Φ : UN → ΩN × RN is proper.

Demonstration. If K is compact in ΩN × RN , we choose un ∈ Φ−1N (K), so

ΦN (un) = (2πλun1 , 2πλun2 , · · · , 2πλunN ; γ1(un), γ2(un), · · · , γN (un)) ∈ K, ∀n ∈ N.

We assume that there exists (2πλ1, 2πλ2, · · · , 2πλN ; γ1, γ2, · · · , γN ) ∈ K such that λunj → λj and

γj(un) → γj up to a subsequence. So (M(un))n∈N converges to some matrix M ∈ CN×N whose co-efficients are defined as follows

Mkj =

iλk−λj

√|λk||λj | , if k 6= j,

γj − i2|λj | , if k = j.

Lemma 4.5.5 yields that σpp(M) ⊂ C−. We set Q(x) := det(x−M) and u = iQ′

Q − iQ′

Q∈ UN . The Viete

map V is defined in (4.4.11) and V(CN− ) is open in CN . Then there exists

a(n) = (a(n)0 , a

(n)1 , · · · , a(n)

N−1), a = (a0, a1, · · · , aN−1) ∈ V(CN− )

such that Qn(x) = det(x−M(un)) =∑N−1j=0 a

(n)j xj + xN and Q(x) =

∑N−1j=0 ajx

j + xN . We have

limn→+∞

Qn(x) = Q(x), ∀x ∈ R =⇒ limn→+∞

a(n) = a

The continuity of the map ΓN : a = (a0, a1, · · · , aN−1) ∈ V(CN− ) 7→ Πu = iQ′

Q ∈ L2+ yields that

Πun = iQ′nQn

= ΓN (a(n))→ ΓN (a) = iQ′

Q= Πu in L2

+, as n→ +∞.

Since UN inherits the subspace topology of L2(R,R), we have (un)n∈N converges to u in UN . The continuityof the map ΦN shows that ΦN (u) = (2πλ1, 2πλ2, · · · , 2πλN ; γ1, γ2, · · · , γN ) ∈ K.


Proposition 4.5.16. The map ΦN : UN → ΩN × RN is bijective and both ΦN and its inverse Φ−1N are

real analytic.

Demonstration. The analyticity of ΦN is given by proposition 4.4.7 and corollary 4.4.12. The injectivityis given by corollary 4.5.10. Proposition 4.5.14 yields that ΦN : UN → ΩN ×RN is a local diffeomorphismby inverse function theorem for manifolds. So ΦN is an open map. Since every proper continuous map tolocally compact space is closed, ΦN is also a closed map by lemma 4.5.15. Since the target space ΩN×RNis connected, we have ΦN (UN ) = ΩN × RN and ΦN : UN → ΩN × RN is a real analytic diffeomorphism.

Remark 4.5.17. We establish the relation between ΦN : UN → ΩN × RN and ΓN : V(CN− ) → Π(UN )introduced in proposition 4.4.10. We set M : ΩN × RN → CN×N to be the matrix-valued real analyticfunction M(η1, η2, · · · , ηN ; θ1, θ2, · · · , θN ) = (Mkj)1≤k,j≤N with coefficients defined as

Mkj =

2πi

ηk−ηj

√ηkηj, if k 6= j,

θj + πiηj, if k = j.

Then, we set C : M ∈ CN×N 7→ (a0, a1, · · · , aN−1) ∈ CN such that

Q(x) :=

N−1∑j=0

ajxj + xN = det(x−M). (4.5.25)

Since (−1)n−jaj = Tr(Λn−jM) is the sum of all principle minors of M of size (N − j) × (N − j), forevery j = 1, 2, · · · , N , the map C is real analytic on CN×N and C M(ΩN × RN ) ⊂ V(CN− ) by lemma4.5.5, where V denotes the Viete map defined as (4.4.11). In lemma 4.4.10, we have shown that the map

ΓN : a = (a0, a1, · · · , aN−1) ∈ V(CN− ) 7→ Πu = iQ′

Q ∈ Π(UN ) is biholomorphic, where the polynomial Q

is defined as (4.5.25). We conclude by the following identity

Φ−1N = 2Re ΓN C M (4.5.26)

The smooth manifolds Π(UN ) and V(CN− ) are both diffeomorphic to the convex open subset ΩN × RN ,so they are simply connected (see also proposition 4.6.5). At last, we recall Hadamard’s global inversetheorem.

