Khasminskii--Whitham averaging for randomly perturbed KdV equation

2007/10/20 by Sergeï B. Kuksin, Sergei B. Kuksin, Kuksin, Sergei B. +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #Quantum chaos and dynamical systems #math-ph #math.AP #math.MP

paper · pdf · doi:10.48550/arxiv.0710.3869

arxiv created 2007/10/20 · arxiv updated 2009/12/01

Abstract

We consider the damped-driven KdV equation u-νu_xx+u_xxx-6uu_x=√νη(t,x), x∈ S¹, ∫ u dx≡ ∫ηdx≡0, where 0<ν≤1 and the random process η is smooth in x and white in t. For any periodic function u(x) let I=(I₁,I₂,...) be the vector, formed by the KdV integrals of motion, calculated for the potential u(x). We prove that if u(t,x) is a solution of the equation above, then for 0≤ t\lesssimν^-1 and ν→0 the vector I(t)=(I₁(u(t,⋅)),I₂(u(t,⋅)),...) satisfies the (Whitham) averaged equation.

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