Soliton dynamics for the Korteweg-de Vries equation with multiplicative homogeneous noise

We consider a randomly perturbed Korteweg-de Vries equation. The perturbation is a random potential depending both on space and time, with a white noise behavior in time, and a regular, but stationary behavior in space. We investigate the dynamics of the soliton of the KdV equation in the presence of this random perturbation, assuming that the amplitude of the perturbation is small. We estimate precisely the exit time of the perturbed solution from a neighborhood of the modulated soliton, and we obtain the modulation equations for the soliton parameters. We moreover prove a central limit theorem for the dispersive part of the solution, and investigate the asymptotic behavior in time of the limit process.


Introduction
Our aim is to describe the dynamics of a soliton solution of the Korteweg-de Vries equation in the presence of a random potential, depending both on space and time and which is white in time. After the first paper [21] showing "superdiffusion" of the soliton of the KdV equation in the presence of an external force which is a white noise in time (see also [1], [16]), the interest in such questions of soliton dynamics in the presence of either deterministic or random perturbations has recently increased in the mathematical community. In [15], e.g. the question is investigated with the help of inverse scattering methods, for different types of time-white noise perturbations, still for the KdV equation, while in [11], [12], the case of a soliton of the NLS equation is studied, with the presence of a slowly varying deterministic external potential. Random potential perturbations for NLS equations have also been considered in [14] and [9]. The diffusion of solitons of the KdV equation in the presence of additive noise was numerically investigated in [19]. Also, in [5], we studied the soliton dynamics for a KdV equation with an additive space-time noise. Our aim here is to reproduce the analysis of [5] in the case of a random potential, which is stationary in space : the solution of the stochastic equation starting from a soliton at initial time will then stay close to a modulated soliton up to times small compared to ε −2 where ε is the amplitude of the random perturbation (see below). In the present case, where the noise is multiplicative (the random potential) we are then able to analyze more precisely the modulation equations for the soliton parameters and the linearized equation for the remaining (dispersive) part of the solution, and especially its asymptotic behavior in time.
We consider a stochastic KdV equation which may be written in Itô form as where ε > 0 is a small parameter, u is a random process defined on (t, x) ∈ R + × R, W is a Wiener process on L 2 (R) whose covariance operator φφ * is such that φ is a convolution operator on L 2 (R) defined by The convolution kernel k satisfies Considering a complete orthonormal system (e i ) i∈N in L 2 (R), we may alternatively write W as (β i ) i∈N being an independent family of real valued Brownian motions. The correlation function of the process W is then given by The existence and uniqueness of solutions for stochastic KdV equations of the type (1.1) but with an additive noise have been studied in [4], [7], [8]. The multiplicative case with homogeneous noise as described above was considered in [6]: assuming, together with the above condition, that k is an integrable function of x ∈ R allowed us to prove the global existence and uniqueness of solutions to equation (1.1) in the energy space H 1 (R), that is in the space where both the mass are well defined. Note that m and H are conserved for the equation without noise, that is Under the above conditions on k, it was then proved in [6] that for any given initial data u 0 ∈ H 1 (R), there is a unique solution u of (1.1) with paths a.s. continuous for t ∈ R + with values in H 1 (R). Our aim in this article is to analyze the qualitative influence of a noise on a soliton solution of the deterministic equation. More precisely, we study the qualitative behavior of solutions of (1.1) in the limit ε tends to zero, assuming that the initial state of the solution is a soliton of equation (1.6). We recall indeed that equation (1.6) possesses a two-parameter family of solitary waves (or soliton) solutions, propagating with a constant velocity c > 0, with the expression u c, We do not recall here the well-known results concerning the stability of the soliton solutions u c,x 0 in equation (1.6), but we refer to [2], [3], [17] or [18] for a review of the stability questions using PDE methods, or to [13] and [20] for a review of the stability of the solitons with the help of the inverse scattering transform.
