F!JDt HEWLETT~lj PACKARD

Energy Dependent Schrodinger Operatorsand Complex Hamiltonian Systems onRiemann Surfaces

Mark S. Alberl; Gregory G. Luther, Jerrold E. Marsden2

Basic Research Institute in theMathematical SciencesHP Laboratories BristolHPL-BRIMS-96-20August, 1996

energy~dependentSchrodmger Operator;semi-classicalintegrable;evofution equation;monodro~y;complex WICB

We use so-called energy dependent Schrodingeroperators to establish a liiik between special classesof solutions of N-component systems of evolutionequations and finite dimensional Hamiltonian§ystems on the moduli spaces of Riemann surfaces.We also investigate the phase space geometry ofthese Hamiltonian systems and introoucedeformations of the level sets associated toconserved quantities, which results in a new class ofsolutions WIth monodromy for N-component systemso(pde's.After constructing a variety of mechanical systemsrelated to the spatial flows of nonlinear evolutionequations, we investigate their semiclassical limits.IIi particUlar, we obtain semiclassical asymptoticsfor the Bloch ~igenfunctions of t4e ep.ergydependent Schrodmger operators, which IS ofimportance in investigating zero-dispersion limits ofN-component systems ofpoe's.

1 University of Notre Dame, Department of Maths2 Control and Dynamical Systems, Caltech, California

Internal Accession Date Only

Energy Dependent Schrodinger Operators and ComplexHamiltonian Systems on Riemann Surfaces

Mark S. Alber * Gregory G. Luther t

August 27, 1996

Jerrold E. Marsden :I:

SHORT TITLE: Schrodinger Operators and Complex Hamiltonian Systems

Abstract

We use so-called energy dependent Schrodinger operators to establish a link betweenspecial classes of solutions of N-component systems of evolution equations and finitedimensional Hamiltonian systems on the moduli spaces of Riemann surfaces. We alsoinvestigate the phase space geometry of these Hamiltonian systems and introduce de­formations of the level sets associated to conserved quantities, which results in a newclass of solutions with monodromy for N-component systems of pde's.

After constructing a variety of mechanical systems related to the spatial flows ofnonlinear evolution equations, we investigate their semiclassical limits. In particular,we obtain semiclassical asymptotics for the Bloch eigenfunctions of the energy dependentSchrodinger operators, which is of importance in investigating zero-dispersion limits ofN-component systems of pde's.

1 Introduction

There are several circumstances in which semiclassical solutions and techniques are usedfor nonlinear evolution equations. Of particular interest is the zero-dispersion limit. Directasymptotic expansions of the evolution equations typically yield equations of hydrodynamictype. More sophisticated techniques for integrable systems employ limiting techniques andWKB analysis in the context of the inverse scattering theory (see Lax and Levermore I, II,III [1983], Venakidis [1987] and for a survey see Lax, Levermore and Venakidis [1993]). Thestructure of integrable equations can also be exploited using complex geometric asymptoticsin this limit (see Alber [1989, 1991] and Alber and Marsden [1992]).

"Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556. Research partiallysupported by NSF grants DMS 9403861 and 9508711.

tDepartment of Mathematics, University of Notre Dame, Notre Dame, IN 46556, and the Basic ResearchInstitute in the Mathematical Sciences, Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, BristolBS12 6QZ, UK. GGL gratefully acknowledges support from BRIMS, Hewlett-Packard Labs and from NSFDMS under grant 9508711.

tControl and Dynamical Systems 104-44, Caltech, Pasadena, CA 91125. Research partially supportedby NSF grant DMS 9302992

1

2

Jin, Levermore and McLaughlin [1994] use a WKB-type expansion for the associatedscattering equations of the (d)NLS (defocusing nonlinear Schrodinger) equation and obtainequations for the conserved quantities in Riemann invariant form for the semiclassical limit(11, -+ 0). This work was extended in Bronski and McLaughlin [1994] who described theformation of caustics in a numerical context and showed that an extension of hydrodynamictype equations must be used to adequately account for the modulational instability of the(f)NLS (focusing nonlinear Schrodinger) equation.

There have been many important developments in which the methods of complex andalgebraic geometry have been used to investigate the eigenfunctions of Hill's operator in thecontext of integrable equations. For example, Bloch eigenfunctions of Hill's operators, whichare meromorphic on the associated spectral curves play an important role in the inversescattering transform method for nonlinear soliton equations. For details see Ablowitz andSegur [1981], Newell [1985] and Ablowitz and Clarkson [1991].

Connections have also been established between special classes of solutions of many non­linear equations and finite dimensional mechanical systems. Moser [1980,1981] and Knorrer[1982] used one-dimensional Schrodinger equations with finite gap potentials (having a fi­nite number of bands of continuous spectrum) and reflectionless potentials (having a finitenumber of elements of discrete spectrum) to demonstrate a connection between the classicalC. Neumann problem and quasi-periodic and soliton solutions of the KdV equation. Theyalso established a connection between the C. Neumann problem and the Jacobi problem ofgeodesics on quadrics.

For the C. Neumann problem, the spectral parameter appears linearly in the potentialof the corresponding Schrodinger equation: V = u - >.. In contrast, Antonowicz and Fordy[1987a,b, 1988, 1989] and Antonowicz, Fordy and Liu [1991] investigated potentials withpoles in the spectral parameter for what they refer to as energy dependent Schrodingeroperators connected to certain systems of evolution equations. Specifically, they obtainedmulti-Hamiltonian structures for N -component integrable systems of equations related tothe following isospectral eigenvalue problem:

(1.1)

(1.2)M N

K=:Lkj>.jj V=:LVj(x,t)>.j,j=O j=O

where the kj are constants and the Vj(x, t) are functions of the variable x, the parameter tand the spectral parameter>. is complex. This includes the coupled KdV and Dym systems.The presence of a pole in the potential was shown in Alber, Camassa, Holm and Marsden[1994b, 1995] to be essential for the existence of the weak billiard solutions of the nonlinearequations including the Dym equation and equations in its hierarchy. The weak billiardsolutions can be obtained for all N -component systems.

