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Orbital stability of periodic waves for the nonlinear Schrodinger equation

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Niveau: Supérieur, Doctorat, Bac+8
Orbital stability of periodic waves for the nonlinear Schrodinger equation Thierry Gallay Institut Fourier Universite de Grenoble I B.P. 74 38402 Saint-Martin-d'Heres, France Mariana Ha˘ra˘gus¸ Departement de Mathematiques Universite de Franche-Comte 16 route de Gray 25030 Besanc¸on, France Abstract The nonlinear Schrodinger equation has several families of quasi-periodic travelling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work [13] where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss [16, 17], and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile. Running head: Periodic waves in the NLS equation Corresponding author: Thierry Gallay, Keywords: Nonlinear Schrodinger equation, periodic waves, orbital stability

  • structure function

  • then

  • wave

  • has been

  • orbital stability

  • periodic waves

  • dilation invariance

  • has

  • nls equation


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Language English

Orbitalstabilityofperiodicwavesforthenonlinear
Schrodingerequation

ThierryGallay
InstitutFourier
UniversitedeGrenobleI
B.P.74
38402Saint-Martin-d’Heres,France
MarianaHaragus
DepartementdeMathematiques
UniversitedeFranche-Comte
16routedeGray
25030Besancon,France

Abstract
ThenonlinearSchrodingerequationhasseveralfamiliesofquasi-periodictravellingwaves,
eachofwhichcanbeparametrizeduptosymmetriesbytworealnumbers:theperiodofthe
modulusofthewaveprole,andthevariationofitsphaseoveraperiod(Floquetexponent).In
thedefocusingcase,weshowthatthesetravellingwavesareorbitallystablewithintheclassof
solutionshavingthesameperiodandthesameFloquetexponent.Thisgeneralizesaprevious
work[13]whereonlysmallamplitudesolutionswereconsidered.Asimilarresultisobtained
inthefocusingcase,underanon-degeneracyconditionwhichcanbecheckednumerically.The
proofreliesonthegeneralapproachtoorbitalstabilityasdevelopedbyGrillakis,Shatah,and
Strauss[16,17],andrequiresadetailedanalysisoftheHamiltoniansystemsatisedbythe
waveprole.

Runninghead:
PeriodicwavesintheNLSequation
Correspondingauthor:
ThierryGallay,
Thierry.Gallay@ujf-grenoble.fr
Keywords:
NonlinearSchrodingerequation,periodicwaves,orbitalstability

1Introduction

Thispaperisdevotedtothestabilityanalysisofthequasi-periodictravellingwavesolutionsofthe
cubicnonlinearSchrodinger(NLS)equation
i
U
t
(
x,t
)+
U
xx
(
x,t
)+

|
U
(
x,t
)
|
2
U
(
x,t
)=0
,x

R
,t

R
,
(1.1)
where

∈{
1;1
}
and
U
(
x,t
)

C
.Eq.(1.1)isauniversalenvelopeequationdescribingthe
propagationofweaklynonlinearwavesindispersivemedia(see[23]foracomprehensiveintroduc-
tion).Thenonlinearityin(1.1)is“attractive”if

=+1(focusingcase)and“repulsive”if

=

1
(defocusingcase).InbothcasesEq.(1.1)hasafamilyofquasi-periodictravellingwavesofthe
mrofU
(
x,t
)=e
i(
px

ωt
)
V
(
x

ct
)
,x

R
,t

R
,
(1.2)
where
p,ω,c
arerealparametersand
V
:
R

C
isaperiodicfunction.Thesimplestelementsof
thisfamilyarethe
planewaves
,forwhich
V
isidenticallyconstantand
p,ω
satisfythedispersion
relation
p
2
=
ω
+

|
V
|
2
.Itiswell-knownandeasytoverifythattheplanewavesaredynamically
stableinthedefocusingcase,andunstable(if
V
6
=0)inthefocusingcase[25,13].Weshall
thereforeconcentrateonthelessexploredsituationwhere
V
isanontrivialperiodicfunction.In
thatcase,weshallreferto(1.2)asa
periodicwave
,although
U
(
x,t
)isingeneralaquasi-periodic
functionofboth
x
and
t
.
Thenumberofparametersin(1.2)canbereducedifweusethesymmetriesofEq.(1.1).We
recallthattheNLSequationisinvariantunderthefollowingtransformations:

(i)
U
(
x,t
)
7→
U
(
x,t
)e
i
ϕ
,
ϕ

R
(Phaseinvariance);
(ii)
U
(
x,t
)
7→
U
(
x
+
,t
),


R
(Translationinvariance);
2vv(iii)
U
(
x,t
)
7→
e

i
(
2
x
+
4
t
)
U
(
x
+
vt,t
),
v

R
(Galileaninvariance);
(iv)
U
(
x,t
)
7→
U
(
x,
2
t
),
>
0(Dilationinvariance).
If
U
(
x,t
)isaperiodicwaveasin(1.2),wecanusetheGalileaninvariancetotransformitintoa
solutionofthesameformwith
c
=0.Then,usingthedilationinvariance,wecanfurtherassume
that
ω
∈{
1;0;1
}
.Itfollowsthat
U
(
x,t
)=e

i
ωt
W
(
x
),where
W
(
x
)=e
i
px
V
(
x
)isasolutionof
theordinarydierentialequation
W
xx
(
x
)+
ωW
(
x
)+

|
W
(
x
)
|
2
W
(
x
)=0
,x

R
.
(1.3)
Theboundedsolutionsof(1.3)arecompletelyclassiedforallvaluesoftheparameters
ω,
.
Thesimplestonesaretheplanewaves
W
(
x
)=
A
e
i
px
where
p

R
,
A

C
and
p
2
=
ω
+

|
A
|
2
.
Theperiodicwavescorrespondtoquasi-periodicsolutionsof(1.3)oftheform
W
(
x
)=
r
(
x
)e
i
ϕ
(
x
)
,
where
r,ϕ
arerealfunctionswiththepropertythat
r
and
ϕ
x
areperiodicwiththesameperiod.
ItturnsoutthatEq.(1.3)hasafour-parameterfamilyofsuchsolutions,bothinthefocusingand
inthedefocusingcase(seeSections2and4below).Actually,if

=

1,wemustassumethat
ω
=1otherwise(1.3)hasnonontrivialboundedsolutions;if

=+1,Eq.(1.3)hasquasi-periodic

2

solutionsforallvaluesof
ω
,butweshallonlyconsiderthegenericcases
ω
=

1.If
ω<
0,
inadditiontoplanewavesandperiodicwaves,thereexistpulse-likesolutionsof(1.3)whichare
homoclinicas
x
→∞
toaplanewaveor

tothezerosolution.Themostfamousone(if

=1
and
ω
=

1)isthegroundstate
W
(
x
)=2
/
cosh(
x
)whichcorrespondstothesolitarywaveof
thefocusingNLSequation.
Incontrasttotheplanewavesorthesolitarywaveswhichhavebeenextensivelystudied[7,24,
25],relativelylittleseemstobeknownaboutthestabilityofperiodicwaves.
Spectralstability
with
respecttolong-wavedisturbanceshasbeenexaminedbyRowlands[22],whofoundthatperiodic
waveswithreal-valuedproleareunstableinthefocusingcaseandstable(atleastinthelong-wave
regime)inthedefocusingcase.Asimilaranalysishasbeencarriedoutinhigherspacedimensions
[18,20],showinginparticularthatone-dimensionalperiodicwavesarealwaysunstablewithrespect
totransverseperturbations.Toourknowledge,spectralstabilityhasbeenrigorouslyestablished
onlyintwoparticularcases:forsmallamplitudeperiodicwavesofthedefocusingNLSequation
[13],andforperiodictrainsofwidelyspacedsolitonpulsesinthefocusingequationwithaperiodic
potential[3,21].Asforthe
nonlinearstability
,theonlyresultweareawareofisduetoAngulo
[1],whoprovedveryrecentlythatthefamilyofdnoidalwavesofthefocusingNLSequationis
orbitallystablewithrespecttoperturbationswhichhavethesameperiodasthewaveitself,see
also[2]forasimilarstabilityanalysisofthecnoidalwavesoftheKdVequation.Werecallthat
theperiodicwavesofNLSwithreal-valuedprolearecalled“cnoidalwaves”whentheyhavezero
averageoveraperiod(liketheJacobianellipticfunction
cn
),and“dnoidalwaves”whentheyhave
nonzeroaverage(liketheellipticfunction
dn
).
Inthispaper,westudythenonlinearstabilityofallperiodicwavesof(1.1),butwerestrict
ourselvestoaspecicclassofperturbationswhichwenowdescribe.Anyquasi-periodicsolution
of(1.3)canbewrittenintheform
W
(
x
)=e
i
px
Q
per
(2
kx
)
,x

R
,
(1.4)
where
p

R
,
k>
0,and
Q
per
:
R

C
is2

-periodic.Here
k
=
/T
,where
T>
0isthe
minimalperiodof
|
W
|
.Therepresentation(1.4)isnotunique,sincewecanaddto
p
anyinteger
multipleof2
k
(andmodifytheperiodicfunction
Q
per
accordingly),buttheFloquetmultipliere
i
pT
isuniquelyde