Theorem 4.5.18. Suppose X and Y are connected smooth manifolds, then every proper local dif-feomorphism F : X → Y is surjective. If Y is simply connected in addition, then every proper localdiffeomorphism F : X → Y is a diffeomorphism.

Demonstration. For the surjectivity, see Nijenhuis–Richardson [112] and the proof of proposition 4.5.16.If the target space is simply connected, see Gordon [63] for the injectivity.

Remark 4.5.19. Since the target space ΩN ×RN is convex, there is another way to show the injectivityof ΦN without using the inversion formulas in subsection 4.5.2. It suffices to use the simple connectednessof ΩN × RN and Hadamard’s global inverse theorem 4.5.18.


4.5.5 A Lagrangian submanifold

In general, the symplectomorphism property of ΦN is equivalent to its Poisson bracket characterization(4.5.4), which will be proved in proposition 4.5.24. The first two formulas of (4.5.4) given in corollary4.5.13, lead us to focusing on the study of a special Lagrangian submanifold of UN , denoted by

ΛN := u ∈ UN : γj(u) = 0, ∀j = 1, 2, · · · , N, (4.5.27)

where the generalized angles γj : u ∈ UN 7→ Re〈Gϕuj , ϕuj 〉L2 are defined in (4.5.1). A characterizationlemma of ΛN is given at first.

Lemma 4.5.20. For every u ∈ UN , then each of the following four properties implies the others :

(a). u ∈ ΛN .(b). For every x ∈ R, we have Πu(x) = Πu(−x).(c). u is an even function R→ R.(d). The Fourier transform u is real-valued.

Then every element u ∈ ΛN has translation–scaling parameter in (iR)N/SN i.e. u(x) =∑Nj=1

2ηjx2+η2

j, for

some ηj > 0.

Demonstration. (a)⇒ (b) : If u ∈ ΛN , then the matrix M(u) defined in (4.5.5) is an N ×N matrix withpurely imaginary coefficients. Recall the definition of X(u), Y (u) ∈ RN in (4.5.12) :

X(u)T = (√|λu1 |,

√|λu2 |, · · · ,

√|λuN |), Y (u)T = (

√|λu1 |−1,

√|λu2 |−1, · · · ,

√|λuN |−1).

The inversion formula (4.5.11) yields that

Πu(x) = i〈(M(u)− x)−1X(u), Y (u)〉CN = −i〈(M(u) + x)−1X(u), Y (u)〉CN = Πu(−x).

(b)⇒ (c) is given by the formula u = Πu+ Πu. (c)⇒ (d) is given by u(x) = u(x) = u(−x).

(d) ⇒ (a) : Choose λ ∈ σpp(Lu) = λu1 , λu2 , · · · , λuN and ϕ ∈ Ker(λ− Lu). Since both u and its Fourier

transform u are real-valued, we have [(ϕ)∨]∧(ξ) = ϕ(ξ), where (ϕ)∨(x) := ϕ(−x), ∀x, ξ ∈ R. Thus,

Tu((ϕ)∨) = (Tuϕ)∨ =⇒ (ϕ)∨ ∈ Ker(λ− Lu).

We choose the orthonormal basis ϕu1 , ϕu2 , , · · · , ϕuN in Hpp(Lu) as in formula (4.4.9). Proposition 4.2.4

yields that dimC Ker(λ− Lu) = 1. For every j = 1, 2, · · · , N , there exists θj ∈ R such that

(ϕuj )∨ = eiθjϕuj ⇐⇒ (ϕuj )∧(ξ) = eiθj (ϕuj )∧(ξ), ∀ξ ∈ R.

So we set φuj := exp(iθj2 )ϕuj , then its Fourier transform (φuj )∧ is a real-valued function. Recall the definition

of G in (4.3.2) and γj in (4.5.5), then we have

γj(u) = Re〈Gϕuj , ϕuj 〉L2(R) = Re〈Gφuj , φuj 〉L2(R) = − 1

2πIm〈∂ξ[(φuj )∧], (φuj )∧〉L2(0,+∞) = 0.

by using Plancherel formula.


Lemma 4.5.21. The level set ΛN is a real analytic Lagrangian submanifold of (UN , ω).

