Numerical resolution of stochastic focusing NLS equations

Applied Mathematics

Applied Mathematics Letters

Letters 15 (2002) 661-669

www.elsevier.com/locate/aml

Numerical Resolution of Stochastic Focusing NLS Equations A. DEBUSSCHE ENS de Cachan, Avenue R. Schuman, 35170 Bruz, France Universite

L. DI MENZA Paris&d, URA 760, B&t. 425, 91405 Orsay, France laurent.dimenzaQmath.u-psud.fr (Received

July 2001; accepted

Communicated

August 2001)

by R. Temam

Abstract-In this note, we numerically investigate a stochastic nonlinear Schriidinger equation derived as a perturbation of the deterministic NLS equation. The classical NLS equation with focusing nonlinearity of power law type is perturbed by a random term; it is a strong perturbation since we consider a space-time white noise. It acts either as a forcing term (additive noise) or as a potential (multiplicative noise). For simulations made on a uniform grid, we see that all trajectories blow-up in finite time, no matter how the initial data are chosen. Such a grid cannot represent a noise with zero correlation lengths, so that in these experiments, the noise is, in fact, spatially smooth. On the contrary, we simulate a noise with arbitrarily small scales using local refinement and show that in the multiplicative case, blow-up is prevented by a space-time white noise. We also present results on noise induced soliton diffusion. @ 2002 Elsevier Science Ltd. All rights reserved.

Keywords-Stochastic schemes.

partial differential equations, White noise, Blow-up, Finite differences

1. INTRODUCTION The nonlinear propagation

Schrodinger across

nonlinear

equation media

(NLS)

is a well-known

in many physical

model

contexts

used for the study

such as plasma

physics,

of wave

as well

as nonlinear optics or quantum chemistry (see [I] and the references therein). More recently, stochastic perturbations of this deterministic model have been investigated in order to involve inhomogeneities of the media or noisy sources. The natural question arising here is the behaviour of generic solutions of NLS when these additional terms are taken into account (as in [2-41). It is believed in particular that such a perturbation might have an effect on the formation of singularities. Let us recall that in the deterministic case, some solutions are known to blow up for critical or supercritical nonlinearities. This is the case if the energy is initially negative (see [5]). A white noise perturbation is highly singular and might affect this behaviour. Indeed, a white noise contains all possible scales, and therefore, can disturb the energy transfer from large to small scales involved in the blow-up mechanism. Another aspect concerns the persistence of 0893-9659/02/s - see front matter @J 2002 Elsevier Science Ltd. All rights reserved. PII: SO893-9659(02)00025-3

Typeset

by AM-TEX

A. DEBUSSCHE AND L. DIMENZA

662

propagation of “ideal” solitons made for the Korteweg-deVries We consider

in randomly perturbed equation (see [6,7]).

here the stochastic

nonlinear

Schrodinger

iz+g$+lu12%=

&f(U),

media.

We have f(u)

function.

= i in the case of additive

has already

been

equation

(t,x)

ult=o= uo, where u = u(t, x) is a complex-valued

Such a study

E

R+ x R,

(1)

x E R,

The source term involves

noise and f(u)

a random

= ku for the multiplicative

perturbation. case, where g

is a space-time white noise and E stands for the noise amplitude. Such an equation has been studied in [&lo]. Note that in the deterministic case E = 0, the solution u satisfies the two invariance

properties hl(u(t))

= s R”

H(u(t))

= ;

144 x)12 dx = M (uo) >

s,.(IIVu(t, x)l12- --&,u(~,x)~~~~+~)) = dx

The first invariance still holds for the solution in the multiplicative noise since the product has to be understood in the Stratonovitch

H (UC,).