Well known techniques exist for establishing a link between mechanical systems andtheir quantum mechanical counterparts. One such technique uses the WKB method toanalyze asymptotic limits and thereby establish the corresponding semiclassical solutions(modes). In this paper we use this technique to establish certain semiclassical limits of the

(1.3)

(1.4)

3

spatial flows related to integrable nonlinear evolution equations. In this way we establisha connection between special classes of solutions of nonlinear equations and mechanicalsystems to form a semiclassical theory for these nonlinear equations.

In what follows we make several points. We use energy dependent Schrodinger operatorsof the type (1.1) and their associated spectral curves to establish a link between specialclasses of solutions of N -component systems of evolution equations and finite-dimensionalHamiltonian systems on Riemann surfaces. We apply special limiting procedures (involvingthe coalescence of roots of the basic polynomial of the spectral curve) to quasi-periodicsolutions to obtain a system with monodromy. This provides an example of a system ofpde's with monodromy. Prior to this, such effects were related to mechanical systems andtheir quantum counterparts such as the classical and quantum spherical pendulum (seeCushman and Duistermaat [1988], Duistermaat [1988], and Alber and Marsden [1996]).

We also describe several mechanical systems that are associated with different classes ofsolutions of N -component systems. All of these systems admit a geometric interpretationin the form of geodesic flows on n-dimensional pseudo-spheres in the fields of differentpotentials. Therefore they can be linked to the Laplace Beltrami operators in a way similarto the method of geometric optics. These systems are solved using algebraic geometricmethods, and semiclassical approximations for eigenfunctions of Laplace Beltrami operatorshaving the form of complex WKB modes are studied. Using these elements we proceed toinvestigate the monodromy of the semiclassical approximation to the spectrum.

To investigate the zero-dispersion limit of N -component systems one considers the classof generalized Schrodinger equations

L,p = (K:~2 +v),p =0 ,

whereM N

]( = Lkj£j, V = LVj(x,t»J .j=O j=O

These are similar to (1.1) and (1.2) except that the spectral parameters in the expansionsof K and V now have a different form. The parameter £ will be our small parameterin this study. In this paper we investigate the asymptotic limit € -t 0 of the associatedBloch eigenfunctions. Using this idea, we link Jacobi geodesic flows on quadrics and theWKB approximation of the eigenfunctions of the stationary one-dimensional Schrodingerequation.

Finally, we introduce time t into the system as a parameter and describe the dependenceof the WKB modes on this parameter.

2 Hamiltonian Systems on Riemann Surfaces and Solutionsof Nonlinear Equations

First we recall a link between quasi-periodic and soliton solutions of integrable nonlinearequations and Hamiltonian systems on Riemann surfaces (see, for example, Alber, Ca­massa, Holm and Marsden [1994, 1995]) established using algebraic geometric methods

4

(see, amongst others, Ercolani and McKean [1990]). Both the inverse scattering transformmethod for integrable systems (see Ablowitz and Segur [1981]) and method of generatingequations use a link between ends of the gaps of the continuous spectrum and elements ofthe discrete spectrum of the Schrodinger operators and first integrals for integrable systems.Potentials in the Schrodinger operator which yield finite gap structure of the spectrum anddiscrete spectrum are called finite gap potentials and reflectionless potentials respectively.

The quasi-periodic solutions of many integrable nonlinear equations can be described interms of finite dimensional Hamiltonian systems on C 2n . A complete set of first integralsfor such equations can be obtained, for example, by the method of generating equations,as is summarized in Alber, Camassa, Holm and Marsden [1994, 1995]. The method ofgenerating equations introduces a finite dimensional complex phase space c2n and twocommuting Hamiltonian flows. The first Hamiltonian flow gives the spatial evolution, andthe other gives the temporal evolution of special classes of solutions of the original partialdifferential equation. The level sets of the first integrals are Riemann surfaces having branchpoints parameterized by the choice of values of the first integrals.

Consider the finite gap spectrum of the operator (1.1). (For details about the timeevolution of the eigenfunctions of generalized Schrodinger equations of this type and multi­Hamiltonian structures associated with them, see Antonowicz and Fordy [1988, 1989].) Insome cases, such as the KdV equation, A appears as an eigenvalue and one ultimatelyequates the potential with a quasi-periodic solution of the nonlinear equation itself. Inother cases such as the NLS equation, a connection between the potential and solutions ofthe equations is more complicated but can be established using the method of generatingequations.

To carry out the procedure, one begins by looking for a solution A of the Lax system

of the form

L'lj; =0 }

(~~ + [L,A]) 'lj; =0

A - B~ _ ~aB- ax 2 ax '

(2.1)

(2.2)

where L is defined in (1.1). Substituting the given form of A into the Lax system, oneobtains

av Bill V V'at =-2 +2B' J( +B J( , (2.3)

where the prime denotes a/ax. In what follows we will also sometimes use a dot to denotea/at, so V = av/at.

