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

Sophia Antipolis

Description

Traveling waves for the Nonlinear Schrodinger Equation with general nonlinearity in dimension one D. Chiron? Abstract We study the traveling waves of the Nonlinear Schrodinger Equation in dimension one. Through various model cases, we show that for nonlinearities having the same qualitative behaviour as the standard Gross-Pitaevkii one, the traveling waves may have rather different properties. In particular, our examples exhibit multiplicity or nonexistence results, cusps (as for the Jones-Roberts curve in the three-dimensional Gross-Pitaevskii equation), and a transonic limit which can be the modified (KdV) solitons or even the generalized (KdV) soliton instead of the standard (KdV) soliton. Key-words: traveling wave, Nonlinear Schrodinger Equation, Gross-Pitaevskii Equation, Korteweg-de Vries soliton, (mKdV) solitons, (gKdV) soliton. MSC (2010): 34B40, 34C99, 35B35, 35Q55. 1 Introduction In this paper, we consider the Nonlinear Schrodinger Equation in dimension one i ∂? ∂t + ∂2x? + ?f(|?| 2) = 0. (NLS) This equation appears as a relevant model in condensed matter physics: Bose-Einstein condensation and superfluidity (see [28], [16], [18], [1]); Nonlinear Optics (see, for instance, the survey [22]).

• gross-pitaevskii equation


• u1∂xu2 ?


• equation can


• traveling waves


• particular trivial


• bose-einstein condensation


• nonlinearity


• newton equation

Sujets

• Antipolis
• Gross?Pitaevskii equation
• Bose?Einstein condensate
• Nonlinear system

Informations

• Publié par: pefav
• Nombre de lectures: 8
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

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
:[e-mail].

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
}
(