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Numerical study of two dimensional stochastic NLS equations
Marc Barton-Smith, Arnaud Debussche, Laurent Di Menza
To cite this version:
Marc Barton-Smith, Arnaud Debussche, Laurent Di Menza. Numerical study of two dimensional stochastic NLS equations. Numerical Methods for Partial Differential Equations, Wiley, 2005, 21 (4), pp.810-842. �10.1002/num.20064�. �hal-00001470�
stohasti NLS equations
Mar BARTON-SMITH
, Arnaud DEBUSSCHE y
and Laurent DIMENZA z
:CERMICS,ENPC,Cite Desartes, 77455 Champ-sur-Marne
y :ENSCahan, Antennede Bretagne, Campusde Ker-Lann, 35170 Bruz
z :AnalyseNumerique etEDP,Universite Paris-Sud, 91405 Orsay
Abstrat: In this paper, we numerially solve the two-dimensional stohas-
ti nonlinear Shrodinger equation in the ase of multipliative and additive white
noises. The aim is to investigate their inuene on well-known deterministi solu-
tions: stationary states and blowing-up solutions. In the rst ase, we nd that a
multipliativenoisehasadampingeetverysimilartodiusion.However, forsmall
amplitudesofthenoise,thestruture ofsolitarystateisstillloalized.Intheseond
ase, a loal renement algorithmis used to overome the diÆulty arising for the
omputation of singular solutions.Our experiments show that multipliativewhite
noise stopsthe deterministiblow-up whihours intheritialase.This extends
the results of [15℄ in the one-dimensional ase.
Keywords: Stohasti partial dierential equations, multipliative and addi-
tive noise, nonlinear Shrodinger equations, nite dierene shemes, renement
proedure.
1 Introdution
Nonlinear Shrodingerequations (NLS)play animportant role for the understand-
ingofmanyphysialphenomena.Forinstane, NLSappears inwave propagationin
nonlinear media, uid and quantum mehanis orplasma physis. It is well known
that insome ases {inpartiular inthe ase ofa fousing power lawnonlinearity{
NLS equations possess solutionsof speial formwhih are loalizedin spae, prop-
agating at a nite onstant veloity and keeping the same shape. These are alled
solitary waves and in the partiular ase of a vanishing veloity these are alled
stationarywaves(see [10℄and[29℄ forareviewonNLS).Dependingonthe powerof
the nonlinearity, these solitary waves are stable or unstable. Under a ritial value
of the nonlinearexponent,the nonlinearityisalled subritial andinthis ase, the
solitary waves are stable. For larger values (that is in the ritial and superritial
up.
In thispaper,wewish toinvestigatetheinuene ofdierentkindsofnoiseson
solitary wave propagation and on the blow-up mehanism, in the two-dimensional
ase. Noisy terms might represent the eets of inhomogeneities in the medium in
whih the waves propagate, as well as noisy soures or of negleted terms in the
modelizationyielding toNLS equations.They an alsobeonsidered asa modelof
perturbation and it is natural to investigate if the qualitative behaviors desribed
above are robust or not and how noise an hange them. Here two dierent types
of noises willbe studied: additive noise and multipliativenoise. The rst one ats
as an additive random foring term added to the NLS equation and has the form
i dW
dt
; the ase of additive noise is studied in [18℄ where olletive oordinates and
large deviation arguments are used to get informationon the inuene of the noise
on the propagation of solitary waves. The seond one an be seen as a random
potentialtermofthe formiuÆ dW
dt
addedtoNLSequation.Multipliativenoise has
been introdued in the ontext of Sheibe aggregates (see [5℄ and [27℄). Then NLS
is writtenas
du i
d
udt ijuj 2
udt = 8
>
<
>
:
iuÆdW
idW;
(1)
where u =u(t;x;w); t 0 being the time variable,x the spae variableand ! the
random variable.
There are several studies on noisy nonlinear dispersive equations. In [23℄ for
example, thanks to inverse sattering and perturbation tehniques, the authors de-
rive some qualitative informations for small noise for dierent equations like NLS,
Korteweg-de Vries, Sine-Gordon or Klein-Gordon. The relevane of numerial sim-
ulations is also pointed out to obtain some results for more general noises. Suh
simulations have been used in [16℄ and [28℄ to study the inuene of a white noise
ontheKorteweg-deVriesequation.NLSequationswithrandomtermsaredesribed
in [1℄, [2℄ and [19℄ (see also the referenes therein). In these artiles,the noise isei-
ther apotentialoraperturbationofthedispersiveterm orthe nonlinearoeÆient,
it has smooth pathsand again aninverse sattering transformis used. Anumerial
study of the inuene of anoiseon theblow-up for NLShas been performed in[15℄
in the ase of a white noise in spaedimension one. Furthermore,many theoretial
results existabout the stohasti NLS (see for instane [11℄) but validonly for or-
related additiveormultipliativenoises.
