Numerical simulation of the stochastic Korteweg–de Vries equation

https://doi.org/10.1016/S0167-2789(99)00072-XGet rights and content

Abstract

In this work, we numerically investigate the influence of a homogeneous noise on the evolution of solitons for the Korteweg–de Vries equation. Our numerical method is based on finite elements and least-squares. We present numerical experiments for different values of noise amplitude and describe different types of behaviours.

Introduction

The Korteweg–de Vries equation models the propagation of weakly nonlinear dispersive waves in various fields: plasma physics [30], surface waves on the top of an incompressible irrotational inviscid fluid [18], [37], beam propagation [27]. From a mathematical point of view, this equation is recognized as a simple canonical equation for such phenomena since it combines some of the simplest types of dispersion with the simplest types of nonlinearity.

When using a convenient set of coordinates and after rescaling the unknown, it can be written as∂u∂t3u∂x3+u∂u∂x=0,where t≥0 and x∈R. This equation must be supplemented with an initial data and with boundary conditions. These can be of two types. It can be required that u decays to zero at infinity or that u is periodic with a given period L.

In Eq. (1.1), no exterior influence is taken into account. When considering the propagation of ion-acoustic solitons in a noisy plasma, it seems that a noise term has to be added. Indeed, in their experiment, Chang et al. [6] have observed that when a soliton wave is taken as initial condition, the average wave is damped and decays like tα, α>0. This behaviour has been studied theoretically by various authors who consider the stochastic Korteweg–de Vries equation∂u∂t+u∂u∂x3u∂x3ξ̇ when the perturbation term ξ̇ is of white-noise type. The case of a time-dependent noise ξ̇=ξ̇(t) is a particular one since the equation is still integrable. This has been studied by Wadati [34] who proved a decay of the averaged solution 〈u(x,t)〉 with α=3/2. Using perturbation techniques, similar results have been obtained for the time-dependent noise with damping [35]; for a multiplicative noise [15]; for space and time-dependent noise [17]. Also a numerical study [29] has shown that Eq. (1.2), with space–time whitenoise, seems to reproduce the behaviour observed experimentally in [6].

In many others circumstances, apart from the theory of plasmas, the Korteweg–de Vries equation (1.1) is an idealized model in which many effects have been neglected, and it is not unreasonable to model them stochastically: when the time scales of the phenomena modeled by Eq. (1.1) are much larger than the correlations of the noise, it is justified to assume that it is of white-noise type.

Beside the physical motivations above, we think that the understanding of the perturbation of an integrable equation such as Eq. (1.1) by a noise is of great mathematical interest.

In this work, we wish to study the influence of a noise term on the propagation of soliton profile and consider more general quantities than the average 〈u(x,t)〉 which in our opinion does not contain enough information. Indeed, if a soliton is considered as initial condition, it is reasonable to think that it will keep its shape for some time and it seems interesting for instance to measure the average of its maximum amplitude. Clearly the evolution of 〈u(x,t)〉 does not give any indication of that aspect. Similarly, we want to study the phase shift due to the collision of two solitary waves in the presence of noise.

Many articles have also studied the derivation of the forced Korteweg–de Vries equation:∂u∂t3u∂x3+u∂u∂x=f,to the modeling of the generation of waves in the presence of a moving exterior pressure field or past an obstacle and in various circumstances: surface waves [2], [7], [20], [38]; rotating flow [11]; coastal current over a topographic feature [12], [23]; β-plane waves by flows other topography [13], [25], [36]. Equation (1.3) has been studied mathematically in [4]. In general, the forcing f is spatially localized and represents the disturbance. Again, it is reasonable to consider the case of a small random perturbation of this equation. However, here it seems that the noise should also be localized and the interest is in the influence of the noise on the generation of waves. This is studied in [10].

The effect of small random inhomogeneities in the bottom of a fluid has also been studied in [16] where a Korteweg–de Vries–Burgers equation is obtained. When these inhomogeneities are rapidly varying and not small, the shallow water approximation is not valid and the Korteweg–de Vries model is not used (see [24]).

The stochastic equation of interest is here∂u∂t+u∂u∂x3u∂x3ξ̇ which we have obtained from Eq. (1.1) by a normalization and the addition of the noise term ξ(x,t) which is a Gaussian process with correlationsEξ̇(x,t)ξ̇(y,s)=c(x−y)δt−s with γ measuring the amplitude of the noise. It is δ-correlated in time since we shall assume that it is of white-noise type. If it were white also in space, this would result in c(xy)=δxy. However, such an irregular correlation function is difficult to handle and a smoother correlation in space is often used.

Note that since the correlation depends only on the difference xy, the noise is homogeneous. The mathematical construction of ξ will be precisely described below.

In the case of a space-independent noise, c=1, the noise can be removed by a simple change of coordinates and a soliton-type solution can be given. This case has been studied in [34]. Here we are interested in the case of a space-dependent noise. It seems difficult to obtain an explicit solution, nevertheless qualitative information could be obtained from a perturbation method (see [17]) but only with small noise. Therefore, we choose to use numerical simulations to get information.

