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Stability in the energy spae for hains of solitons of the LandauLifshitz
equation
Andr de Laire (Universit Lille 1)
Joint work with Philippe Gravejat (ole polytehnique)
Shrdinger equations and appliations
CIRM June 18th 2014
Andr de Laire Stability for hains of solitons of the LL equation 1/1
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Outline
Andr de Laire Stability for hains of solitons of the LL equation 2/1
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The LandauLifshitz equation
The dynamis of magnetization in a ferromagneti material is given by the
LandauLifshitz equation.
The magnetization is a diretion eld:
~m(x , t) : RD R S2 R3, ~m = (m
1
,m2
,m3
)
|~m| = (m21
+m22
+m23
)1/2 = 1
Andr de Laire Stability for hains of solitons of the LL equation 3/1
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Models
3D ferromagneti materials
2D: Thin materials.
1D: Cesium nikel triuoride CsNiF
3
, the oupling between the Ni ions is muh
stronger than the other ouplings in the lattie.
Andr de Laire Stability for hains of solitons of the LL equation 4/1
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The LandauLifshitz equation
t ~m = ~m ~heff (~m)
precession
~heff (~m): eetive magneti eld
Andr de Laire Stability for hains of solitons of the LL equation 5/1
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The LandauLifshitz equation
t ~m = ~m ~heff (~m)
precession
~m (~m ~heff (~m))
damping
~heff (~m): eetive magneti eld
> 0 : Gilbert damping oeient > 0, w.l.o.g. 2 + 2 = 1
Andr de Laire Stability for hains of solitons of the LL equation 5/1
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Andr de Laire Stability for hains of solitons of the LL equation 6/1
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The anisotropi undamped Landau-Lifshitz equation
~heff (~m) = ~m m3e3,
t ~m = ~m (~m m3e3),
~m(x , t) : RD R S2, ~m = (m
1
,m2
,m3
)
Andr de Laire Stability for hains of solitons of the LL equation 7/1
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The anisotropi undamped Landau-Lifshitz equation
~heff (~m) = ~m m3e3,
t ~m = ~m (~m m3e3),
~m(x , t) : RD R S2, ~m = (m
1
,m2
,m3
)
tm1 = m2(m3 m3) +m3Sm2
tm2 = m3m1 m1(m3 + m3)
tm3 = m1m2 +m2m1
Andr de Laire Stability for hains of solitons of the LL equation 7/1
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The anisotropi undamped Landau-Lifshitz equation
~heff (~m) = ~m m3e3,
t ~m = ~m (~m m3e3),
~m(x , t) : RD R S2, ~m = (m
1
,m2
,m3
)
The equation is hamiltonian. The onserved Hamiltonian is the Landau-Lifshitz energy
E(~m(t)) :=1
2
RN
|~m(x , t)|2 dx +
2
RN
m3
(x , t)2 dx = E(~m(0)), t R.
The anisotropy parameter 0:
= 0 : isotropi ase, Shrdinger map equation
> 0 : easy-plane or planar anisotropy lim|x|
m3
(x , t) = 0
In the sequel, we onsider = 1 and solutions m with nite Landau-Lifshitz energy.
Andr de Laire Stability for hains of solitons of the LL equation 7/1
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The hydrodynami framework
When the map m := m1
+ im2
does not vanish, it may be written as
m =
1 m23
exp i.
The funtions v = m3
and w = are solutions to the hydrodynami Landau-Lifshitzequation
tv = div((1 v2)w
),
tw = (
v(|w |2 1) +v
1 v2+
v |v |2
(1 v2)2
)
.(HLL)
The linearized equation around the zero solution is dispersive with dispersion relation
2 = |k |4 + |k |2.
Dispersion veloity is at least equal to the sound speed cs = 1.
Andr de Laire Stability for hains of solitons of the LL equation 8/1
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Solitons
We look for traveling waves solutions propagating along the x1
-axis with speed c, i.e.
~m(x) = ~u(x
1
ct, x2
, . . . , xD).
Then the prole
~u satises
c1
~u + ~u (~u u
3
e
3
) = 0.