case considered with a real sense. Our numerical study

aims at understanding the noise influence on blow-up phenomena occurring in the critical case (0 = 2) or supercritical case (a > 2), and on the propagation of spatially localized structures. The numerical simulation for this kind of equation is delicate because the white noise k does not have a prescribed correlation length. For instance, the choice of the mesh parameters cannot be linked to a characteristic length associated to the noise, since it has a zero correlation length. Also, such a noise is highly irregular. We will first briefly introduce the stochastic framework for this equation. We then describe our numerical method emphasizing on the discretization of the noise term. Finally, we will show our main numerical results. We first use a fixed uniform grid and observe that blow-up occurs in all cases, even if the initial datum corresponds to a global deterministic such a grid, the numerical noise can be reinterpreted as the discretization

solution. However, on of a spatially smooth

noise. Hence, it is natural to observe this behaviour which agrees with previously obtained theoretical results (91. Only a refinement strategy can give a noise with arbitrarily small scales. We use such an algorithm locally to minimize the computational cost. In this way, we simulate the effect of a space-time white noise. We see that in the multiplicative case, the formation of singularities is strongly perturbed and blow-up is prevented. Moreover, we recover the soliton diffusion phenomenon already known in the non-blow-up situation (see [11,12]).

2. STOCHASTIC

FRAMEWORK

In order to have a correct interpretation of (l), we define here some tools used in next section (see [8] for a more precise statement). A probability space (R, F’, P) being given together with Wiener process associated to a filtration (Ft’t)~ze, (W(t)) ~0 on L2(R) is said to be a cylindrical the stochastic basis (R,F,P, (.ZF(t))tlc) if for any orthonormal basis (ei)%eN of L2(R), setting is a sequence of independent real Brownian motions on pi(t) = (W(t),%) (i E N, t > O), @i)iEN (0, &P,(3(t))tlo).

Thus, W(t,x,w)

= C&(t,w)e,(x),

t > 0,

x E R,

w E R.

(2)

iEN

We define the space time white linear operator a., @f(x) =

noise 2 by k = s.

Furthermore,

k(x, yl).f(~) dy,

f E L2(W,

sR

given a kernel

k and the

(3)

NLS Equations we define the Wiener delta

correlated

663

I%’ = QW with covariance

process

operator

in time and has the space correlation

@a*. In this case, k = $$

is

c(x, Y) = JR Ic(x, .z)~(Y, Z) dz. If

function

kernel, the noise is homogeneous in space and c(z, Y) = c(x - y). U&Y) = k(z-Y) is a convolution In the case of a space time white noise, k(z, Y) = c(x, y) = 6,_, and @ = Id. Equation (1) can be interpreted as i duf ($$ + ~u~~~u)= df with df = E di$’ for the additive noise and df = amdW for the multiplicative noise, where o means Stratonovitch product. It can be proved (see [&lo]) that if @ is a Hilbert-Schmidt defined

on a interval

operator

[0, r*(w))

r*(w) or r*(w) = 00. In the one-dimensional as in the deterministic weaker

assumptions

into H1(R), then

from L2(R)

there exists a unique

such that either the H1 norm of u goes to infinity

context:

subcritical

case 0 < 2, the solution

r*(w) = co, (see [S] for other results

solution

when t tends to is always global

in the subcritical

case under

on a).

3. NUMERICAL We solve (1) numerically

by means

RESOLUTION

of a finite difference

scheme.

We compute

the approximate

values zl$’ of the solution at grid points (zj,&) (t, = nAt, xj = Ax, At, and Ax being the mesh steps of time and space). Given L > 0, computations will be done on the spatial domain ] - L, L[ with

Neumann

boundary

Crank-Nicolson

]U;+r[’ step,

the discretized

- ]U,n12

($,% + 3

a nonlinear

system

(2) and

with

such that

= (.c3,(&t..--&(tn))/At&, and

fJ

g05fdf

=

(pJ(&+l)

random

variables

of the Nag library.

=

(4)

4+1/2

to solve.

j=-J

’

3

The main

,...,

problem

In the case of an additive

respect

to time

variable.

We consider j = -J

ej = l~(j-l,2)az,(j+1,2)az,/J, and

11-Jaz,(-5+1,2)*2)l~~,

pendent

use the symmetric

J,

n>O,

here is the way of noise,

fJ+1’2

can

of

integrating

(ej) of L2(-L,L)

.:+1>

value of the noise term.

be seen as an approximation

using

We then

discretization

at each time

computing

at each end of the segment.

l~3n+112(0+1) - lM2(u+1)

i +2(0+1) giving

conditions

eJ

j =

=

basis =

From (5), we then have jjn+li2

1[(J-lj2)~z,J~z)/~~.