Demonstration. The map γ : u ∈ UN 7→ (γ1(u), γ2(u), · · · , γN (u)) ∈ RN is a real analytic submersionby proposition 4.5.14. So the level set ΛN is a properly embedded real analytic submanifold of UN anddimR ΛN = N . The classification of the tangent space Tu(UN ) is given by formula (4.4.1). If u(x) =∑Nj=1

2ηjx2+η2

j, for some ηj > 0, every tangent vector h ∈ ΛN is an even function by lemma 4.5.20. So h is

real valued and we have

Tu(ΛN ) =

N⊕j=1

Rfuj , where fuj (x) =2[x2−η2

j ]

[x2+η2j ]2. (4.5.28)

We have (fuj )∧(ξ) = −2π|ξ|e−ηj |ξ|. Then by definition of ω, we have

ωu(h1, h2) =i

2π

∫R

h1(ξ)h2(ξ)

ξdξ =

i

2π

∫R

h1(ξ)h2(ξ)

ξdξ ∈ iR, ∀h1, h2 ∈ Tu(ΛN ). (4.5.29)

Since the symplectic form ω is real-valued, we have ωu(h1, h2) = 0, for every h1, h2 ∈ Tu(ΛN ). SincedimR(ΛN ) = N = 1

2 dimR UN , ΛN is a Lagrangian submanifold of UN .

4.5.6 The symplectomorphism property

Finally, we prove the assertion (b) in theorem 4.5.2, i.e. the map ΦN : (UN , ω) → (ΩN × RN , ν) is

symplectic, where ω(h1, h2) := i2π

∫Rh1(ξ)h2(ξ)

ξ dξ, for every h1, h2 ∈ Tu(UN ) and

ΩN × RN = (r1, r2, · · · , rN ;α1, α2, · · · , αN ) ∈ R2N : r1 < r2 < · · · < rN < 0, ν =

N∑j=1

drj ∧ dαj .

We set ΨN = Φ−1N : ΩN × RN → UN , let Ψ∗Nω denote the pullback of the symplectic form ω by ΨN , i.e.

for every p = (r1, r2, · · · , rN ;α1, α2, · · · , αN ) ∈ ΩN × RN , set u = ΨN (p) ∈ UN ,

(Ψ∗Nω)p (V1, V2) = ωu(dΨN (p)(V1),dΨN (p)(V2)), (4.5.30)

for every V1, V2 ∈ Tp(ΩN × RN ). The goal is to prove that

ν := Ψ∗Nω − ν = 0. (4.5.31)

Recall that the coordinate vectors ∂∂r1

∣∣p, ∂∂r2

∣∣p, · · · , ∂

∂rN

∣∣p; ∂∂α1

∣∣p, ∂∂α2

∣∣p, · · · , ∂

∂αN

∣∣p

form a basis for the

tangent space Tp(ΩN × RN ). We have the following lemma.

Lemma 4.5.22. For every u ∈ UN , set p = ΦN (u) ∈ ΩN × RN . Then we have

dΦN (u)(XIk(u)) =∂

∂αk

∣∣∣p, ∀k = 1, 2, · · · , N. (4.5.32)


Demonstration. Fix u ∈ UN and p = ΦN (u), for every h ∈ Tu(UN ), we have dΦN (u)(h) ∈ Tp(ΩN ×RN ).For every smooth function f : p = (r1, r2, · · · , rN ;α1, α2, · · · , αN ) ∈ ΩN × RN 7→ f(p) ∈ R, then

(dΦN (u)(h)) f = d(f ΦN )(u)(h) =

N∑j=1

(dIj(u)(h)

∂f

∂rj

∣∣∣p

+ dγj(u)(h)∂f

∂αj

∣∣∣p

). (4.5.33)

For every k = 1, 2, · · · , N , we replace h by XIk(u), where XIk denotes the Hamiltonian vector field of thek th action Ik defined in (4.5.1), thus the Poisson bracket formulas (4.5.23) yield that

∂f

∂αk

∣∣∣p

=

N∑j=1

(Ik, Ij(u)

∂f

∂rj

∣∣∣p

+ Ik, γj(u)∂f

∂αj

∣∣∣p

)= (dΦN (u)(XIk(u))) f.

Lemma 4.5.23. For every 1 ≤ j < k ≤ N , there exists a smooth function cjk ∈ C∞(ΩN × RN ) suchthat

ν =∑

1≤j<k≤N

cjkdrj ∧ drk,∂cjk∂αl

∣∣∣p

= 0, ∀j, k, l = 1, 2, · · · , N, (4.5.34)

for every p = (r1, r2, · · · , rN ;α1, α2, · · · , αN ) ∈ ΩN × RN .