Taking
~u and using that
~u (~u~u) = ~u(~u ~u)~u(~u ~u) = ~u|~u|2 ~u,
we obtain
~u = |~u|2~u+ u23
~u u
3
e
3
+ c~u 1
~u. (TWc)
Andr de Laire Stability for hains of solitons of the LL equation 9/1
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Solitons
We look for traveling waves solutions propagating along the x1
-axis with speed c, i.e.
~m(x) = ~u(x
1
ct, x2
, . . . , xD).
Then the prole
~u satises
c1
~u + ~u (~u u
3
e
3
) = 0.
Taking
~u and using that
~u (~u~u) = ~u(~u ~u)~u(~u ~u) = ~u|~u|2 ~u,
we obtain
~u = |~u|2~u+ u23
~u u
3
e
3
+ c~u 1
~u. (TWc)
Trivial solutions: onstants in S1 {0}.
It is enough to onsider c 0.
Andr de Laire Stability for hains of solitons of the LL equation 9/1
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Papaniolaou and Spathis (1999) investigated their existene and qualitative
properties for D = 2, 3. They derived numerially a branh of traveling waves forsubsoni speeds |c| < 1.
Lin and Wei (2010) proved the existene of small speed travelling waves with a
vortex-antivortex pair when D = 2.
d.L. (2014): the non-existene of non-onstant subsoni small energy travelling
waves for D = 2, 3, 4.
d.L. (2014): also smoothness and their algebrai deay at innity for D 2.
Andr de Laire Stability for hains of solitons of the LL equation 10/1
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Some numerial results for D = 2 (PapaniolaouSpathis)
0 15 30 45 60
15
30
p
E
c 0
c 1
E = p
c 0.78
Andr de Laire Stability for hains of solitons of the LL equation 11/1
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Some numerial results for D = 2 (PapaniolaouSpathis)
Funtion u3
for c = 0.5.
Andr de Laire Stability for hains of solitons of the LL equation 12/1
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Some numerial results for D = 2 (PapaniolaouSpathis)
Figure (a): speed c = 0.2. Figure (b): speed c = 0.95.
Andr de Laire Stability for hains of solitons of the LL equation 13/1
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From LandauLifshitz to Shrdinger
Let
~u be a solution of (TWc). Using the stereographi variable
=u1
+ iu2
1 + u3
,
we have that satises
+1 ||2
1+ ||2 + ic
1
=2
1 + ||2( ),
whih seems like a perturbed equation for the traveling waves for a Nonlinear Shrdinger
equation (GrossPitaevskii equation).
Andr de Laire Stability for hains of solitons of the LL equation 14/1
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From LandauLifshitz to Shrdinger
Let
~u be a solution of (TWc). Using the stereographi variable
=u1
+ iu2
1 + u3
,
we have that satises
+1 ||2
1 + ||2 + ic
1
=2
1+ ||2( )
that seems like a perturbed equation for the traveling waves for a nonlinear Shrdinger
equation (GrossPitaevskii equation):
+ (1 ||2) ic1
= 0. (GP)
Andr de Laire Stability for hains of solitons of the LL equation 14/1
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Andr de Laire Stability for hains of solitons of the LL equation 15/1
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Solitons for D = 1
In dimension one, the traveling wave solutions are the following expliit dark solitons.
Lemma (The one-dimensional ase)
Let D = 1 and c 0. Assume that ~u is a nontrivial nite energy solution of (TWc).Then 0 c < 1 and (up to invarianes) the solution is given by
u1
= c seh(
1 c2 x), u2
= tanh(
1 c2 x), u3
=
1 c2 seh(
1 c2 x).
Andr de Laire Stability for hains of solitons of the LL equation 16/1
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Multisolitons for D = 1
The Landau-Lifshitz equation is integrable by means of the inverse sattering method. In
partiular, there are expliit formulae for multi-solitons. They behave like a sum of
ordered solitons as t .
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Andr de Laire Stability for hains of solitons of the LL equation 17/1
05-08.aviMedia File (video/avi)
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Multisolitons for D = 1
The Landau-Lifshitz equation is integrable by means of the inverse sattering method. In
partiular, there are expliit formulae for multi-solitons. They behave like a sum of
ordered solitons as t .
