The Korteweg-de Vries Equation with Multiplicative Homogeneous Noise

Bouard, Anne de; Debussche, Arnaud

doi:10.1142/9789812770639_0004

Cited by 14 publications

(19 citation statements)

References 15 publications

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“…In this section, we give a proof of the existence of modulation parameters and the estimate on the exit time (2.4). The arguments are similar to those in [7] but we repeat them for the sake of completeness. The following lemma gives the evolution of the charge Q and of the energy H by (1.5).…”

Section: Modulation and Estimates On The Exit Timementioning

confidence: 61%

“…We will see then that the remaining part is of order one with respect to ε. The proof of the theorem is rather similar to those in [6,7], but for the sake of completeness we repeat it in the next section. We remark that the proof of theorems 2 and 3 will be completed in subsection 6.3 (see Remark 2.1 below).…”

Section: Resultsmentioning

confidence: 92%

“…As in [6,7], we now take the L 2 inner product of Eqs. (4.2) and (4.3) with φ µ 0 and make use of the orthogonality conditions (2.5) and (2.6), we obtain the equations for the modulation parameters y ε , z ε , a ε and b ε from the identification of drift parts and that of martingale parts.…”

Section: Modulation Equationsmentioning

confidence: 99%

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Modulation analysis for a stochastic NLS equation arising in Bose–Einstein condensation

Bouard

Fukuizumi

2009

Asymptotic Analysis

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We study the asymptotic behavior of the solution of a model equation for Bose-Einstein condensation, in the case where the trapping potential varies randomly in time. The model is the so called Gross-Pitaevskii equation, with a quadratic potential with white noise fluctuations in time whose amplitude ε tends to zero. The initial condition of the solution is a standing wave solution of the unperturbed equation. We prove that up to times of the order of ε −2 , the solution decomposes into the sum of a randomly modulated standing wave and a small remainder, and we derive the equations for the modulation parameters. In addition, we show that the first order of the remainder, as ε goes to zero, converges to a Gaussian process, whose expected mode amplitudes concentrate on the third eigenmode generated by the Hermite functions, on a certain time scale.

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Section: Modulation and Estimates On The Exit Timementioning

confidence: 61%

Section: Resultsmentioning

confidence: 92%

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Modulation analysis for a stochastic NLS equation arising in Bose–Einstein condensation

Bouard

Fukuizumi

2009

Asymptotic Analysis

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Bouard

Debussche

2009

Electron. J. Probab.

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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.1991 Mathematics Subject Classification. 35Q53, 60H15, 76B25, 76B35 .

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“…In this subsection, we prove Theorem 1.6. Following [19], we use a truncated version of (1.4). The main idea is to apply an appropriate cut-off function on the nonlinearity to obtain a family of truncated SNLS, and then prove global well-posedness of these truncated equations.…”

Section: 2mentioning

confidence: 99%

Stochastic nonlinear Schrödinger equations on tori

Cheung

Mosincat

2018

Stoch PDE: Anal Comp

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We consider the stochastic nonlinear Schrödinger equations (SNLS) posed on d-dimensional tori with either additive or multiplicative stochastic forcing. In particular, for the one-dimensional cubic SNLS, we prove global well-posedness in L 2 (T). As for other power-type nonlinearities, namely (i) (super)quintic when d = 1 and (ii) (super)cubic when d ≥ 2, we prove local well-posedness in all scaling-subcritical Sobolev spaces and global well-posedness in the energy space for the defocusing, energy-subcritical problems.Date: March 8, 2018. 2010 Mathematics Subject Classification. 60H15. Key words and phrases. stochastic nonlinear Schrödinger equations; well-posedness. 1. The multiplicative noise given by the Stratonovich product u • φξ with real-valued ξ is relevant in physical applications, as it conserves the mass of u (i.e. t → u(t) 2 L 2x (T d ) is constant) almost surely. Our analysis can handle either the Itô or the Stratonovich product, and we choose to work with the former for the sake of simpler exposition.Here, P ≤N is the Littlewood-Paley projection onto frequencies {n ∈ Z d : |n| ≤ N }, p ≥ 2(d+2) d , and ε > 0 is an arbitrarily small quantity 3 . However, such Strichartz estimates are not strong enough for a fixed point argument in mixed Lebesgue spaces for the deterministic NLS on T d . To overcome this problem, we shall employ the Fourier restriction norm method by means of X s,b -spaces defined via the norms(1.7)The indices s, b ∈ R measure the spatial and temporal regularities of functions u ∈ X s,b , and F t,x denotes Fourier transform of functions defined on R × T d . This harmonic analytic 2. Here, W s,r (T d ) denotes the L r -based Sobolev space defined by the Bessel potential norm:, where n := 1 + |n| 2 . When r = 2, we have H s (T d ) = W s,2 (T d ).3. More recently, Killip and Vişan [25] removed the arbitrarily small loss of ε derivatives in (1.6) when p > 2(d+2) d . However, we do not need this scale-invariant improvement in our results.

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The Korteweg-de Vries Equation with Multiplicative Homogeneous Noise

Cited by 14 publications

References 15 publications

Modulation analysis for a stochastic NLS equation arising in Bose–Einstein condensation

Modulation analysis for a stochastic NLS equation arising in Bose–Einstein condensation

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

Stochastic nonlinear Schrödinger equations on tori

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