Demonstration. The proof is divided into three steps. The first step is to prove that for every p ∈ ΩN×RNand every V ∈ Tp(ΩN × RN ),

νp(∂

∂αl

∣∣∣p, V ) = 0, ∀l = 1, 2, · · · , N. (4.5.35)

In fact, let u = ΨN (p) ∈ UN and p = (r1, r2, · · · , rN ;α1, α2, · · · , αN ), so rl = rl(p) = Il ΨN (p). Then

(Ψ∗Nω)p(∂

∂αl

∣∣∣p, V ) = ωu(dΨN (p)

(∂

∂αl

∣∣∣p

),dΨN (p)(V )) = ωu(XIl(u),dΨN (p)(V ))

by (4.5.32). Thus (Ψ∗Nω)p(∂∂αl

∣∣∣p, V ) = −dIl(u)(dΨN (p)(V )) = −d(Il ΨN )(p)(V ). On the other hand,

νp(∂

∂αl

∣∣∣p, V ) =

N∑j=1

(drj ∧ dαj)

(∂

∂αl

∣∣∣p, V

)= −drl(p)(V )

Thus (4.5.35) is obtained by ν = Ψ∗Nω − ν.

Since ν is a smooth 2-form on ΩN × RN , we have

ν =∑

1≤j<k≤N

(ajkdαj ∧ dαk + bjkdrj ∧ dαk + cjkdrj ∧ drk),

for some smooth functions ajk, bjk, cjk ∈ C∞(ΩN × RN ), 1 ≤ j < k ≤ N . The second step is to prove

that ajk = bjk = 0 on ΩN × RN , for every 1 ≤ j < k ≤ N . In fact, we have drj ∧ drk( ∂∂αl

∣∣∣p, V ) = 0,

drj ∧ dαk(∂

∂αl

∣∣∣p, V ) = −1k=ldr

j(p)(V ) and dαj ∧ dαk(∂

∂αl

∣∣∣p, V ) = 1j=ldα

k(p)(V )− 1k=ldαj(p)(V ).


Then, let l ∈ 2, · · · , N be fixed, for every 1 ≤ j < k ≤ N , we have∑1≤l<k≤N

alkdαk(p)(V )−∑

1≤j<l≤N

(ajldαj(p)(V ) + bjldr

j(p)(V )) = νp(∂

∂αl

∣∣∣p, V ) = 0. (4.5.36)

Then we replace V by ∂∂rj

∣∣∣p

and ∂∂αj

∣∣∣p

respectively in formula (4.5.36), then ajl = bjl = 0, for every

j = 1, 2, · · · , l − 1.

It remains to show that cjk depends on r1, r2, · · · , rN , for every 1 ≤ j < k ≤ N . The symplectic form ω

is closed by proposition 4.4.4 and ν = dκ is exact, where κ =∑Nj=1 r

jdαj . So

dν = d(Ψ∗Nω)− dν = Ψ∗N (dω) = 0.

The exterior derivative of ν =∑

1≤j<k≤N cjkdrj ∧ drk is computed as following

0 =∑

1≤j<k≤N

N∑l=1

(∂cjk∂αl

dαl ∧ drj ∧ drk +∂cjk∂rl

drl ∧ drj ∧ drk).

Since the family drj ∧ drk ∧ dαl1≤j<k≤N,1≤l≤N⋃drj ∧ drk ∧ drl1≤j<k<l≤N is linearly independent

in Ω3(UN ), we have∂cjk∂αl

= 0, for every 1 ≤ j < k ≤ N and l = 1, 2, · · · , N .

Since the 2-form ν is independent of α1, α2, · · · , αN , it suffices to consider points p = (r, α) ∈ ΩN × RNwith α = 0. We shall prove that ν = 0 by introducing the following Lagrangian submanifold of ΩN ×RN ,

ΩN × 0RN = (r1, r2, · · · , rN ; 0, 0, · · · , 0) ∈ R2N : r1 < r2 < · · · < rN < 0.

End of the proof of formula (4.5.31). The submersion level set theorem implies that ΩN × 0RN is aproperly embedded N -dimensional submanifold of ΩN × RN . We have ΩN × 0RN = ΦN (ΛN ), whereΛN is the Lagrangian submanifold of (UN , ω) defined by (4.5.27). For every q ∈ ΩN × 0RN , set v =ΨN (q) ∈ ΛN , we claim at first that

Tq(ΩN × 0RN ) =

N⊕j=1

R∂

∂rj

∣∣∣q

= dΦN (v)(Tv(ΛN )). (4.5.37)

In fact, every tangent vector V ∈ Tq(ΩN × 0RN ) is the velocity at t = 0 of some smooth curveξ : t ∈ (−1, 1) 7→ ξ(t) = (ξ1(t), ξ2(t), · · · , ξN (t); 0, 0, · · · , 0) ∈ ΩN × 0RN such that ξ(0) = q, i.e.