Let us consider as in [5] the solution u ε (t, x) of equation (1.1) which is such that u ε (0, x) = ϕ 0 (x) where c 0 > 0 is fixed. Then, in Section 2, we show, as we did in [5] for the additive equation that up to times Cε −2 , where C is a constant, we may write the solution u ε as where the modulation parameters c ε (t) and x ε (t) satisfy a system of stochastic differential equations and the remaining term εη ε is small in H 1 (R). We then prove in Section 3 that the process η ε converges as ε goes to zero, in quadratic mean, to a centered Gaussian process η which satisfies an additively driven linear equation, with a conservative deterministic part; we also investigate the behavior of the process η as t goes to infinity and prove that η is in some sense a Ornstein-Uhlenbeck process, with a unique Gaussian invariant measure. In addition, the parameters x ε (t) and c ε (t) may be developed up to order one in ε and we get where B 1 and B 2 are correlated real valued Brownian motions; keeping only the order one terms in those modulation parameters, we then obtain a diffusion result on the modulated soliton similar to the result obtained by Wadati in [21], but with a different time exponent (see Section 4). In all what follows, (., .) will denote the inner product in L 2 (R), and we denote by T x 0 the translation operator defined for ϕ ∈ C(R) by (T x 0 ϕ)(x) = ϕ(x + x 0 ). Note that since the process W is stationary in space, for any x 0 ∈ R the process T x 0 W is still a Wiener process with covariance φφ * . Indeed by (1.3),

Modulation and estimate on the exit time
In this section, we prove the following theorem.
Theorem 2.1. Assume that the kernel k of the noise satisfies (1.2) together with k ∈ L 1 (R) and let c 0 be fixed. For ε > 0, let u ε (t, x), as defined above, be the solution of (1.1) with u(0, x) = ϕ c 0 (x). Then there exists α 0 > 0 such that, for each α, 0 < α ≤ α 0 , there is a stopping time τ ε α > 0 a.s. and there are semi-martingale processes c ε (t) and x ε (t), defined a.s. for t ≤ τ ε α , with values respectively in R + * and R, so that if we set εη In addition, for α 0 sufficiently small, and any α ≤ α 0 , there is a constant C > 0, depending only on α and c 0 , such that for any T > 0, there is an ε 0 > 0, with, for each ε < ε 0 , It was noticed heuristically in [5], and proved in [10] that in the additive case, the use of the modulation parameters x ε (t) and c ε (t) was necessary in order to get the estimate (2.1). Indeed, it was proved in [10] that if we denote byτ ε,n α = inf{t > 0, u ε,n (t, .) − ϕ c 0 1 > α}, where u ε,n is here the solution of equation (1.1), but with an additive noise that becomes stationary in space as n goes to infinity (see [10] for a precise statement) then there exists a constant C(α, c 0 ) which depends on α and c 0 but not on T such that It is not clear that (2.2) is still true in the present multiplicative case, because the proof involves a controlability problem with a potential which -up to now -is open.
Note also that the decomposition given in Theorem 2.1 is not unique, and is determined by the choice of specific orthogonality conditions (see the proof below). In particular, contrary to the additive case, we will be able here to investigate the asymptotic behavior in time of the limit process by choosing one particular decomposition of the form given in Theorem 2.1. This is the object of Section 3.3. Proof of Theorem 2.1 The proof follows closely the proof of Theorem 2.1 in [5] and we refer to [5] for more details. The parameters x ε (t) and c ε (t) are obtained thanks to the use of the implicit function Theorem. These are then local semi-martingales defined as long as |c ε (t) − c 0 | < α and u ε (t, . + x ε (t)) − ϕ c 0 1 < α, and setting one has for each ε > 0, almost surely, In order to estimate the exit time we make use , as in [5], of the functional defined for u ∈ H 1 (R), where H and m are defined respectively in (1.4) and (1.5). Note that ϕ c 0 is a critical point of Q c 0 . We denote by L c 0 the linearized operator around ϕ c 0 , that is The next lemma, which is proved with the use of the Itô Formula, using the same regularization procedure as in [4], gives the evolution of H and m for the solution u ε of (1.1) with u ε (0) = ϕ c 0 : The existence of such a constant is a classical result (see [2] or [3]). Then it is easy to show (see [5]) that there is a constant C(α 0 ) > 0 such that for any t < τ ε α , then by (2.9), the translation invariance of Q c 0 , and Lemma 2.2 The term |c ε (τ ) − c 0 | is then estimated thanks to the orthogonality condition (η ε , ϕ c 0 ) = 0 and the evolution of m(u ε (τ )) given in Lemma 2.2; one obtains, for some constants µ > 0 and C > 0, depending only on c 0 and α 0 (with α ≤ α 0 ) Hence, choosing α 0 sufficiently small one gets With this estimate in hand, together with (2.11), the conclusion of Theorem 2.1 follows with the same arguments as in the proof of Proposition 3.1 in [10]. These arguments rely on classical exponential tail estimates for stochastic integrals, after noticing that u ε (s) 1 ≤ C, a.s. for s ∈ [0, τ ε α ∧ T ] and α ≤ α 0 , so that the quadratic variation of each of the integrals involved in the above estimates are bounded above by CT .

