Traveling waves for the Nonlinear Schrodinger Equation with general nonlinearity in dimension one

TravelingwavesfortheNonlinearSchro¨dingerEquation
withgeneralnonlinearityindimensionone
D.Chiron
∗

Abstract
WestudythetravelingwavesoftheNonlinearSchro¨dingerEquationindimensionone.Through
variousmodelcases,weshowthatfornonlinearitieshavingthesamequalitativebehaviourasthestandard
Gross-Pitaevkiione,thetravelingwavesmayhaveratherdifferentproperties.Inparticular,ourexamples
exhibitmultiplicityornonexistenceresults,cusps(asfortheJones-Robertscurveinthethree-dimensional
Gross-Pitaevskiiequation),andatransoniclimitwhichcanbethemodified(KdV)solitonsoreventhe
generalized(KdV)solitoninsteadofthestandard(KdV)soliton.

Key-words:
travelingwave,NonlinearSchro¨dingerEquation,Gross-PitaevskiiEquation,Korteweg-deVries
soliton,(mKdV)solitons,(gKdV)soliton.
MSC(2010):
34B40,34C99,35B35,35Q55.

1Introduction
Inthispaper,weconsidertheNonlinearSchro¨dingerEquationindimensionone
Ψ∂i
+
∂
x
2
Ψ+Ψ
f
(
|
Ψ
|
2
)=0
.
(NLS)
t∂Thisequationappearsasarelevantmodelincondensedmatterphysics:Bose-Einsteincondensationand
superfluidity(see[28],[16],[18],[1]);NonlinearOptics(see,forinstance,thesurvey[22]).Thenonlinearity
f
maybe
f
(
%
)=
±
%
or
f
(
%
)=1
−
%
,inwhichcase(NLS)istermedtheGross-Pitaevskiiequation,or
f
(
%
)=
−
%
2
(see,
e.g.
,[23])inthecontextofBose-Einsteincondensates,andmoregenerallyapurepower.
Theso-called“cubic-quintic”(NLS),where,forsomepositiveconstants
α
1
,
α
3
and
α
5
,
f
(
%
)=
−
α
1
+
α
3
%
−
α
5
%
2
and
f
hastwopositiveroots,isalsoofhighinterestinphysics(see,
e.g.
,[5]).Weshallfocusontheone
dimensionalcase,whichappearsquiteofteninNonlinearOptics.Inthiscontext,thenonlinearitycantake
variousforms(see[22]):
%
0

11

%
2
−
%
2

f
(
%
)=
−
α%
ν
−
β%
2
ν
,f
(
%
)=
−
1
ν
−
%ν
,f
(
%
)=
−
α%
1+
γ
tanh
20
...
(1)
2(1+
%
0
)(1+
%
0
)
σ
where
α
,
β
,
γ
,
ν
,
σ
aregivenconstants(thesecondone,forinstance,takesintoaccountsaturationeffects).
Forthefirstnonlinearityin(1),weusuallyhave
αβ<
0,henceitisclose,insomesense,tothecubic-
quinticnonlinearity.Therefore,itisnaturaltoallowthenonlinearitytobequitegeneral.Inthecontextof
Bose-EinsteincondensationorNonlinearOptics,thenaturalconditionatinfinityappearstobe
|
Ψ
|
2
→
r
02
as
|
x
|→
+
∞
,
∗
LaboratoireJ.A.Dieudonne´,Universite´deNice-SophiaAntipolis,ParcValrose,06108NiceCedex02,France.
e-mail
:chiron@unice.fr.

1

where
r
0
>
0issuchthat
f
(
r
02
)=0.Weshallassumethroughoutthepaper
f
assmoothasrequired.
ForsolutionsΨof(NLS)whichdonotvanish,wemayusetheMadelungtransform
Ψ=
a
exp(
iφ
)
andrewrite(NLS)asanhydrodynamicalsystemwithanadditionalquantumpressure

∂
t
a
+2
∂
x
φ∂
x
a
+
a∂
x
2
φ
=0

∂
t
ρ
+2
∂
x
(
ρu
)=0

∂
2
a
or

∂
2
(
√
ρ
)

(2)
x
∂
t
φ
+(
∂
x
φ
)
2
−
f
(
a
2
)
−
x
=0


∂
t
u
+2
u∂
x
u
−
∂
x
(
f
(
ρ
))
−
∂
x
√
=0
ρawith(
ρ,u
)
≡
(
a
2
,∂
x
φ
).WhenneglectingthequantumpressureandlinearizingthisEulersystemaroundthe
particulartrivialsolutionΨ=
r
0
(or(
a,u
)=(
r
0
,
0)),weobtainthefreewaveequation