Equation (2.3) is called the generating equation. Expanding B and V in the parameterA and equating like powers produces different hierarchies of integrable systems. In manycases these expansions are polynomial in A with coefficients depending on functions Vj anda finite number of their space derivatives as well as one time derivative. Note that a fixedpotential V produces a recurrence relation for one hierarchy of evolution equations. Forinstance, if we take B to be polynomial of degree n, defining B = Bn = 2:1=0 bjAn-j, weobtain the nth system of evolution equations with dependent variables Vj (see (1.2)).

5

Stationary solutions of any member of the hierarchy such that 8Vj 8t = 0 depend onB and V, but the evolution variable, t, becomes a parameter. Setting 8Vj8t = 0 in (2.3),multiplying by BK and integrating once produces the stationary generating equation,

K(-B"B + B;') +2B'V = CIA) , (2.4)

where C(>.) is a polynomial of>. of order (2n +N). A similar equation was first obtained inconnection with the KdV equation in Gelfand and Dikey [1975] by investigating asymptoticsof resolvents of the Sturm-Liouville equations.

The evolution of B is obtained using a dynamic generating equation. At each instant,that is for each value of the parameter t, B is a solution of the the stationary generatingequation (2.4). Differentiating (2.4) with respect to t and requiring consistency with (2.3)yields,

(2.5)

where Bl = 2:;=0 bj>.n-j is a solution of the dynamical generating equation (2.3). Thisdetermines the time evolution of the functions B. A proof is given, for example, in Alberand Alber [1987] and Alber et al. [1994]. Here the n is calculated using the number of rootsof the spectral polynomial. These may be endpoints of gaps or isolated poles correspondingto solitons. Thus, n is the dimension of the solution space and [labels the equation in thehierarchy.

To show how the generating equations yield the quasi-periodic n-gap solutions of evolu­tion equations, one introduces the roots, /l, of B = Bn = TI]=1(>' - /lj(x,t)). Substituting>. = /lj, j = 1, ... , n one by one into (2.4) one obtains a system of ordinary differentialequations for the spatial flow of the /l-variables:

C(J.Li)K(J.Li) , j = 1, ... , n .

, 1/lj = TIn ( )r::f.j /li - /lr

The motion in t is produced by substituting>. = J.Li with B this time in (2.5) so that,

(2.6)

(2.7)C(J.Li) .K (/li) , J = 1, ... , n .

. Bl(J.Li)/li = n

TIr::f.j(/li - J.Lr)

Solutions of this system of equations for J.Li(x, t) can be related to the Vj(x, t) to obtainsolutions of the original hierarchies of evolution equations generated by (2.3). Basic factsabout these systems are therefore of great interest. The systems (2.6) and (2.7) have aHamiltonian structure with

(2.8)n D(j.L') (P'l- _ C(~j~)

H - '""' J J K(~1 . - 1- L.J TIn ( )' J - , ... , n ,

j=1 r::f.j J.Lj - J.Lr

where D(J.Li) = 1 and D(J.Lj) = Bl (J.Lj) in the stationary and dynamical cases, respectively.The two Hamiltonian flows have the same set of first integrals, whose zero level sets are,

p7 _ C(/lj)J - K(/li) , j = 1, ...,n. (2.9)

(2.10)

6

One can think of e2n as being the cotangent bundle of en, with configuration vari­ables Itl' ... ,ltn and with canonically conjugate momenta PI, ... , Pn. The two commutingHamiltonians on e 2n both have the form

1 .. 2H = 2g)]Pj +V(ltb ... ltn) ,

where gji is a Riemannian metric on en. The two Hamiltonians are distinguished bydifferent choices of the metric but they have the same set of first integrals (2.9). Thus, twocommuting flows are obtained on the symmetric product of n copies of the Riemann surface

defined by the first integrals. It is called the spectral curve associated with the completelyintegrable problem. Branch points of the Riemann surface are given by the roots of C(>')and K(>.). (For details about completely integrable systems on complex tori see Ercolaniand McKean [1990]). These Riemann surfaces can be regarded as complex Lagrangiansubmanifolds. This is called the It-representation of the problem. Sometimes the It variablesare also called Dirichlet eigenvalues or an auxiliary spectrum. The It variables move alongcycles on the corresponding Riemann surface in accordance with the equations (2.6) and(2.7), which define their motion over the basic cuts on the Riemann surface whose endpointsare the endpoints of the gaps in the spectrum of the associated operator (1.1).

In the case of quasi-periodic and soliton solutions, one obtains compact and noncompactRiemann surfaces (see for example Alber and Marsden [1992, 1994] and §3 and §4 of thispaper). The It-representation yields complex angle representations on Jacobi varieties ina way described in Alber et. at. [1994, 1995]. This yields, amongst other things, a repre­sentation of the It-variables in terms of () and r-functions (see McKean[1979] and Mumford[1983]). In the last step, the solution of the initial nonlinear equation is expressed in termsof the J.L-variables.

In the J.L-representation the calculation of modulation equations is facilitated. Whithamequations can be obtained in terms of holomorphic and meromorphic differentials. Theydescribe the slow modulation of branch points on the Riemann surfaces (see Flaschka,Forest and McLaughlin [1980]). Notice that the monodromy effect to be described in §4 isimportant in investigating Whitham equations of the N -component systems.

In the most general case, the J.L-representation describes a geodesic flow for the metricds2 = 2:j=l gjjJ.Li

2 on complex Riemannian manifolds. Classification of all such metricswhich yield separation of variables in Hamilton-Jacobi equations and integrability of thegeodesic flows has been investigated recently in Benenti et at. [1995]. In the context oftheirpaper J.L-variables are viewed as generalized elliptic coordinates.

