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

Abstract

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 .

Vortex solutions in Bose-Einstein condensation under a trapping potential varying randomly in time

“…We will see then that the remaining part is of order one with respect to ε. The proof of this fact is rather similar to Theorem 2.1 in [10], Theorem 2.1 in [11], and Theorem 2 in [13], and we only mention the differences from the previous works. The decomposition is in the form…”

“…In this section, we give the outline of proof for the existence of the modulation parameter and the estimate on the exit time (2.21). The arguments are similar to those in [11,13]. The following lemma shows the equalities of the charge Q and of the energy H by the evolution of (2.8).…”

“…The method we will use, so called collective coordinates approach, consists in writing that the main part of the solution is given by a modulated vortex and in finding then the modulation equations for the vortex parameters. Such ideas to analyze the asymptotic behavior have been used by many authors in the physics literature, as well as in the study of mathematical problems, but mainly for the ground states (see, for example, in the deterministic case [22,23,38] and in the stochastic case [10,11,13,15]). The stability property of the vortex solutions in the deterministic equation is required in order to apply such collective coordinates approach, and we will here take advantage of the fact that the m-equivariance property of the solutions is preserved by the noise.…”

Benjamin–Ono soliton dynamics in a slowly varying potential

“…Dynamics of near double solitons for mKdV under semiclassical perturbation of mKdV was studied by Holmer, Perelman, & Zworski [14]. De Bouard and Debussche [9] considered a stochastic perturbation of KdV with multiplicative white noise. Pocovnicu [31,32] and Gérard & Grellier [12] considered the cubic-Szego equation.…”

Amplitude equations for SPDEs with quadratic nonlinearities forced by additive and multiplicative noise

“…Stochastic partial differential equations (SPDEs) with quadratic nonlinearities of the type du = Audt + B(u, u)dt + G(u)dW (t), (1.1) are used to study some physical phenomenon such as hydrodynamic turbulence (Burgers' equation) [11], surface erosion (Kuramoto-Sivashinsky equation) [18], amorphous thin-film growth [23], propagation of solitons (Korteweg-de Vries equation) [9,10] and Rayleigh-Bénard convection [2].…”