V f =d

dt

∣∣∣t=0

(f ξ) =

N∑j=1

ξ′j(0)∂f

∂rj

∣∣∣q, ∀f ∈ C∞(ΩN × RN ). (4.5.38)

So the first equality of (4.5.37) is obtained. Then we set η(t) = ΨN ξ(t), ∀t ∈ (−1, 1). For everyg ∈ C∞(UN ), we replace f by g ΨN ∈ C∞(ΩN × RN ) in (4.5.38) to obtain that

dΨN (q)(V )g = V (g ΨN ) =d

dt

∣∣∣t=0

(g η) = η′(0)g.


Since η is a smooth curve in the Lagrangian section ΛN such that η(0) = v, we have dΨN (q)(V ) = η′(0) ∈Tv(ΛN ). So formula (4.5.37) holds. Since ν =

∑Nj=1 drj∧dαj , the submanifold ΩN×0RN is Lagrangian.

For every p = (r1, r2, · · · , rN ;α1, α2, · · · , αN ) ∈ ΩN × RN and every V1, V2 ∈ Tp(ΩN × RN ), where

Vm =

N∑j=1

(a

(m)j

∂

∂rj

∣∣∣p

+ b(m)j

∂

∂αj

∣∣∣p

), a

(m)j , b

(m)j ∈ R, m = 1, 2,

we choose q = (r1, r2, · · · , rN ; 0, 0, · · · , 0) ∈ ΩN × 0RN and W1,W2 ∈ Tq(ΩN × 0RN ), where

Wm =

N∑j=1

a(m)j

∂

∂rj

∣∣∣p, m = 1, 2.

We set v = ΨN (q) ∈ ΛN . We have proved that cjk(p) = cjk(q), then (4.5.34) yields that

νp(V1, V2) =∑

1≤j<k≤N

a(1)j a

(2)k cjk(p) = νq(W1,W2) = ωv(dΨN (v)(W1),dΨN (v)(W2)),

because νq(W1,W2) = 0. The identification (4.5.37) yields that hm := dΨN (v)(Wm) ∈ Tv(ΛN ), form = 1, 2. Consequently, we have νp(V1, V2) = ωv(h1, h2) = 0.

Formula (4.5.31) is equivalent to Φ∗Nν = ω, so ΦN : (UN , ω) → (ΩN × RN , ν) is a symplectomorphism.Finally, we recall a basic property in symplectic geometry : the three formulas in (4.5.4) are equivalentto the symplectomorphism property of ΦN .

Proposition 4.5.24. If ΦN : (UN , ω)→ (ΩN × RN , ν) is a diffeomorphism,

ΦN (u) = (I1(u), I2(u), · · · , IN (u); γ1(u), γ2(u), · · · , γN (u)), ∀u ∈ UN ,

for some smooth functions Ij , γj on UN , then each of the following three properties implies the others :

(a). ΦN : (UN , ω)→ (ΩN × RN , ν) is a symplectomorphism, i.e. Φ∗Nν = ω.(b). For every j, k = 1, 2, · · · , N , we have Ij , Ik = γj , γk = 0 and Ij , γk = 1j=k on UN .(c). For every k = 1, 2, · · · , N , we have

dΦN (u)(XIk(u)) =

∂

∂αk

∣∣∣ΦN (u)

, dΦN (u)(Xγk(u)) = − ∂

∂rk

∣∣∣ΦN (u)

, ∀u ∈ UN .

Demonstration. (a) ⇒ (b). For any smooth function f : ΩN × RN → R, its Hamiltonian vector field isgiven by

Xf (p) =

N∑j=1

∂f

∂rj(p)

∂

∂αj

∣∣∣p− ∂f

∂αj(p)

∂

∂rj

∣∣∣p, ∀p ∈ ΩN × RN . (4.5.39)

If Φ∗Nν = ω, then XfΦN (u) = dΨN (p) Xf (p), if p = ΦN (u), where ΨN = Φ−1N . The Poisson bracket of

two smooth functions f, g on ΩN × RN is given by

f, gν(p) = (Xfg)∣∣∣p

= νp(Xf (p), Xg(p)) =

N∑j=1

∂f

∂rj(p)

∂g

∂αj(p)− ∂f

∂αj(p)

∂g

∂rj(p). (4.5.40)

4.6. APPENDICES 133

Then f ΦN , g ΦN = f, gν ΦN on UN . It suffices to choose f, g ∈ Ij ΨN , γj ΨN1≤j≤N .