A central limit theorem
This section is devoted to the proof of the next theorem: Theorem 3.1. Under the assumptions of Theorem 2.1, let α < α 0 be fixed. Then we can findc ε (t) andx ε (t) satisfying the conclusion of Theorem 2.1 such that ifη ε is defined as in Theorem 2.1, for any T > 0, the process (η ε (t)) t∈[0,T ] converges in L 2 (Ω; L ∞ (0, τ ε α ∧ T ; L 2 (R))) to a Gaussian processη satisfying the additive linear equation withη(0) = 0, whereW is the Wiener process with covariance φφ * given byW = T c 0 t W , and Q is a projection operator. Moreover, for a > 0 sufficiently small compared to c 0 , the process w(t, x) = e axη (t, x) is a well defined H 1 valued process, of Ornstein-Uhlenbeck type, which converges in law to an H 1 -valued Gaussian random variable as t goes to infinity.
The conclusion of Theorem 3.1 will be obtained in three steps. The first step consists in estimating the modulation parameters obtained in Theorem 2.1, in terms of η ε , using the equations for those parameters; then the convergence of η ε as ε tends to zero is proved, and finally in the third step, a slight change in the modulation parameters is performed, in order that the limit process η may be written as an Ornstein-Uhlenbeck process.
From now on, we assume that α is fixed and sufficiently small, so that the conclusion of Theorem 2.1 holds, and we denote τ ε α by τ ε .
3.1. Modulation equations. Since we know that the modulation parameters x ε (t) and c ε (t) are semi-martingale processes adapted to the filtration generated by (W (t)) t≥0 , we may a priori write the stochastic evolution equations for those parameters in the form where y ε and a ε are real valued adapted processes with a.s. locally integrable paths on [0, τ ε ), and b ε , z ε are predictable processes with paths a.s. in L 2 loc (0, τ ε ; L 2 (R)). We then proceed as in [5] : the Itô-Wentzell Formula applied to u ε (t, x + x ε (t)), together with equation (1.1) for u ε and the first equation of (3.2) for x ε give a stochastic evolution equation for u ε (t, x + x ε ). On the other hand, the standard Itô Formula together with the second equation of (3.2) for c ε give an equation for the evolution of ϕ c ε (t) . Replacing then ϕ c ε (t) + εη ε (t, x) for u ε (t, x + x ε (t)) in the first equation leads to the following stochastic equation for the evolution of η ε (t) : where L c 0 is defined in (2.5). Now, taking the L 2 -inner product of equation (3.3) with ϕ c 0 , on the one hand, and with ∂ x ϕ c 0 on the other hand, then using the orthogonality conditions (2.3) and the fact that L c 0 ∂ x ϕ c 0 = 0, and finally identifying the drift parts and the martingale parts of each of the resulting equations lead to the same kind of system that we previously obtained in [5]; namely, setting then one gets for the drift parts and O(|c ε − c 0 | + η ε 1 ) is uniform in ε, t and ω as long as t ≤ τ ε . Concerning the martingale parts, one gets the equation
Proof The proof is exactly the same as the proof of Corollary 4.3 in [5], once noticed that, a.s.
where we have used the Parseval equality in the fourth line.

3.2.
Convergence of η ε . Let us first assume that η ε has a limit as ε goes to zero, and take formally the limit as ε goes to zero in the preceding equations. Then, as was noticed above, Moreover, formally, η satisfies the equation It is easy to show that (3.16) has a unique adapted solution η with paths a.s. in C(R + , H 1 ) satisfying η(0) = 0. Moreover using the fact that (∂ c ϕ c 0 , ∂ x ϕ c 0 ) = 0, one easily gets from the above equation that (η, ϕ c 0 ) = (η, ∂ x ϕ c 0 ) = 0, ∀t > 0. Next, we make use of the following lemmas, whose proofs are obtained in the same way as the corresponding Lemmas in [5]. Lemma 3.3. Let η be the solution of (3.16) with η(0) = 0. Then, for any T > 0, there is a constant C depending only on c 0 , T and k 1 such that E η(t) 4 1 ≤ C, ∀t ≤ T.