Inthisartile,wewanttodoasimilarstudyasin[15℄indimensiontwo.Werst
reall,inSetion2,somebasioneptssuhasthestohastiframeworkandgeneral
well-posedness theoretial results. We also present the nite dierenes numerial
method,emphasizing on the noise disretization. In Setion 3, we study the eets
of bothadditiveandmultipliativenoisesonstationarywavesinthesubritialand
the physial modelorresponds to the ritial ase, =1, and the stationary wave
is not stable. It results that the propagation an be studied only on a short time
interval. Thus, we have hosen tosimulate also asubritial nonlinearity- =1=2
-allowingthepropagationoverlong timeinterval.Wendthat multipliativenoise
hasadampingeetthatanbeomparedforlargetimeswiththedampingobserved
for Ginzburg-Landau models. In Setion 4, we numerially investigate the noise
inuene on blow-up formation in the ritial ase. Only multipliative noise will
beonsidered here, sine additivenoise has no real eet onthe blow-up. Even for
the deterministiase, thenumerialmethodhastobeonsistentwithsmallspatial
sales of the blow-up struture. A loal renement algorithm is given, similar to
the one given in [15℄ in the one-dimensional ase, and tested rst for deterministi
blow-up. Renement riteria have to give reasonable omputational osts in our
two-dimensional experiments. Note that a lot of works for the omputation of the
blow-up of deterministi NLS (see [3℄, [4℄, [29℄, [30℄ and [31℄) or Korteweg-de Vries
havebeendone([7℄,[8℄).Even iftheyonern deterministiequationsandarebased
on nite elements, they are very helpful to nd the orret tehniques to ompute
blow-upinour stohasti ases.Stohasti testsare nallyperformedwith dierent
kinds ofblowing-upsolutions.The twodimensional ase studiedhere ismuhmore
diÆultthantheonedimensionalasestudiedin[15℄,espeiallyfortheomputation
of singularsolutions.Indeed,therenementmethodismuhmorediÆulttoderive
here. Bad riteria for renements yield expensive omputationalosts orvery poor
results. In Setion 4, we try to give details on the diÆulties enountered and the
remedies we found. Moreover, the blow-up is muh more severe in dimension two
anditisdiÆulttodetetthe eetofanoise.Weexpet thatamultipliativenoise
always preventsthe formationofsingularities.However, if the blow-upistoostrong
we rsthave tosimulateastrongly fousingsolutionreahingveryhigh amplitudes
and in some ases we havenot been able toestablish this fat.
2 General onsiderations on the equations and on
the numerial sheme
2.1 Set up of the problem
The equations whih willbestudied here are the following:
8
>
>
>
>
<
>
>
>
>
:
du i
d
udt ijuj 2
udt= 8
>
<
>
:
iuÆdW
idW;
u(0)=u
0 :
(2)
Dirihletboundaryonditions willbeonsidered onasquare domainDof R 2
, u
0 is
the initialondition,W isa realvaluedWiener proess onL 2
(D)assoiated witha
lteredprobabilityspae (;F;P;fF
t g
t0
).The rstkindof noiseisreferred asthe
(see [5℄), whereasthe seond oneisreferred asthe additivease. When the noiseW
is aylindrialWiener proess, it an be writtenas
W(t;x;!)= 1
X
k=0
k (t;!)e
k
(x); t0; x2D; !2: (3)
where (
k )
k2N
are real independent brownian motions (
k )
k2N
and (e
k )
k2N is an
orthonormal Hilbert basis of L 2
(D).
More generally,foralinearoperatoronL 2
(D),aWienerproesswithovari-
ane operator is given by
W(t;x;!)= 1
X
k=0
k
(t;!)e
k
(x); t0; x2D;! 2:
In general,the series above do not onverge in L 2
(D). This is true only when is
a Hilbert-Shmidtoperator.