(b)⇒ (c). We do the same calculus as in lemma 4.5.22 to obtain that

dΦN (u)(XIk(u)) =

∂

∂αk

∣∣∣ΦN (u)

, dΦN (u)(Xγk(u)) = − ∂

∂rk

∣∣∣ΦN (u)

, ∀u ∈ UN . (4.5.41)

(c) ⇒ (a). The same calculus as in lemma 4.5.22 yields that (c) ⇒ (b). Formula (4.5.41) implies thatXI1

, XI2, · · · , XIN

;Xγ1, Xγ2

, · · · , XγN forms a basis in X(UN ). Since the 2-covectors (Φ∗Nν)u and ωu

coincide at every couple of elements of this basis, they are the same, so Φ∗Nν = ω.

4.6 Appendices

We establish several topological properties of the N -soliton manifold UN without using the action–anglemap ΦN : UN → ΩN × RN . The Viete map V : (β1, β2, · · · , βN ) ∈ CN 7→ (a0, a1, · · · , aN−1) ∈ CN isdefined by

N∏j=1

(X − βj) =

N−1∑k=0

akXk +XN . (4.6.1)

Proposition 4.6.1. Endowed with the Hermitian form H introduced in (4.4.15), (Π(UN ),H) is a simplyconnected Kahler manifold which is biholomorphically equivalent to V(CN− ).

Proposition 4.6.2. The N -soliton manifold UN is a universal covering manifold of the following N -gappotential manifold for the BO equation on the torus T := R/2πZ as described by Gerard–Kappeler [54],

UTN = v = h+ h ∈ L2(T,R) : h : y ∈ T 7→ −eiyQ

′(eiy)

Q(eiy)∈ C, Q ∈ C+

N [X], (4.6.2)

where C+N [X] consists of all monic polynomial Q ∈ C[X] of degree N , whose roots are contained in the

annulus A := z ∈ C : |z| > 1. The fundamental group of UTN is (Z,+).

Remark 4.6.3. The real analytic symplectic manifold UTN is mapped real bi-analytically onto CN−1×C∗

by the restriction of the Birkhoff map constructed in Gerard–Kappeler [54]. The union of all finite gappotentials

⋃N≥0 U

TN is dense in L2

r,0(T) = v ∈ L2(T,R) :∫T v = 0. However

⋃N≥1 UN is not dense in

L2(R, (1 + x2)dx). We refer to Coifman–Wickerhauser [28] to see solutions with sufficiently small initialdata and the case of non-existence of rapidly decreasing solitons.

The simple connectedness of UN is proved in subsection 4.6.1. Then we establish a real analytic coveringmap UN → UT

N in subsection 4.6.2.

4.6.1 The simple connectedness of UNThanks to the biholomorphical equivalence between the Kahler manifolds Π(UN ) and V(CN− ) establishedin lemma 4.4.10, it suffices to prove the simple connectedness of the subset V(CN− ), where V denotes theViete map defined by (4.6.1). Since every fiber of the Viete map is invariant under the permutation of


components, we introduce the following group action. Equipped with the discrete topology, the symmetricgroup SN acts continuously on CN by permuting the components of every vector :

σ : (β0, β1, · · · , βN−1) ∈ CN 7→ (βσ(0), βσ(1), · · · , βσ(N−1)) ∈ CN , ∀σ ∈ SN . (4.6.3)

A subset A ⊂ CN is said to be stable under SN if⋃σ∈SN

σ(A) = A. We recall the basic property of theViete map V and the action of symmetric group SN .

Lemma 4.6.4. The Viete map V : CN → CN is a both open and closed quotient map. For everyA ⊂ CN , A is stable under SN if and only if A is saturated with respect to V, the quotient space A/SNis homeomorphic to V(A).

We set ∆ := (β, β, · · · , β) ∈ CN : ∀β ∈ C. The goal of this subsection is to prove the following result.