We will use this interpretation of the J.L-variables in the next section to clarify a connec­tion to mechanical systems. Namely, the procedure outlined in this section also producesmany important classical as well as new integrable mechanical systems with Hamiltoniansof the form (2.10) for particular choices of Band V. These expansions can also be usedto obtain semiclassical solutions of the eigenvalue problem represented by L in (1.1) andyields asymptotics for the Bloch eigenfunctions. These finite dimensional representations

(3.1)

(3.2)

7

also permit the analysis of a connection between caustic envelopes in the phase space ofthese mechanical systems and Maslov-Keller indices in the semiclassical approximation ofthe spectrum ofthe Laplace-Beltrami operators (see Alber [1989,1991], Alber and Marsden[1992] and §6 and §7 of this paper).

3 The Family of Homoclinic Orbits Described as Hamilto­nian Flows on a Riemann surface

Here we recall the J-t representation for the quasi-periodic and homoclinic Hamiltonian flowsof the classical C. Neumann problem of mechanics.

Devaney [1978] investigated homoclinic orbits ofthe C. Neumann problem. Moser [1981]studied equilibrium solutions possessing stable and unstable manifolds for this mechanicalproblem in connection with the spectral theory of finite gap and reflectionless potentialsof Schrodinger operators and quasi-periodic and soliton solutions of the KdV equation. Inparticular, it was shown that equilibrium solutions are associated with the reflectionlesspotentials which correspond to a set of negative double points of the discrete spectrum.

Here we show that a special limiting process applied to the quasi-periodic J-t representa­tion leads to the introduction of a new set of first integrals and exponential Hamiltoniansfor homoclinic orbits. (See Alber and Marsden [1994] for details). This approach will beapplied in the next section to the 2-component Dym system to describe special classes ofsolutions with monodromy.

The system of first integrals introduced by Devaney and Uhlenbeck (see Devaney [1978])for the C. Neumann problem, namely

Fj(Y, if) : TSn-+ R ,

F o( .)_ ~+~",(ifjYk-ifkYj)2] Y, Y - Y] 2 LJ 12 -1~ ,j = 1, ... , n

k#i k ]

plays a central role in Devaney's [1978] description of transversal homoclinic orbits. Here

10 < 1} < ... < 1n and 10 = 0 .

Namely, he proved that the first integrals are identically zero along the orbits; i.e.,

Fi(y,if) =0, j = 1, ... ,n. (3.3)

(3.4)

We shall use this observation together with the J.L representations for the quasi-periodicflows of the C. Neumann problem, i.e., solutions of the system

OJ.Li _ 2 J- I1~=} (J.Li - m r ) I1k~~ (J.Li - ak) . - 1~ - ()' J - , ... , n .uX Ili#i J.Li - J.Li

In accordance with the general result described in §2, this system is defined on the thesymmetric product

f: (~x ... x ~)/(Jn (3.5)

8

of n copies of the Riemannian surface

n+1 n

~: p2 = - II (J.' - ak) II (JL - mr) ;k=l r=l

(3.6)

in fact, r is a Lagrangian submanifold of the phase space C 2n • In this setting, the firstintegrals of the problem can be represented in the form

(3.7)

Here,aj = -21J, j =0, ... , n ,

and the constants m r and aj are the endpoints of the gaps in the spectrum of the associatedSchrodinger equation. Condition (3.3) and formula (3.7) yield the following choice of firstintegrals mj:

mj = aj = bj, j = 1, ..., n; ao = 0 , (3.8)

which corresponds to a homoclinic orbit, meaning that all roots of the basic polynomial ofthe Riemannian surface (3.6) are double negative roots, except ao = O.

The system (3.4) corresponding to the case of a singular spectrum (3.8) coincides withthe system of equations describing profiles of the soliton solutions of the KdV equation;that is, with

OJLj =2V-JLj TIk=l(JLj - bk) .!:l TI ( )' J = 1, ... , n .uX ii:j JLj - JLi

This system may be rewritten as

(3.9)

(3.10)

Summing with respect to j = 1, ... , n for each r and integrating, one obtains the followingangle variables

~ 1 f~j dJLj 0Or = L '2 }"o ..;=JIj( . _ b )) = X + Or, r = 1, ... , n .

j=l ~j /-LJ /-LJ r

The corresponding action function and action variables can be chosen as follows

~ 11~j 10g(JLj - br )dJLjS = L 2 0 ~ ,Ir = ar , r = 1, ... , n .

j=l~} V -JLJ

(3.11)

(3.12)

This results in Or = -oS/oIr and so leads to the following form of the first integrals:

j = 1, ... ,n. (3.13)

(3.14)

9

This gives a new Hamiltonian on C 2n that is of exponential type for the system (3.9)describing the family of homoclinic orbits, namely

H = 'Li=l (e2';=;;Pj - C(J-Lj)) ,ilr#j(J-Lj - J-Lr)

where C(J-L) = ilk=l (J-L - ak). Note that the flow of (3.1.1) is defined on the noncompact levelsets of (3.13) and it is linearized in terms of the angle variables (3.11).