Numerical schemes for Eq. (1.1) have been developed and efficient schemes are available (see [3], [31] and the reference therein). Among these, [31] proposes a scheme which keeps the integrability property of Eq. (1.1). In [3], high orders in both space and time methods are used and a very efficient scheme is developed in order to study the blow up for the generalized Korteweg–de Vries equation.

In our situation, the solution is not expected to be smooth either in space or in time, and we have chosen a scheme based on finite elements of order one and the least-squares method. This type of scheme is expected to be robust, an interesting property when dealing with noise, and has been introduced in [5]. We have improved the scheme of [5] and obtained a method which can simulate the solutions of Eq. (1.1) in a very satisfactory way.

We will present numerical results which tend to show that the presence of the noise in Eq. (1.4) creates a noisy background in the profile of the solution. At the beginning this background is uniform but it progressively organizes itself and we observe the formation of nonlinear right-going waves. When a soliton profile is taken as initial data, it seems that it is not affected by the noise at the beginning. Then the solitary wave interacts with the waves created by the noise and these inelastic interactions result in a modification of its amplitude and velocity. However, the wave is not destroyed and propagates for very long time as can be seen on the level curves of the solution. We have observed this phenomenon on several computed trajectories. Concerning the behavior before the creation of the waves by the noise, we have been able to simulate many trajectories and to compute averages. For instance, we have computed the average of the velocity of the solitary waves for different values of γ. We find a value which is very close to the velocity of the wave for the deterministic equation. Therefore, it seems that the noise does not affect strongly the velocity of the soliton. We have also computed the averaged invariant quantities and the averaged phase shift in the collision of two solitons. The noise strongly affects the invariants. We observe a linear evolution with respect to time but it seems that the phase shift is not changed on the average. Let us recall that in the deterministic theory, the integrability of the system (1.1), which is a consequence of an infinity of conservation laws, implies elastic soliton interactions. Now, though the invariants are totally destroyed and as far as elasticity is concerned (i.e. velocity unchanged and a phase shift), the noise does not affect on average those interactions. However, the standard deviation of the phase shift increases with the amplitude of the noise, indicating that the interaction loses its elastic aspect when we consider trajectories only.

In the case of periodic boundary conditions, the mathematical construction of ξ can be described as follows. We introduce a cylindrical Wiener process W on L2(]0,L[) by settingW(t,x)=i=0βi(t)ei(x),where {ei}i∈N is an orthonormal basis of L2(]0,L[) and i}i∈N is a sequence of independent brownian motions. These brownian motions are random processes βi(t,ω), t≥0, ω∈Ω, defined on a stochastic basis (Ω,F,P,{Ft}t≥0) such that for each ts and for each i, βi(t)−βi(s) is a Ft-measurable Gaussian random variable independent of Fs, and therefore of βi(s).

It is well-known that the brownian motions are nowhere differentiable with respect to time. Also, it is easy to see that the series defining W does not converge in L2(]0,L[). Formally the space–time white-noise is defined asη̇(x,t)=dWdt and has the property of being δ-correlated in space and time:Eη̇(x,t)η̇(y,s)=δx−yδt−s.

For Φ a linear map from L2(]0,L[) to L2(]0,L[), we setW̃=ΦW=i∈NΦeiβi,andξ̇=dW̃dt=Φη̇.

For instance, if Φ is an operator of the formΦf=]0,L[k(x,y)f(y)dy,f∈L2(]0,L[)then the correlation is given byEξ̇(x,t)ξ̇(y,s)=δt−s]0,L[k(x,z)k(y,z)dzand the noise is homogeneous if the kernel k is of the formk(x,y)=k(x−y).

The mathematical form (or Ito form) of Eq. (1.4) isdu+u∂u∂x3u∂x3dt=γΦdW.

For the boundary condition, x∈R and u decays at infinity, the description is the same with ]0,L[ being replaced by R. It has been shown in [9], [26] that under suitable smoothness assumptions on the data, Eq. (1.7) possesses a unique solution. For instance, in [26], it is shown that, in the case of a localized noise, it suffices that Φ is a Hilbert–Schmidt operator from L2(I) into L2(I), where I is the perturbed region. This assumption amounts requiring that kL2(I×I) in Eq. (1.6).

Our numerical method is described in Section 2 where we show that it gives good results for the deterministic equations on known solutions: propagation of solitons, collision of solitons, splitting of a Gaussian profile. The results of the stochastic simulation are presented in Section 3, for the case of Eq. (1.4).

Section snippets

The numerical method

We now describe the numerical scheme which we use to simulate our stochastic equation. Due to the lack of regularity of the noise, it is expected that solutions are smooth neither in time (not differentiable) nor in space (not H1). Therefore, we consider low order discretization. It follows that we lose important properties such as the conservation of the L2(R) norm. However, we test our method on the deterministic equation and show that it behaves very well. It does not introduce any numerical

Behaviour of solitons in the presence of noise

We now present numerical experiment on Eq. (1.7). We use the numerical method described in Section 2 and the same time and space discretization as in Section 2.3.

Acknowledgements

The authors warmly thank Professor Y. Kutznetzov for having suggested to them the study of this problem.

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