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Andr de Laire Stability for hains of solitons of the LL equation 17/1
02-05-08.aviMedia File (video/avi)
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Chains of solitons
For c 6= 0, the soliton u =(1 u2
3
) 12
exp i is represented in the hydrodynami
framework by the pair
vc =(vc ,wc
)=
(u3
, x),
where
vc(x) =(1 c2)
1
2
osh((1 c2)1
2 x) ,
and
wc (x) =c vc (x)
1 vc(x)2.
Andr de Laire Stability for hains of solitons of the LL equation 18/1
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Chains of solitons
For c 6= 0, the soliton u =(1 u2
3
) 12
exp i is represented in the hydrodynami
framework by the pair
vc =(vc ,wc
)=
(u3
, x),
where
vc(x) =(1 c2)
1
2
osh((1 c2)1
2 x) ,
and
wc (x) =c vc (x)
1 vc(x)2.
A hain of solitons is dened as a perturbation of a sum of solitons
Sc,a,s(x) =N
j=1
sj vcj (x aj) =N
j=1
(sj vcj (x aj ), sj wcj (x aj)
),
with a = (a1
, . . . , aN) RN, c = (c
1
, . . . , cN) (1, 1)Nand s = (s
1
, . . . , sN) {1}N.
The hain is well-prepared when the speeds are ordered a
ording to the positions, i.e.
c1
< < cN ,
for
a1
< < aN .
Andr de Laire Stability for hains of solitons of the LL equation 18/1
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Andr de Laire Stability for hains of solitons of the LL equation 19/1
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The Cauhy problem for the original Landau-Lifshitz equation
The energy spae is dened as
E(R) ={m : R S2, s.t. m L2(R) and m
3
L2(R)}.
Theorem (Zhou-Guo '84, Sulem-Sulem-Bardos '86)
Let m0 E(R). There exists a global solution m L(R, E(R)) to
tm +m (xxm m3e3) = 0, (LL)
with initial datum m0.
Remark: no uniqueness
Theorem (Existene and uniqueness of smooth solutions)
Let k 1 and m0 E(R), with [m0] Hk(R). There exists a unique global solutionm : R R S2 to (LL) with initial datum m0, suh that
tm L([T ,T ],Hk1(R)) and (m
3
, xm) L([T ,T ],Hk(R))2,
for any positive number T . The Landau-Lifshitz energy is onserved along the ow.
See Sulem-Sulem-Bardos '86, Chang-Shatah-Uhlenbek '00, MGahagan '04.
Andr de Laire Stability for hains of solitons of the LL equation 20/1
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The Cauhy problem for the hydrodynami Landau-Lifshitz equation
In the hydrodynami framework, the Landau-Lifshitz energy is equal to
E(v ,w) =1
2
R
( (xv)2
1 v2+ (1 v2)w2 + v2
)
.
The non-vanishing spae is dened as
NV(R) ={v = (v ,w) H1(R) L2(R), s.t. max
xR|v(x)| < 1
}.
Another onserved quantity is the momentum:
P(v ,w) =
R
vw .
Theorem 1 (d.L-Gravejat, 2014)
Let v0 = (v0,w0) NV(R).
(i) There exists a number Tmax
> 0 and a unique solution v C0([0,Tmax
),NV(R)) to(HLL) with initial datum v
0
suh that there exist smooth solutions vn C(R [0,T ])
to (HLL) satisfying
vn v in C0([0,T ],NV(R)), as n , for any T < T
max
.
(ii) Tmax
is haraterized by limtTmax
maxxR |v(x , t)| = 1 if Tmax
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Andr de Laire Stability for hains of solitons of the LL equation 22/1
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Minimizing haraterization of solitons
For c 6= 0, the soliton vc is a minimizer of the variational problem
Emin
(pc ) = inf{E(v), v : R C s.t. P(v) = pc
}.
The speed c is related to the momentum pc through the formula
pc = 2 artan( (1 c)
1
2
c
)
.
The Euler-Lagrange equation is y
E(vc) = cP
(vc).
p
E
E = p
2
Andr de Laire Stability for hains of solitons of the LL equation 23/1
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Orbital stability of well-prepared hains of solitons
Theorem 2 (d.L.-Gravejat 2014)
Let c0 (1, 1)N , s0 {1}N , and v0 = (v0,w0) NV(R). Assume that
c0
1
< < 0 < < c0N .