If isdened through a kernel K
u(x)= Z
D
K(x;y)u(y)dy; for u2 H;
then the spatial orrelation funtion isgiven by:
C
(x;y)= Z
D
K(x;z)K(z;y)dz:
ThespaeandtimeorrelationofW beingformallygivenbyE
dW
dt
(t;x);
dW
dt (s;y)
and, stillformally,we have:
E
dW
dt
(t;x);
dW
dt (s;y)
=C
(x;y)Æ
t s :
Wesee that this type ofnoise is always unorrelated-or white- intime. If=I
d ,
i.e. if W is a ylindrial Wiener proess, the noise is also white in spae and the
spatial orrelation C
(x;y) isthe Diramass Æ
x y .
Theorrelationfuntionisaphysiallymeasurablequantity;aorrelationwhih
is the Dira mass Æ
x y Æ
t s
indiatesa white noiseboth intime and spae.
Let usalso remarkthat it is oftenwritten _ = dW
dt
so that equation (2)beomes:
du
dt i
d
u ijuj 2
u= 8
>
<
>
:
iuÆ_
i:_
(4)
For NLS,the energy and mass are respetively dened by:
H(u)= 1
2 Z
D
kru(x)k 2
dx
1
2(+1) Z
D ju(x)j
2(+1)
dx;
M(u)=
D ju(x)j
2
dx:
It iswell-known(see forexample[29℄) thatthese quantitiesare invariantforthe de-
terministi NLS.Withanadditivenoise,noneof themisonserved. ForaStratono-
vith multipliativenoise, onlythe mass isonserved.
2.2 Main theoretial results
Wethink that itis importantto reall the theoretialresults on the NLSequation.
Hopefully, this enables the reader to understand the issue at stake. We begin with
the deterministi NLS equation.
Theorem 2.1. For u
0 2 H
1
(R d
), the deterministi NLS equation (that is = 0)
on D = R d
is loally well-posed if 0 <
2
d 2
for d > 2 or for any if d = 1 or
2. Besides the solution is global if d <2. Moreover, for d 2 and u
0 2 H
1
(R d
)
suh that H(u
0
)<0 and xu
0 2L
2
(R d
), then the solution blows-up at a nite time.
The proof of this result as well as many improvements an be found in [10℄
and [29℄. Note that if d 2 there also exist solutions suh that H(u
0
) > 0 but
blow up ina nite time.For evident reasons, itis not possible tosimulate the NLS
equation on R d
and we have to restrit our omputations to a bounded domain.
However, if we only simulate spatially loalized solutions and the omputational
domainDissuÆiently large,weexpet thatthe numerialsolutionisveryloseto
the solutiononR d
.Anotherpointisthat inthe ased=2onsideredinthisartile,
itanbeshownthatinthesubritialasetheNLSequationadmitsauniqueglobal
solutionon bounded star-shaped domains(see [9℄). Moreover, Kavianhas shown in
[22℄thataninitialdatawithnegativeenergyonastar-shapeddomainwithDirihlet
ondition also gives ablowing-upsolution inthe ritialand superritialases.
For the NLS equations with additive noise idW, with a Hilbert-Shmidt
operator fromL 2
(R d
)toH 1
(R d
), wehavethe followingtheorem, proved in[11, 12℄:
Theorem 2.2. Assume that 0 <
2
d 2
if d>2 or 0 if d 2. If u
0
is a F
0
measurable random variable with values in H 1
(R d
), then there exists a unique solu-
tion u(u
0
;:) to NLS with additive noise with ontinuous H 1
(R d
) valued paths. This
solution is dened on a random interval [0;(u
0
;!)
, where (u
0
;!) is a stopping
time suh that we almost surely have lim
t!(u0;!) ju(t)j
H 1
= 1 or (u
0
;!) = 1.
If d < 2 then (u
0
;!) = 1 almost surely. Moreover, if d 2, then for any
u
0 2H
1
(R d
) suh that xu
0 2L
2
(R d
) and any t>0
P((u
0
)<t)>0:
FormultipliativenoiseiuÆdW,wehavetoassumethataHilbert-Shmidt
operator from L 2
(R d
) toH 1
(R d
) and alsothat is -radonifying operator fromH
to W 1;
(R d
)(with >2d),then we have the followingtheorem (see [11, 14℄):
Theorem 2.3. Assumethat
2
< <
d 2
or <
d 1
ifd>3,or0< <2ifd=3,
or 0< if d =1 or 2, then there exist r 2 and p be suh that 2
r
=d(
1
2 1
p ) and
for any u
0
with values in H 1
(R d
) there exists a stopping time(u
0
;!)and a unique
solution of NLS with multipliative noise starting from u
0
whih isalmost surely in
C([0;T℄;H 1
(R d
))\L r
((0;T);W 1;p
(R d
)) for any T < . Moreover we almost surely
have: limsup
t!(u
0
;!) ju(t)j
H
1 = 1 or (u
0
;!) = 1. If d <2 then (u
0
;!) = 1
almostsurely. Moreover,ifd>2and isHilbert-Shmidtfrom L 2
(R d
)toH 2
(R d
),
then for any u
0 2H
2
(R d
) suh that jxj 2
u
0 2L
2
(R d
) and any t>0
P((u
0
)<t)>0:
If d =2, for u
0
as above with suÆiently negative energy, there exists
t >0 suh
that
P((u
0 )<
t)>0:
Again, these resultsdonot orrespond withour situationsine our simulations
will be performed on a bounded domain. However, we think that the results pre-
sented belowgivea goodidea of the behaviorof the solutionsof NLS equationson
R 2
.