-J+l,.

an orthonormal

+ 1,. . . , J - 1, e-J

. . , J-1, .f::1’2 =

(P-~(t~+l)-P-~(t~))/at~~,

-,bJ(tn))/Atdm. Since ~~‘1’2 = (&(tn+i) -,&&))/m are indewith normal law N(O, l), we use the random numbers generator routine At each time increment,

we compute

a new vector (x~~““,

. . . , );“J+“‘)

approximating (~“:l’~, . . . , TJ”+“~). Note that from (5), we see that the numerical noise has the form Qnurn dW where anurn is the orthogonal projector on the space spanned by e-J,. . . , eJ. The corresponding correlation function is

I

1 Z’ -&,

[(j-i)A~,(j+i)Ax)

ifz,yE

if x,y E [-JAz,

Glum(x - Y) = ( x,Y~

I

0,

((J-+x,JAx],

otherwise.

(-J

+ 1/2)As)

forsomej, or

A. DEBUSSCHEAND L. DIMENZA

664 This confirms

that

we have an approximation

of a space time white noise.

Indeed,

it is easily

seen that cnum is an approximation of &,, the Dirac mass at 2 - y. However, it also shows that the numerical noise has length scales always larger than Ax and in that sense, it is not white. The derivation the following

of the noise contribution discretization

and f3?+1/2 = x,“+1’2(~~ in this latter discrete

term

+ u7+‘)/2dm

mass invariant

First,

case.

product

to be real.

NUMERICAL

some numerical

tests

we consider

It can be proved

enables

The nonlinear

us to preserve

system

case.

information

of the random

In this stochastic term.

Case

sigma+

with the method

described

of the noise on the blow-up context,

Thus,

Multiplicative

Ntot of tests

noise, 1000 trajectories solution

/

(H
‘I

‘1

2 E

0.4

epsilon epsilon epsilon

I”,

= 0.01 = 0.05 = 0.1

Additive Case sigma+

I

1000

noise,

local deterministic

trajectories solution

(HcO)

1.0

I

I I 0.8!

0.6

‘I :1

0.4

-

epsilon epsilon epsilon

= 0.01 = 0.05 = 0.1

0.0 0.04 I..-. Figure 1. Supercritical additive noise

I

/

0.2

‘; 0.05

of NLS

only one test will not give us significant

local deterministic

0.6

c

the

in the previous

of the solutions

we have to make a large number

z

z

that

is solved with

0.6

B

case

RESULTS

obtained

the influence

in the deterministic

1.0

Finally,

in the additive

(see [13]).

algorithm

we want to study

because

case is similar.

= ~~‘1’2/~~

the Stratonovitch

if the noise is assumed

4.

We now present

fJy+“’

in the multiplicative

case, our way of discretizing

use of a fixed-point

section.

in the multiplicative

of the random

0.06

case 0 = 3, local deterministic

solution,

multiplicative

and

NLS Equations

starting

from the same initial

itative

behaviour

n(t) stands maximal

for the number amplitude

We first present to a deterministic

cases. similar

solutions

a threshold

value.

f(t)

Using

the notation

the qual-

= n(t)/NtOt

still exist at time t, that

that

1 the profile of f in the multiplicative a symmetry

on the trajectory,

always

in order to understand

We then define the quantity

where

is, for which the 2, f(t)

of Section

is an

some tests in the supercritical case 0 = 3 with an initial data ~0 corresponding blow-up (in this case, we choose a Gaussian data such that H(uo) < 0). of all the curves

brings

This confirms behaviour.

and additive

with respect

the noise delays or accelerate

made with an initial

with small amplitude) noise

of numerical

is less than

We plot in Figure of E. We notice computations

is, N tot trajectories)

(that

solution.

of P(T* > t).

approximation

Depending

data

of the random

665

the solution

into

the theoretical However,

blow-up.

data whose deterministic

also give a decreasing results

the symmetry

solution

is global

regime

in both

(a Gaussian

multiplicative

is lost and a multiplicative

Case

sigma=3.