There exist four numbers > 0, > 0, A > 0 and L > 0, suh that, if
v
0 Sc0,a0,s0
H1L2
= 0 < ,
for positions a0 RN suh that
min
{a0
k+1 a0
k , 1 k N 1}= L0 > L,
then there exist a unique solution v C0(R+,NV(R)) to (HLL) with initial datum v0
, as
well as N funtions ak C1(R+,R), with ak(0) = a
0
k , suh that
a
k(t) c
0
k
A
(0 + e
L0),
and
v(, t) S
c0,a(t),s0
H1L2
A(0 + e
L0),
for any t R+.
Andr de Laire Stability for hains of solitons of the LL equation 24/1
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Orbital stability of a single soliton
Corollary 3 (d.L-Gravejat 2014)
Let c0 (1, 0) (0, 1), a0 R, s0 {1}, and v0 NV(R). There exist two numbers > 0 and A > 0 suh that, if
v
0 s0 vc0( a0)
H1L2
< ,
then there exist a unique solution v C0(R+,NV(R)) to (HLL) with initial datum v0
, as
well as a funtion a C1(R+,R), with a(0) = a0
, suh that
a
(t) c0 A,
and
v(, t) s0 v
c0( a(t))
H1L2
A,
for any t R+.
Andr de Laire Stability for hains of solitons of the LL equation 25/1
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Andr de Laire Stability for hains of solitons of the LL equation 26/1
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Constrution of smooth solutions to (HLL)
It follows from the existene of smooth solutions to (LL).
Lemma
Let k 4 and set
NVk (R) ={v = (v ,w) Hk+1(R) Hk (R), s.t. max
xR|v(x)| < 1
}.
Given a pair v0 NVk(R), there exists a number T
max
> 0 and a unique solution
v L([0,Tmax
),NVk (R)) to (HLL) with initial datum v0. The maximal time Tmax
is
haraterized by the ondition
lim
tTmax
max
xR|v(x , t)| = 1, if T
max
< +.
Andr de Laire Stability for hains of solitons of the LL equation 27/1
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Continuity of the ow map
Given a smooth solution v = (v ,w) to (HLL), we dene the map
=1
2
(xv
(1 v2)1
2
+ i(1 v2)1
2w)
exp i,
where the phase is given by
(x , t) =
x
v(y , t)w(y , t) dy .
We also set
F (v ,)(x , t) =
x
v(y , t)(y , t) dy .
The map is solution to the nonlinear Shrdinger equation
it+ xx+2||2+
1
2
v2
Re((1 2F (v ,))
)(1 2F (v ,)) = 0,
while the funtion v satises the system{
tv = 2x Im((2F (v ,) 1)
),
xv = 2Re((1 2F (v ,))
).
(Generalized Hasimoto: Chang, Shatah and Uhlenbek 2000)
Andr de Laire Stability for hains of solitons of the LL equation 28/1
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The Strihartz estimates for the linear Shrdinger equation provide the ontinuity of the
ow map orresponding to this system.
Lemma
Let T > 0. Given two smooth solutions (v1
,1
) and (v2
,2
) to the previous systemwith initial datum (v0j ,
0
j ), there exist a number > 0, depending only on v0
j L2 and0j L2 , and a universal onstant K suh that
v
1
v2
C0([0,T ],L2)
+
1
2
C0([0,T ],L2)
K(v
0
1
v02
L2
+0
1
02
L2
),
for any T [0,min{,T}].
Theorem 1 derives from expressing this ontinuity property in terms of the pair
v = (v ,w).
Andr de Laire Stability for hains of solitons of the LL equation 29/1
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Orbital stability of a single soliton
The strategy is similar to the one developed by Martel-Merle-Tsai '02, '05, for the
Korteweg-de Vries and the nonlinear Shrdinger equations (see also
Bthuel-Gravejat-Smets '12).
It results from the following quantiation of the minimizing nature of a soliton.