The homoclinic angle representations (3.11) are defined on a noncompact Jacobi variety.1. The variety .1 is defined by a generalized Abel Jacobi map (3.11), as in Ercolani [1989].This map is associated with ~n, the symmetric product of n copies of the Riemann surface

~: (3.15)

(3.16)

Let the double covering of Rn be denoted ~n. This covering is defined by the followingchange of variables

,2 - II. • {32 - b· JO - 1 n<"j - -r], j - - ]' -, ••• , •

The Hamiltonian (3.14) defines a dynamical system on ~n, which lifts to a dynamicalsystem on ~n. An analysis of the angle representation (3.11) shows that the system has ahomoclinic point b = (bI, ... , bn ) on ~n as x -+ 00 (or x -+ -00) and, correspondingly, twoheteroclinic points {3+ and {3- on ~n associated with the following values of J-Lj:

j = 1, ... ,n. (3.17)

(For details see Alber and Marsden [1994a].)The stable WS (and unstable W'U) manifolds of the point b are coincident and consist ofthe orbits in ~n that are forward (and backward) asymptotic to the homoclinic point b.On the other hand, the unstable manifold of the point {3+ connects it to the point {3- in~n and similarly for the stable manifold; these heteroclinic manifolds cover the homoclinicmanifold in ~n.

4 Solutions with Monodromy for the 2-component DymSystem

For the C. Neumann problem, the spectral parameter appears linearly in the potential ofthecorresponding Schrodinger equation. However, in other systems of interest, this dependenceis more complicated. In this section we discuss such systems; specifically, we consider Hamil­tonian systems on Riemann surfaces that are associated with energy dependent Schrodingeroperators. In these systems, the dependence of the potential on the spectral parameter canhave a pole. We also consider N -component systems of equations, using the hierarchy ofthe 2-component Dym systems as an example. In particular, we construct special solutionswith monodromy for these systems of equations.

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Using the method described in §2 and §3 with a potential of the form

v(x, t)V = u(x, t) +). +-).- ,

and the dynamical recurrence chain for coefficients bj obtained from the dynamical gener­ating equation (2.3) by setting B(x, >') = bo(x)>. + b1(x) and equating coefficients of thesame power of >., one obtains the following system of coupled pde's which is called the firstmember of the hierarchy of the 2-component Dym systems,

_vu = ~u'" - ~uU' + v' , }vt 4 2

VV I 1 I-vt = -u v - 2uV .

(4.1)

Notice that the choice of polynomial BI(X, >') =bo(x )>.1 +...+bl of the l'th order yields thel'th integrable system of evolution equations of the 2-component Dym hierarchy.

In what follows, we will introduce a Hamiltonian structure for the set of quasi-periodicsolutions of the equations from the hierarchy of 2-component Dym systems. Certain lim­iting procedures applied to this Hamiltonian system will then yield special solutions withmonodromy.

One obtains j.L-representations for the l'th system from the hierarchy after substitutingE = J.Lj into the generating equations (2.4) and (2.5):

where

j = 1, ...,n,

j = 1, ...,n,

(4.2)

(4.3)

(4.4)

2n+2

C(J.Lj) = -L6 II (J.Lj - mk) ,k=l

1 labels the number of a system in the 2-component Dym hierarchy, and Lo is a constant.In the case 1= lone obtains J.L-representations for the system (4.1). Each ofthe J.L-variablesis defined on a copy of the lliemann surface

R : p 2 = C (J.L) .J.L

Recall that the J.L variables move along cycles on the corresponding Riemann surface (4.4)over the basic cuts between m2j and m2j-l, j = 1, ... ,n.

Equations (4.2) and (4.3) define Hamiltonian systems with the Hamiltonians

D(J.L') (P'l- _C(J.Lj))n ) ) J.Lj

H=L n )'j=l nr~j(J.Lj - J.Lr

(4.5)

11

where D(J.Lj) = 1 and D(J.Lj) = Bl(J.Lj) in the stationary (4.2) and dynamical (4.3) cases,respectively. These have the following common set of first integrals

P2 _ C(J.Lj) ._J' - -- , J - 1, ... , n .

J.Lj(4.6)

(4.8)

Now we construct a system of angle variables to linearize quasi-periodic Hamiltonian flowsusing a method similar to the one described in §3.

Here we are dealing with a degenerate system because the genus of the associated Rie­mann surface is (n + 1) and yet we have only n J.L-variables. Related to this, there is also adegeneracy in the problem of inversion.

Namely, after rearranging systems of equations (4.2) and (4.3), summing and usingLagrange-type interpolation formulas, one obtains the following expressions

(4.7)

where k = 1, ... , nand bk,b bk,Z are constants. After integrating (4.7), one obtains the anglevariables:

Ok = ~ l lJoJ J.Lj dJ.LjL.J = bk,lX + bk,zt + O~, k = 1, ... , n ,j=l 1Jo0 JC(J.Lj)J.Lj

where Oz are constants. The above integrals are taken along cycles aj over basic cuts onthe Riemann surface

(4.9)

which has genus 9 = n+ 1. The system (4.8) is equivalent to a degenerate Jacobi problem ofinversion. This degeneracy can be resolved by solving first the problem of inversion in termsof Riemann e functions on the (n +1)-dimensional Jacobian and then projecting on then-dimensional subvariety. (For details about Jacobi problem of inversion see for exampleErcolani and McKean [1990] and Alber and Alber [1987].)

On the other hand, the monodromy effect can be described as follows. Setting mzn+z =mZn+l = b in the initial Hamiltonian systems (4.2) and (4.3) yields in the manner describedabove, a system of angle variables

k = 1, ... , n - 1 ,

(4.10)

(4.11)

12

where C(J.Li) - n~~l (J.L - mk) and (1k,1l (1k,2 are constants. This can be viewed as aproblem of inversion with a singularity at J.L = b, resulting in a phenomenon of monodromyfor the Hamiltonian system.

We define a Jacobian fibration with the base parametrized by roots of the basic polyno­mial C(A)>' of the Riemann surface (4.9). Now, moving balong a certain closed loop in thespace of parameters (the base of the fibration) can lead to a nontrivial shift in the actionangle variables.