global determnstic

solution

~~ epsilon epsilon epsilon

0.4

0.2

(respectively,

_

0.8

Additive noise, 1000 trajectories 1.0

;-

‘, I 1 ‘, i 0.8: ! ‘, : ‘/ / ‘/ 0.6

g c 0.4

-

Case

sigma=3.

global deterministic

~

Solution

epsilon epsilon epsilon

data the

case, we have observed

= 0.1 = 0.3 = 0.5

0.6

that

and additive

Multiplicative noise, 1000 trajectories l.Or.

time.

numerical

2). We conclude

proved in [9]. In the critical

values

blow-up

More surprisingly,

profile for f (see Figure

an explosive

case for different

to the deterministic

= 0.1 = 0.3 = 0.5

L

, , ) /

Figure 2. Supercritical case u = 3, global deterministic solution, multiplicative and additive noise.

additive)

A. DEBUSSCHEAND L. DIMENZA

666

Multiplicative noise, sigma=2 Comparison refined/non refined 4.0 /-

_

Coarse mesh Local refinement - - - Thinner grid (dt/2,dx/2)

I 3.0

i

t

Additive noise, sigma=2, epsilon = 0.05 Comparison refined/non refined

- -

1.0 L 0.00

Coarsemesh Local refinement Thinner grid (dt/16,dx/16)

0.20

0.40

0.60

I Figure 3. Influence of local refinement on the profile of the solution amplitude and comparison with tests made on uniform grids, multiplicative case, and additive case, C = 2.

noise has a tendency to delay (respectively, accelerate) blow-up. The average value of r* is an increasing (respectively, decreasing) function of E. Moreover, if the initial data corresponds to a global deterministic solution, a multiplicative noise is not able to bring the solution into an explosive regime. However, these tests are made on a uniform grid: the numerical noise has finite correlation lengths given by the time and space steps. In that sense, the noise cannot be considered as a white noise. It is convenient to use a local mesh refinement algorithm in order to simulate a noise with arbitrarily small correlation lengths without increasing too much the computational time (one can see [14] for refinement methods applied to Korteweg-deVries equation). We implement the following strategy. As soon as the solution amplitude becomes too large at one point, we add 2K new space points (K points at each side) and evaluate the values of the solution using linear interpolation. We then choose the new time step At/2 and so on. At each new point, we simulate an independent random variable. Note that in (4), we of course use the local values of the mesh parameters At and Ax. Moreover, in the transition zone separating coarse mesh

NLS Equations

667

Soliton diffusion, subcritical case sigma=1 Multiplicative

-

e&on epsilon epsilon

noise,

100 trajectories.

= 0.01 = 0.05 = 0.1

:’

Tfin = 10

;

/

/

I i

\

Soliton diffusion, critical case sigma=2 Multiplicative 1.5

r----.--

noise,

100 trajectories.

...

---

Tfin = 2

.._~~~~_

‘\

~

epsilon epsilon epsilon epsilon

1.0

= = = =

I/

I! I

0 0.05 0.1 0.2

1 /I, ; / /

Soliton diffusion, supercritical 1.5

Multiplicative

, .-

~

1.0

I

epsilon epsilon epsilon epsilon

noise,

--._-

100 trajectories,

case sigma=3 Tfin = 0.3

= 0 = 0.1 = 0.2 ~0.3

Figure 4. Soliton diffusion for CT= 1 (subcritical 0 = 3 (supercritical case).

case), CT= 2 (critical case), and

A. DEBUSSCHEAND

668

and thin

mesh,

refinement

the discretization

criterium

reference

of the second-order

is /121(t)lloo/llU(to)II,

time (if refinement

L. DI MENZA

space derivative

is changed

> CY,where c~ is a prescribed

in (4).

is made at time t, we set to := t). In the stochastic

The

and to is a

parameter

case, the value of

will involve the local values of mesh parameters Ax and At. We set K = 30 and (Y = 1.2. f.Y2 The same tests as before performed for CJ = 2 show that in the multiplicative case, all the trajectories

remain

bounded

(see Figure

3 where

the amplitude

one trajectory).

In fact, the solution

first follows an explosive

of the numerical

noise counterbalance

the blow-up

the simulations notice

that

smallest refinement

performed

if the simulation

values in the latter algorithm

In the additive the supercritical

on a uniform is made

gives reliable

grid where

on a uniform

computation, results

formation.

we observe

regime.