Lemma 3.4. Let η ε be the solution of (3.3), defined for t ∈ [0, τ ε [, obtained thanks to the modulation procedure of Section 2. Then, for any T > 0, The above lemmas show that Indeed, the expression of c ε (t) − c 0 given by (3.2) together with (3.10) and (3.11) imply easily with C = C(α, c 0 , T, k 1 ). Then, (3.17) is deduced form Lemma 3.4 for q = 2, and follows for all other values of q from the uniform boundedness of |c ε (t) − c 0 | on [0, T ∧ τ ε ]. Note that an immediate consequence of (3.17) is the fact that We will finally need the next lemma.
Proof Here again, it is sufficient to consider the case q = 1. We recall that Z ε l satisfies equation (3.8). First, it is clear that On the other hand, in view of (3.9), denoting F 0 l (t) the formal limit of F ε l (t), one has Then, the Itô Formula applied to the function using equation (3.2) for dx ε (t), together with (3.10), (3.11), and (3.17) lead to the conclusion of Lemma 3.5. Now, in order to prove that where η is the solution of (3.16) with η(0) = 0, it suffices to set v ε = η ε − η, to deduce from (3.16) and (3.3) the equation for dv ε and to apply the Itô Formula to get the evolution of |v ε | 2 L 2 . We do not give the details of those tedious, but easy computations. Finally, the use of the following estimates : on the one hand, and which is obtained as in the proof of Lemma 3.5 on the other hand, together with Lemma 3.3 to 3.5 allow to get the conclusion, that is the convergence of η ε to η in L 2 (Ω, L ∞ (0, τ ε ∧T ; L 2 (R))).

Complements on the limit equation.
First of all, we note that the modulation equations may be written at order one in ε as Note that W 1 and W 2 are real valued Brownian motions, which are independent since because the operator φ * commutes with spatial derivation. Now, we want to investigate the asymptotic behavior in time of the process η. However, in the present form, the process η does not converge in law as t goes to infinity; this is due to the fact that the preceding modulation does not exactly correspond to the projection of the solution u ε on the (two-dimensional) center manifold, in which case the remaining term would belong to the stable manifold around the soliton trajectory. We now show that by slightly changing the modulation parameters, we can get a new decomposition of the solution u ε which is defined on the same time interval as before, but which fits with the preceding requirements. For that purpose, we first need to recall a few facts from [18].
The generalized nullspace of the operator ∂ x L c 0 (that is the operator arising in the linearized evolution equation in the soliton reference frame) is spanned by the functions ∂ x ϕ c 0 and ∂ c ϕ c 0 , with the equality and there are constants θ 1 and θ 2 (with θ 1 = (ϕ c 0 , ∂ c ϕ c 0 )) such that if we set then the generalized nullspace of −L c 0 ∂ x is spanned byg 1 andg 2 and We also set, for a > 0, Then the operator A a defined for a > 0 by A a = e ax ∂ x L c 0 e −ax has a well defined generalized nullspace spanned by f a 1 , f a 2 and the spectral projection on this nullspace is given by P w = 2 k=1 (w, g a k )f a k where w = e ax v, and v is an L 2 function. Moreover, if Q = I − P , then Q is the spectral projection on the stable manifold of A a , and under the condition 0 < a < c 0 /3, there are constants C > 0 and b > 0 such that (3.20) e Aat Qw 1 ≤ Ce −bt w 1 , ∀t > 0, ∀w ∈ H 1 , where e Aat is the C 0 -semi-group generated by A a (see Theorem 4.2 in [18]). Now, let η be the solution of (3.16) with η(0) = 0, and consider w(t, x) = e ax η(t, x). Note that the orthogonality condition (η, ϕ c 0 ) = 0 implies (w, g a 2 ) = 0, so that P w = λ(t)f a 1 with λ(t) = (w(t), g a 1 ) a real valued stochastic process whose evolution is given by where we have used (3.16) and the fact that A a P w = 0 and λ(0) = 0. Hence, λ(t) is bounded in L 4 (Ω; L ∞ (0, T ∧ τ ε )) by Lemma 3.3. Let us setx ε (t) = x ε (t) − ελ(t) for t ∈ [0, τ ε [. Then (3.22) u ε (t, x +x ε (t)) = ϕ c ε (t) (x) + εη ε (t, x) withη ε (t, x) = 1 ε (ϕ c ε (t) (x − ελ(t)) − ϕ c ε (t) (x)) + η ε (t, x − ελ(t)).
Hence, it follows from Lemma 3.3, 3.4 and the above bound on λ that Moreover, this covariance operator converges as t goes to infinity and it follows that w 2 converges in law in H 1 to a Gaussian random variable. The end of the statement of Theorem 3.1 follows, settingQv = e −ax Qe ax v.