We demonstrate this using the x-flow. Consider first the case n = 1. The limitingprocess m3, m4 -+ b yields the following angle variable

flt = riLl J.Ll dJ.Ll = Lox + ()~ .JiL~ (J.Li - b)V-J.Ll(J.Ll - ml)(J.Ll - m2)

(4.12)

where

jiLl JC4(J.Ll )J.Ll dJ.Ll jiL2 JC4(J.L2)J.L2 dJ.L2

5= +iL~ J.Ll - b iLg J.L2 - b

is an action function and

The variables J.Ll and J.L2 move along cycles al and a2 over the cuts [mIl m2] and [m3' m4] onthe Riemann surface W 2 = J.LC4(J.L). There is also a singularity at J.L = b. Transport of a sys­tem of canonical action-angle variables, which linearize the Hamiltonian flow, along a certainloop in the space of parameters (b, mi) in a way similar to the case of the spherical pen­dulum and some other integrable systems with monodromy (see, for example, Duistermaat[1980] and Bates and Zou [1993]) will result in a nontrivial shift, which is a manifestationof the monodromy phenomenon. This can be demonstrated as follows. Canonical actionsare calculated in terms of periods of the differential

along cycles ai on the Riemann surface; for details see Arnold [1978]. Now suppose initiallythat b does not belong to any of the cycles al and a2. Then we move b along a closed loopon the Riemann surface around one of the branch points mi. At some moment b becomes abranch point itself, which results in a shift of the action variable that is given by the residueof the integrand at J.L = b.

13

Flaschka, Forest and McLaughlin [1980] proposed the use of J.L-representations in calcu­lations of modulation equations and showed that in case of the KdV equation, Whithamtheory can be described in terms of holomorphic and meromorphic differentials on themoduli spaces of Riemann surfaces. Namely, they described the slow modulation of branchpoints of the Riemann surfaces.

The monodromy effect described in this section provides an example of a singularity ofWhitham type equations. It has to be taken into account in the investigation of modula­tional theory of the N-component systems.

5 Mechanical Systems and Nonlinear Equations

In §2 we recalled the link between quasi-periodic solutions of integrable nonlinear equationsand Hamiltonian systems on Riemann surfaces established using generating equations andalgebraic-geometric methods. Here we expand on this by generating the Euler-Lagrangeequations for some well known mechanical problems. This establishes a link between suchsystems and the stationary Hamiltonian flows for classes of solutions of integrable nonlinearpde's. (See, for example, Alber et al. [1994,1995] for additional information.) This will beused in §6 to study the semiclassical theory of such solutions.

One obtains the Euler-Lagrange equations, using the method of generating equationsdescribed in §2 and by changing variables in B(x, >') from J.L'S to q's:

We also denote

n n+1 n+1

B(x, >.) = TI(>. - J.Li(X, t)) = I: TI (>. -Ir)qJ(x) .i=l i=l r=l,r#:i

(5.1)

W(>.) = C(>.) = n;~tN(>'-li)]((>.) n~r (>. - ar) ,

where>' is a complex parameter and the constants Ii and ar will be shown to be determinedby the parameters and first integrals of the particular mechanical system.

The expression (5.1) can be viewed as a definition of generalized elliptic J.L coordinateson Riemann surfaces. Coordinates qJ = xJ/Ii are normalized cartesian coordinates of aparticular mechanical system. Equating coefficients for the >.n in (5.1) yields

n+1 n+1 x~

~ q~ =~ ...1. = 1 .LJ 1 LJ I.i=l i=l 1

(5.2)

The parameters Ii can be complex and so (5.2) defines a pseudo-sphere in Cn+1 and providesa constraint for the Euler-Lagrange equations.

In terms of the q-variables, our mechanical problems may be thought of as problemsof geodesics on the pseudo-sphere (5.2) in the presence of a potential. In §6 we link themto the Laplace Beltrami operators in a way similar to the method of geometric optics andstudy semiclassical approximations for eigenfunctions of Laplace Beltrami operators havingthe form of complex WKB modes.

14

Substituting B and V into (2.4) and setting). = Ir one by one, a system of equations isgenerated for the functions qj(x). Below we obtain several different integrable systems fordifferent choices of the potential V(x, ).).

Recall (from Moser [1981]) the classical C. Neumann and Jacobi problems. The C.Neumann problem for the motion of a particle on the n-sphere in the field of a quadraticpotential corresponds to V = u(x) -).. Substituting (5.1) into (2.4), and setting). = Ir oneby one, yields a system of Euler-Lagrange equations,

q'j - qj(u -Ij) =0, j = 1, ... , n +1 ,

corresponding to the Lagrangian

Notice that the Euler-Lagrange equations and constraint Lj~i qj2 = 1 yield the followingexpression for u:

n+l n+l

u(x) = I:ljqj2- I:q?j=l j=l

The function u plays the role of Lagrange multiplier.For the Jacobi problem of geodesics on n-dimensional quadrics (free motion on quadrics),

one takes

and the Euler-Lagrange equations are

where Ij are semiaxis of the ellipsoid. Here the Lagrangian and u are as follows

Next we consider a linear combination ofthe potentials for the C. Neumann and Jacobiproblems. (Mechanical systems associated with this potential were discussed in Braden[1982]). Here we are interested in the connection between this problem and the spatial flowfor a particular class of solutions of the system of coupled pde's (4.1).Here, V = u(x) +). +v(x)/). and the Euler-Lagrange equations are