∂
t
a
¯+
r
0
∂
x
u
¯=0

∂
t
u
¯
−
2
r
0
f
0
(
r
02
)
∂
x
a
¯=0
withassociatedspeedofsound
q
c
s
≡−
2
r
02
f
0
(
r
02
)
>
0
provided
f
0
(
r
02
)
<
0(thatistheEulersystemishyperbolicintheregion
ρ
'
r
02
),whichwewillassume
throughoutthepaper.Thespeedofsoundplaysacrucialroleintheexistenceoftravelingwavesfor(NLS).
TheNonlinearSchro¨dingerequationformallypreservestheenergy
ZE
(Ψ)
≡|
∂
x
Ψ
|
2
+
V
(
|
Ψ
|
2
)
dx,
R2r0Zwhere
V
(
%
)=
f
(
s
)
ds
,aswellasthemomentum.Themomentumisnoteasytodefineindimension
%oneformapsthatvanishsomewhere(see[6],[7]).However,ifΨdoesnotvanish,wemayliftΨ=
a
e
iφ
,and
thenthemomentumisdefined(see[8])by
Z
r
2
Z
P
(Ψ)
≡h
i
Ψ
,∂
x
Ψ
i
1
−
0
dx
=(
a
2
−
r
02
)
∂
x
φdx,
R
|
Ψ
|
2
R
where
h∙
,
∙i
denotestherealscalarproductin
C
.
1.1Thetravelingwaves
Thetravelingwavesplayanimportantroleinthelongtimedynamicsof(NLS)withnonzeroconditionat
infinity.Thesearesolutionsof(NLS)oftheform
Ψ(
t,x
)=
U
(
x
−
ct
)
where
c
isthespeedofpropagation.Theprofile
U
hastosolvetheODE
22∂
x
U
+
Uf
(
|
U
|
)=
ic∂
x
U
(TW
c
)
togetherwiththecondition
|
U
(
x
)
|→
r
0
as
x
→±∞
.Wemaywithoutlossofgeneralityassumethat
c
≥
0
(otherwiseweconsiderthecomplexconjugate
U
insteadof
U
).Moreover,weshallrestrictourselvestofinite
energytravelingwaves,inthesensethat
∂
x
U
∈
L
2
(
R
)and
|
U
|
2
−
r
02
∈
L
2
(
R
).Inwhatfollows,(nontrivial)
travelingwavethenmeansa(nontrivial)solutionto(TW
c
)with
|
U
(
x
)
|→
r
0
as
x
→±∞
andfiniteenergy.
Let
U
besuchatravelingwave.Takingthescalarproductof(TW
c
)with2
∂
x
U
,wededuce
∂
x
|
∂
x
U
|
2
−
V
(
|
U
|
2
)=0in
R
,

2

()7)8(

ecneh|
∂
x
U
|
2
=
V
(
|
U
|
2
)in
R
,
(3)
inviewoftheconditionatinfinityandsince
U
hasfiniteenergy.Similarly,denoting
U
=
U
1
+
iU
2
and
takingthescalarproductof(TW
c
)with
iU
and
U
respectivelyyields,forsomeconstant
K
,
c22U
1
∂
x
U
2
−
U
2
∂
x
U
1
=(
|
U
|−
r
0
)+
K
in
R
(4)
2afterintegrationand
h
U,∂
x
2
U
i
+
|
U
|
2
f
(
|
U
|
2
)=
−
c
(
U
1
∂
x
U
2
−
U
2
∂
x
U
1
)in
R
.
(5)
Equation(4)allowsonetocomputethephaseof
U
when
U
doesnotvanish.Indeed,oneachintervalwhere
U
doesnotvanishonemaywrite
U
=
a
e
iφ
and(4)becomes
ca
2
∂
x
φ
=
η
+
K.
(6)
2Sincewerestrictourselvestotravelingwaveswithfiniteenergy,wemusthave
K
=0in(4).Indeed,
|
U
|→
R
r
0
>
0as
|
x
|→
+
∞
,hen
R
ce
U
hasalifting
U
=
a
e
iφ
with
a
≥
r
0
/
2forlarge
|
x
|
,say
|
x
|≥
R
,and
since
{|
x
|≥
R
}
(