Khasminskii--Whitham averaging for randomly perturbed KdV equation

Abstract

We consider the damped-driven KdV equation $$\dot{u}-\nu u_{xx}+u_{xxx}-6u{u}_x=\sqrt{\nu }\eta (t,x),x\in S^1,\int udx\equiv \int \eta dx\equiv 0,$$ where $0<\nu \le 1$ and the random process $\eta$ is smooth in $x$ and white in $t$. For any periodic function $u(x)$ let $I=(I_1,I_2,...)$ 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 $0\le t\lesssim {\nu }^{-1}$ and $\nu \to 0$ the vector $I(t)=(I_1(u(t,\cdot )),I_2(u(t,\cdot )),...)$ satisfies the (Whitham) averaged equation.