Proposition 4.6.5. For every open simply connected subset A ⊂ CN , if A is stable under the symmetricgroup SN and A

⋂∆ 6= ∅, then V(A) is an open simply connected subset of CN .

Demonstration. Let A ⊂ CN be a nonempty open simply connected subset that is stable by SN . Thesubset B := V(A) is open, connected and locally simply connected, then it admits a universal coveringspace E and a covering map π : E → B. The triple (E, π,B) is identified as a fiber bundle over B whosemodel fiber F is discrete. The target is to show that F has cardinality 1.

Let (P, q, B) denote the fiber product (Husemoller [77]) of bundles (A,V, B) and (E, π,B), defined by

P = A×B E := (β, e) ∈ A× E : π(e) = V(β), q : (β, e) ∈P 7→ V(β) = π(e) ∈ B. (4.6.4)

The total space P is equipped with the subspace topology of the product space A× E and projectionsonto the first factor and onto the second factor are denoted respectively by

p : (β, e) ∈P 7→ β ∈ A, W : (β, e) ∈P 7→ e ∈ E. (4.6.5)

Both p and W are continuous functions on P and the following diagram commutes.

P E

A B

p q

W

V

π

We claim two properties concerning the projections p and W.

i. W : P → E is an open quotient map and p : P → A is a covering map whose model fiber is F.ii. Equipped with the discrete topology, the symmetric group SN acts continuously on P by permutingcomponents of the first factor

σ : (β, e) ∈P 7→ (σ(β), e) ∈P, ∀σ ∈ SN ,

where σ ∈ GLN (C) is defined by (4.6.3). Hence the quotient map W : P → E is closed.

Thanks to the simple connectedness of the base space A, the covering space P is the disjoint unionof its connected components (Ak)k∈F and the restriction of the covering map p|Ak

: Ak → A is a

4.6. APPENDICES 135

homeomorphism. Since P is locally path-connected, every component Ak is both open and closed, thenW|Ak

: Ak → E an open closed quotient map. So is the lift gk := W|Ak (p|Ak

)−1 : A → E. Notethat π gk = V and SN stabilizes every element of ∆. We choose β ∈ A

⋂∆ and b := V(β). Since the

fiber V−1(b) = β is a singleton, so is the fiber π−1(b). Hence |F| = 1 and the universal covering mapp : E → B is a homeomorphism. So B is simply connected.

Remark 4.6.6. Let F be a closed submanifold of a smooth connected manifold M without boundary offinite dimension. If dimRM−dimR F ≥ 3, then the inclusion map i : M\F →M induces an isomorphismbetween the fundamental groups i∗ : π1(M\F, x) → π1(M,x), for every x ∈ M\F (see Theoreme 2.3 inP.146 of Godbillon [62]). Note that the closed submanifold ∆ ⊂ CN has real dimension 2. When N ≥ 3,the condition A

⋂∆ 6= ∅ cannot be deduced by the other three conditions in the hypothesis of proposition

4.6.5 : A is open, simply connected and stable by SN .

As a consequence, V(CN− ) is open and simply connected because CN− is an open convex subset of CNwhich is stable under the symmetric group SN and ∆

⋂CN− = (z, z, · · · , z) ∈ CN : Imz < 0. Together

with lemma 4.4.10, we finish the proof of proposition 4.6.1.

4.6.2 Covering manifold

The Szego projector on L2(T,C) is given by ΠTv(x) =∑n≥0 vne

inx, for every v ∈ L2(T,C) such that

v(x) =∑n∈Z vne

inx with vn = 12π

∫ 2π

0v(x)e−inxdx. Equipped with the subspace topology of ΠT(L2(T,C))

and the Hermitian form

HT(v1, v2) = 〈D−1ΠTv1,ΠTv2〉L2(T) =

1

2π

∫ 2π

0

D−1ΠTv1(x)ΠTv2(x)dx,

the subset ΠT(UTN ) is a Kahler manifold, which is mapped biholomorphically onto V(A N ) with A =

C\D(0, 1) = z ∈ C : |z| > 1 in Gerard–Kappeler [54].

Proposition 4.6.7. There exists a covering map π : V(CN− )→ V(A N ).

