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Soliton dynamics for the Korteweg-de Vries equation with multiplicative homogeneous noise. (English) Zbl 1185.35222
Summary: 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.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
76B25 Solitary waves for incompressible inviscid fluids
35C08 Soliton solutions
35B40 Asymptotic behavior of solutions to PDEs
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