Then,

grid with a similar

blow-up.

is plotted

scales

different

from

large scales.

At and Ax being behaviour.

This phenomenon

taken

This suggests

large computational

for

the small

This is completely

the noise only contains

while avoiding

case, the noise still generates

of the solution

We

as the that the

times. can also be seen for

case 0 = 3.

We now investigate the case of the propagation of stationary states u(t, x) = U(x) exp iwt, (w > 0) for the deterministic equation when a multiplicative noise is considered. In the onedimensional case, we are faced with a differential equation with additional boundary conditions at infinity. This equation can be explicitly solved: for w = 1, it is well known that U(x) = ((cr + l)sech2(ax))1/2”. Here again, we intend to see the noise influence on the propagation of this localized in space structure, starting from the Cauchy data uo = U. Here, we compute the approximate value fi(t, x) = (u(t, x)) of the expectation of u(t, x) simply given by the arithmetic mean among all the trajectories. In Figure 4, the profiles of ti(t,x) are plotted at a prescribed final time t = tfin for different values of E in the subcritical case (a = l), the critical case (a = 2), and the supercritical case (a = 3). A diffusion mechanism can be observed in all cases even though for u = 2 and u = 3, the solution blows up in the absence of noise. Furthermore, this diffusion is more relevant when the noise amplitude is large. However, the average structure still propagates in the medium. Other tests performed in the nonintegrable case u = 3/2 give the same results and also show that the approximate expectation amplitude decreases as t-l.

REFERENCES 1. P.L. Sulem and C. Sulem, The Nonlinear Schrddinger Equation, Self-Focusing and Wave Collapse, AMS, Springer-Verlag, (1999). 2. 0. Bang, P.L. Christiansen, F. If and K.O. Rasmussen, White noise in the two-dimensional nonlinear SchrGdinger equation, Appl. Math. 57, 3-15 (1995). 3. K.O. Rasmussen, Y.B. Gaididei, 0. Bang and P.L. Christiansen, The influence of noise on critical collapse in the nonlinear SchrGdinger equation, Phys. Rev. A 204, 121-127 (1995). 4. T. Ueda and W.L. Kath, Dynamics of optical pulses in randomly birefringent fibers, Physica D 55, 166-181 (1992). 5. T. Cazenave, An introduction to nonlinear SchrGdinger equation, In Textos de Me’todos Matemciticos, Voolume 26, Instituto de Matemitica, UFRJ, Rio de Janeiro, (1989). 6. J. Printems, Aspects thkoriques et numkriques de l’kquation de Korteweg-deVries stochastique, Ph.D. Thesis, University Paris-Sud Orsay (1998). 7. A. Debussche and J. Printems, Numerical simulation of the stochastic Korteweg-de Vries equation, Physica D 134, 220-226 (1999). 8. A. DeBouard and A. Debussche, A stochastic nonlinear Schradinger equation with multiplicative noise, Comm. Math. Phys. 205, 161-181 (1999). 9. A. DeBouard and A. Debussche, On the effect of a noise on the solutions of the focusing supercritical Schradinger equation (Preprint). 10. A. De Bouard, A. Debussche and L. Di Menza, Theoretical and numerical aspects of stochastic nonlinear SchrGdinger equations, Monte-Carlo Meth. and Appl. 7 (l/2), 55-63 (2001). 11. M. Scalerandi, A. Roman0 and C.A. Condat, Korteweg-deVries solitons under additive stochastic perturbations, Phys. Rev. E 58, 4166-4173 (1998). 12. V. Konotop and L. Vazquez, Nonlinear Random Waves, World Scientific, River Edge, NJ,, (1994). 13. A. Debussche and L. DiMenza, Numerical resolution of a nonlinear stochastic focusing SchrBdinger equation (Preprint).

669

NLS Equations 14.

J.L.

Bona,

V.A.

for the generalized

Dougalis,

0. Karakashian

Korteweg-deVries

and

equation,

R. T&Kinney, Phil.

‘Prans.

Conservative Royal

Sot.

high

London

order A 351,

numerical 107-164

schemes (1995).