Remark 4.6.8. Consider the cubic Szego equation on the torus (see Gerard–Grellier [47, 49, 51, 52])

i∂twT = ΠT(|wT|2wT), (t, x) ∈ R× T, (4.6.6)

and the cubic Szego equation on the line (see Pocovnicu [119, 120]), we set ΠR := Π in (4.1.12),

i∂twR = ΠR(|wR|2wR), (t, x) ∈ R× R. (4.6.7)

The manifold of N -solitons for the cubic Szego equation on the line is not simply connected. Let M(N)R

denote all rational functions of the form wR : x ∈ R 7→ P (x)Q(x) ∈ C where P ∈ C≤N−1[X] and Q ∈ CN [X]

is a monic polynomial such that Q−1(0) ⊂ C− and P,Q have no common factors. Then M(N)R is aKahler manifold of complex dimension 2N . So is the subset M(N)T consisting of all rational functions

of the form wT : x ∈ T 7→ P (eix)Q(eix) ∈ C where P ∈ C≤N−1[X] and Q ∈ CN [X] is a monic polynomial such

that Q−1(0) ⊂ A and P,Q have no common factors. Both of them have rank characterization of Hankeloperators by Kronecker-type theorem (see Lemma 8.12 in Chapter 1 of Peller [117], p. 54). So the manifoldM(N)R (resp. M(N)T) is invariant under the flow of equation (4.6.7) (resp. of equation (4.6.6)) and the(generalized) action–angle coordinates of equation (4.6.7) (resp. of equation (4.6.6)) are defined in some


open dense subset of M(N)R (resp. of M(N)T). Moreover, if N ≥ 2 then M(N)R is simply connectedby proposition 4.6.5 and remark 4.6.6. There exists a holomorphic covering map M(N)R → M(N)T byfollowing the construction in proposition 4.6.7. The manifold of N -solitons for the cubic Szego equationon the line is an open dense subset of M(N)R

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Titre: Comportement en grand temps et intégrabilité de certaines équations dispersivessur l'espace de Hardy

Mots clés: Équation de Schrödinger non linéaire, Projecteur de Szeg®, Équation de BenjaminOno, Espace de Hardy, Onde progressive, Coordonnées d'actionangle

Résumé: On s'intéresse dans cette thèse àtrois modèles d'équations hamiltoniennes dis-persives non linéaires: l'équation de Schrödingercubique défocalisante ltrée par le projecteur deSzeg® ΠT, qui enlève tous les modes de Fourierstrictement négatifs, sur le tore T := R/2πZ(NLSSzeg® cubique), l'équation de Schrödingerquintique focalisante ltrée par le projecteur deSzeg® ΠR sur la droite R (NLSSzeg® quin-tique) et l'équation de BenjaminOno (BO) surla droite. Ces trois modèles nous donnentl'occasion d'étudier les propriétés qualitatives de

certaines ondes progressives, le phénomène decroissance de normes de Sobolev, le phénomènede diusion non linéaire et certaines propriétésd'intégrabilité de systèmes dynamiques hamil-toniens. Le but de cette thèse est de compren-dre l'inuence des opérateurs non locaux ΠT etΠR sur des équations de type de Schrödingeret d'adapter les outils liés à l'espace de Hardysur le cercle et sur la droite. Dans le troisièmemodèle, la théorie de l'intégrabilité permet defaire le lien avec certains aspects algébriques etgéométriques.

Title: Long time behavior and integrability of some dispersive equations on the Hardy

space

Keywords: Non linear Schrödinger equation, Szeg® projector, BenjaminOno equation, Hardyspace, Traveling wave, Actionangle coordinates

Abstract: We are interested in three non lin-ear dispersive Hamiltonian equations: the de-focusing cubic Schrödinger equation ltered bythe Szeg® projector ΠT that cancels every neg-ative Fourier modes, leading to the cubic NLSSzeg® equation on the torus T := R/2πZ; thefocusing quintic Schrödinger equation, which isltered by the Szeg® projector ΠR, leading to thequintic NLSSzeg® equation on the line R; andthe BenjaminOno (BO) equation on the line.These three models allow us to study their qual-

itative properties of some traveling waves, thephenomenon of the growth of Sobolev norms,the phenomenon of non linear scattering andsome properties about the complete integrabil-ity of Hamiltonian dynamical systems. The goalof this thesis is to investigate the inuence of thenon local operators ΠT and ΠR on some one-dimensional Schrödinger-type equations and toadapt the tools of the Hardy space on the torusand on the line. In the third model, the integra-bility theory allows to establish the connectionwith some algebraic and geometric aspects.

Université Paris-Saclay

Espace Technologique / Immeuble Discovery

Route de l’Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

comportement en grand temps et intégrabilité de certaines

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