q'j - qj (u + Ij + ~) = 0 ,

15

with the constraint l:']~i qj2 = 1. The Lagrangian and the potential are as follows:

n+1 n+l 1u(x) = - L: qj2 - L: I jqj 2 + 2

'-I '-I "n+! !i.]- ]- L...-j=l Ij

Here we are using1

v(x) = -----n(I:j;;i if) 2 '

which is obtained from a recurrence chain generated by (2.4).Now consider the flow for the shallow water equation (see Alber et a1. [1994]). Here

and

qj - qj (~ + 1) =0.The Lagrangian and potential are as follows

The next example is provided by the spatial flow for a generalization of the shallow waterequation. Calogero and Degasperis [1982] studied an interesting new class of integrablemechanical systems. We generalize this class by considering a 2-component case whoseassociated system of coupled pde's has the form of a coupled system of equations of shallowwater type. Here,

v =u(x)+ v~) ,

and

qj-q;(u+~)=o.The Lagrangian and potential are as follows

See Appendix 1 for the mechanical system associated with the 2-component KdV system.

16

6 Complex Semiclassical Solutions

Recall that solutions of integrable systems can be reduced to finite-dimensional Hamiltoniansystems with the commuting x and t flows as outlined in §2. Hamiltonian flows for a varietyof important integrable mechanical systems are constructed using the same techniques andare linked to the spatial flows for particular classes of solutions of the nonlinear pde's.

Each of these flows can be characterized by a Hamiltonian of the form (2.10). Herewe discuss the semiclassical limits of the associated Laplace-Beltrami operators for thesesystems. In what follows, we recall from Alber [1989, 1991] and Alber and Marsden [1992],a method of complex geometric asymptotics for integrable Hamiltonian flows on Riemannsurfaces. We will use geometric asymptotics to describe the quantization conditions ofBohr-Sommerfeld-Keller (BSK) type. Then we will investigate the dependence of theseconditions on the parameters (i.e., the first integrals) of the system. This dependence nearsingularities produces effects caused by the classical and semiclassical monodromy.

Let us consider quadratic complex Hamiltonians of the form (2.10) defined on e 2n . Wethink of e 2n as being the cotangent bundle of en, with configuration variables J1.1,· .. , J1.nand with canonically conjugate momenta Pll"" Pn .

Notice that the Hamiltonians for quasi-periodic solutions of the systems considered inthe previous sections are indeed of this form. We consider the functions gjj as componentsof a (diagonal) Riemannian metric, construct the associated Laplace-Beltrami operator, andthen the stationary Schrodinger equation

(6.1)

defined on the n-dimensional complex Riemannian manifold en. Here "Vj and "Vj arecovariant and contravariant derivatives defined by the tensor gjj and w (which is the inverseof Planck's constant h) and E (the energy eigenvalue) are parameters. Note also that ingeneral, the metric tensor is not constant, and even may have singularities, so that thekinetic term in the expression for H is not purely quadratic.

Now we establish a link between (6.1) and the Hamiltonian system (2.10) by means ofgeometric asymptotics; namely, we consider the following function that is similar to the wellknown Ansatz from WKB theory:

U(Zll"',Zn) = LAk(J1.ll".,J1.n)exp[iwSk(J1.ll ... ,J1.n)] ,k

n n

L II Ukj(llj) =L II (Akj(llj) exp[iwSkj(llj))) , (6.2)k j=l k j=l

which is a multivalued function of several complex variables defined on en. If, instead, oneconsiders U to be defined on the covering space of the Jacobi variety of the problem, thenU becomes single valued. The functions present in this expression together with r, whichdenotes a vector of Maslov indices, will be determined below. Note that k also labels theclassical paths between initial and current points in the configuration space.

Substituting (6.2) in (6.1), equating coefficients for wand w 2 and integrating, we obtain

17

the amplitude function A, which is a solution of the transport equation, in the form

A- Ao- J(DdetJ) ,

(6.3)

where D = JIlf:::191/ is the volume element of the metric and J is the Jacobian of thechange of coordinates from the JL-representation to the angle 9-representation. We also findthat the phase function S is a solution of the Hamilton-Jacobi equation

(6.4)

so that it coincides with the action function. Finally, we describe the dependence of theWKB modes (6.2) on t as a parameter, by introducing commuting t-flow for the JL variablesdescribed by (2.7).

7 Semiclassical Monodromy

Now we can apply the constructions that have been developed above to the case ofthe specialclass of solutions of the 2-potential Dym system for n = 2 (see §4). The Hamiltonian andcomplex geometric asymptotic solution in this case have the form:

(7.1)

and

(7.2)

where

(7.3)

(7.4)

(7.5)

Tl= 1 .jCS(JL2) dJL2

hI JL2 - b

T2= 1 .jCS(JL2) dJL2

h 2 JL2 - b

The amplitude A has singularities at the branch points JLl = mIl m2, JL2 = m3, m4 and atan additional singular point b on the associated Riemann surface. Each time a trajectoryapproaches one of these singularities, we continue it to complex x and go around a small

18

circle in the complex plane enclosing the singularity. This results in a phase shift (±i1r/2)of the phase function S, which is common in geometric asymptotics. The indices kI and k2

keep track of the number of oriented circuits for J.lI and J.l2 around aI and a2. The complexmode (7.2) is defined on the covering space of the complex Jacobi variety. Note that in thereal case, this complex mode is defined on the covering space of a real subtorus. Keepingthis in mind, quantum conditions of BSK type can be imposed as conditions on the numberof sheets of the covering space of the corresponding lliemann surface for each coordinate J.lI

and J.l2:

~rl +wk,T, = 2rN, , }II (7.6)

2"T2 +wk2T2 = 21rN 2 •

Here N}, N 2 are integer quantum numbers related to each other and to integer indices k}, k2

and TI, T2 as follows

w = 21rNI = _1_ (21rN2_ 1rT2) , (7.7)kITI k2T2 2

which is an asymptotic formulae for the eigenvalues of the stationary Schrodinger equation(6.1). (For details about Maslov-Keller indices and quantum conditions see amongst othersRobbins [1991] and Alber [1989, 1991].) Maslov-Keller indices play an important role innumerical study of the semiclassical limit of the NLS equations presented in Jin, Levermoreand McLaughlin [1994].