Notethat,the noisehasastrongeet onthe blow-upmehanism.Contraryto
the deterministisituation,inthesuperritialase, anyinitialdatagivesasingular
solution. This is also true in the ritial ase with additive noise. However, this
assumes aspatially smooth noise.Wewill see inSetion4 that if the noise iswhite
in spae, the situationis ompletelydierent.
2.3 The numerial method
Our shemeisbased onaCrank-Niolsonnite diereneshemeinspaeand time
onauniformgridwith(M+1) 2
pointsonthesquaredomain[0;x
max
℄ 2
.Thisimpliit
sheme was hosen beause the energy and the mass are onserved indeterministi
ase(seebelowforthedenitionofthenumerialenergyandmass).Thetimestepis
Æt and u n
is the numerialsolution atthe disrete time nÆt. The step of the square
grid is h and u
kj
is the numerial solution at the point (kh;jh). The numerial
sheme is the following:
i u
n+1
kj u
n
kj
Æt
+ 1
2h 2
(u n+1
k+1j 2u
n+1
kj +u
n+1
k 1j +u
n
k+1j 2u
n
kj +u
n
k 1j )
+ (u n+1
kj+1 2u
n+1
kj +u
n+1
kj 1 +u
n
kj+1 2u
n
kj +u
n
kj 1 )
+NL n+
1
2
kj
= W
n+
1
2
kj
where
NL n+
1
2
kj
= 1
2(+1) ju
n+1
kj j
2+2
ju n
kj j
2+2
ju n+1
kj j
2
ju n
kj j
2
!
u n+1
kj +u
n
kj
W n+
1
2
kj
= 8
>
>
>
>
>
<
>
>
>
>
>
: 1
2h p
Æt w
n+
1
2
kj (u
n+1
kj +u
n
kj
) formultipliativenoise
1
h p
Æt w
n+
1
2
kj
foradditive noise.
(5)
The w n+
1
2
kj
are independent real normal random variables. Atually, for additivean
delta orrelated - or equivalently a spae-time white - noise, this numerial noise
W n+
1
2
kj
should be the approximationof
1
h 2
Æt Z
D
kj Z
(n+1)Æt
nÆt
dWdx; (6)
where D
kj
is the elementary square domainaround x
kj
given by
D
kj
=
(k 1
2
)h; (k+ 1
2 )h
(j 1
2
)h; (j+ 1
2 )h
:
Then with the denition (3) of Setion2.1we get,
1
h 2
Æt Z
D
kj Z
(n+1)Æt
nÆt
dWdx = 1
h 2
Æt Z
D
kj Z
(n+1)Æt
nÆt
X
m2N e
m (x)d
m (s)dx
= 1
h 2
Æt X
m2N Z
D
kj e
m (x)dx
!
Z
(n+1)Æt
nÆt
d
m (s):
Let us hoose the Hilbert basis suh that the e
m
are the funtions e
kj
= 1
h
D
kj
vanishing outside D
kj
, ompleted by an innite number of funtions in order to
have aHilbertianbasis. Then by orthogonality,we have
Z
D
kj e
l ;m
(x)dx=0
if (l;m)6=(k;j) and we get
1
h 2
Æt Z
D
kj Z
(n+1)Æt
nÆt
dWdx = 1
h 2
Æt Z
D
kj e
kj (x)dx
!
Z
(n+1)Æt
nÆt
d
kj (s)
= 1
hÆt Z
(n+1)Æt
nÆt
d
kj
(s) (7)
= 1
hÆt (
kj
((n+1)Æt)
kj
(nÆt)):
(8)