Lemma
Let c (1, 0) (0, 1), and set Hc = E(vc) cP
(vc). There exists a number c > 0suh that
Hc(), L2 c2
H1L2 ,
for any pair H1(R) L2(R) suh that
xvc , L2 = P(vc), L2 = 0.
When = v vc satises the two orthogonality onditions, we have
E(v) cP(v)= E(vc) cP(vc) +1
2
Hc(), L2 +O(3H1L2
)
E(vc) cP(vc) +c2
2H1L2 +O(3H1L2
).
The orbital stability of vc is then a onsequene of the onservation of E and P.
The argument extends to a well-prepared sum of solitons through a suient separation
of the ontributions of eah soliton.
Andr de Laire Stability for hains of solitons of the LL equation 30/1
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The modulation parameters
In order to guarantee the two orthogonality onditions, we introdue modulation
parameters. Given > 0 and L > 0, we set
V(, L) ={
v NV(R), s.t. infak+1>ak+L
v Sc,a,sH1L2 < }
.
Lemma
There exist numbers 1
> 0, 1
> 0 and L1
> 0 suh that, given a solution
v C0([0,T ],H1(R) L2(R)) to (HLL), with v(, t) V(, L), for < 1
and L > L1
,
there exist two funtions a C1([0,T ],RN) and c C1([0,T ], (1, 1)N) suh that themap
(, t) = v(, t) Sc(t),a(t),s,
satises the orthogonality onditions
xvck (t),ak (t), (, t)
L2=
P
(vck (t),ak (t)), (, t)
L2= 0,
for any 1 k N and any t [0,T ]. Moreover, we have
(, t)H1L2 +N
k=1
|ck (t) c0
k | = O(),
N
k=1
(|ak (t) ck (t)|+ |c
k(t)|
)= O((, t)H1L2) +O(e
1
L), for t [0,T ].
Andr de Laire Stability for hains of solitons of the LL equation 31/1
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Loalizing the ontributions of eah soliton
For small enough, we introdue the funtions 1
= 1, N+1 = 0, and
j (x) =1
2
(
1 + tanh(
(
x aj1(t) + aj (t)
2
)))
,
for 2 j N, and we set
F(v) = E(v)N
j=1
c0
j Pj (v), where Pj (v) =
R
(j j+1
)vw .
Lemma
There exist numbers > 0, 2
> 0, 2
> 0 and L2
> 0 suh that, given a solution
v C0([0,T ],H1(R) L2(R)) to (HLL), with
v(, t) V(, L),
for < 2
and L > L2
, we have for t [0,T ]:
F(v) N
j=1
(E(vc0
j) c0j P(vc0
j))+
2
H1L2+O
( N
j=1
|cj (t) c0
j |2
)
+O(exp(
2
L)).
However, the funtional F(v) is no longer onserved along the ow.
Andr de Laire Stability for hains of solitons of the LL equation 32/1
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The monotoniity formula
The onservation law for the momentum writes as
t(vw
)=
1
2
x
(
v2 + w2
(1 3v2
)+
3 v2
(1 v2)2(xv)
2
)
1
2
xxx ln(1 v2
).
Lemma
There exist numbers 3
> 0, 3
> 0 and L3
> 0 suh that, given a solution
v C0([0,T ],H1(R) L2(R)) to (HLL), with
v(, t) V(, L),
for < 3
and L > L3
, we have
F (t) O(exp(
3
(L+ t)),
for any t [0,T ].
This allows us to obtain Theorem 2.
Andr de Laire Stability for hains of solitons of the LL equation 33/1
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Perspetives
Asymptoti stability of the solitons (Yakine Bahri, PhD Thesis).
Constrution and orbital stability of solitons in higher dimension.
Asymptoti stability in higher dimension?
Andr de Laire Stability for hains of solitons of the LL equation 34/1
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Thank you for your attention!
Andr de Laire Stability for hains of solitons of the LL equation 35/1
IntroductionThe LandauLifshitz equation
SolitonsThe anisotropic undamped Landau-Lifshitz equationConnection with the Schrdinger equation
Chain of solitons (D=1)Solitons and Multisolitons
The Cauchy problemOrbital stability of solitonsSketch of the proof of mains theoremsSketch of the proof of Theorem 1Sketch of the proof of Theorem 2
Perspectives and open problems