Notice that the quantum conditions (7.6) include a monodromy part after transportalong a closed loop in the space of parameters.

8 The Stationary Schrodinger Equation, Bloch functionsand Complex WKB Solutions

Lastly we contrast semiclassical approximations for the Bloch eigenfunctions to the approachof §6 which uses a lliemannian metric associated with the complex Hamiltonian systemassociated with finite-dimensional solution space.

Namely we show how to obtain semiclassical approximations for the Bloch eigenfunctionsof the operators defined in (1.3) and (104) and used in the inverse scattering transformtheory (see Ablowitz and Segur [1981]). The equation (1.3) is a natural generalization of aone-dimensional stationary Schrodinger equation

which we rewrite

where

h2 fj27j;- --- + (V(x) - -\)7j; = 0 ,2m fJx2

Vex, -\, E) = U(x, -\), U(x, -\) = 2m(u(x) - -\) ,f.

(8.1)

(8.2)

(8.3)

19

and E = 11,2. In our notation E will be the complex parameter, and we will treat it indepen­dently from the energy >.. As usual, the WKB method is used to approximate 'If; for smallvalues of the parameter E. In what follows we will use (8.2) to demonstrate semiclassicaltheory for the operators defined in (1.3).

We look for a solution of (8.2) in the form of a Bloch function

(8.4)

where B = B(x, E) is a function of x and E, G = G(E) is a function of E, and Bo is a constantdefined by the initial data. This gives a solution of (8.2) if and only if

B/2

- B"B + - + 2B2V = G ,2

(8.5)

as can be checked by direct substitution. This equation was first obtained in connection withthe KdV equation in Gelfand and Dikey [1975] by investigating asymptotics of resolventsof the Sturm-Liouville equations.

Notice that (8.5) is just the stationary generating equation (2.4). We choose B(x, E) inthe form of a polynomial of E and G(E) to be a rational function with constant coefficients:

n

B(X,E) = L:bl(X)En-1 ,1=0

(8.6)

The equation (8.5) with potential (8.3) coincides with the generating equation for thegeodesic flow on an n-dimensional quadric. It was shown (see Cewen [1990] and Alberet al. [1994,1995]) to provide an x-flow for a partial differential equation of Dym type. Theequation (8.5) yields a chain of recurrence relations between the coefficients bl and q. (fordetails see Alber et aI. [1994, 1995]). In particular we have

and therefore

2m(V - E) .

As E - 0, the functions G(E) and B(€) are asymptotic to C2n and bn • This yields the€

following asymptotics for the Bloch function

'If; = 'If;o 1 exp (± ~ l X

J2m(u - E)dx) ,(u - E)4 yE Xo

(8.7)

which is precisely a well-known WKB solution of the one-dimensional Schrodinger equation(8.1). A similar procedure produces asymptotics for Bloch functions corresponding to po­tentials with poles can be carried out using recurrence relations and geodesic flows on the

20

associated Riemann surfaces. For example, a WKB solution in the case of the 2-potentialDym system described above has the form

7/;0 (i 1:1: )7/; = 1" exp ± r; J2mvdxV4 v € :1:0

(8.8)

The Bloch function contains all necessary information about the finite gap spectrum and theassociated Riemann surfaces. It is an ideal object for applying zero-dispersion limit theory.In the future, we intend to apply this approach to the semiclassical theory of (d)NLS and(f)NLS equations.

9 Conclusions

In this paper we have used so-called energy dependent Schrodinger operators to establisha link between special classes of solutions of N -component systems of evolution equationsand finite dimensional Hamiltonian systems on moduli spaces of Riemann surfaces. Thisyields in particular, a new class of solutions with monodromy for N -component systems ofpde's.

We also have used a connection between classes of solutions of nonlinear equations andfinite dimensional mechanical systems to investigate the semiclassical theory of nonlinearequations. In the future, we intend to apply this approach for WKB-theory for the equations(1.3) and (1.4) in connection with the the zero-dispersion limit of N-component systemssuch as the coupled NLS equations.

10 Appendix 1

1. (The spatial flow for the 2-component KdV system.) The discrete version of thissystem corresponds to the Toda and relativistic Toda lattices. (For details see Alber[1987].) Here

v = ,X2 +v(x)'x +u(x) ,

and the Euler-Lagrange equations are

qj' - qj (u +vIj + I;) = 0 .

The Lagrangian and potential are as follows

21

Here we are using the expression

n+1

V(X) =2I: Ijq; ,j=l

which is obtained from the recurrence chain generated by (2.4) with K = O. Usingthe recurrence chain generated by the dynamical generating equation (2.4) and settingB(x, >.) = bo(x)>. +b1(x) one obtains the following integrable system of coupled pde'sof KdV type:

au = ~Vlll _ v'u - ~vu' }at 4 2'

av , 3 ,- = u - -vv .at 2

(10.1)

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