Orbital stability of periodic waves for the nonlinear Schrodinger equation

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