Khasminskii--Whitham averaging for randomly perturbed KdV equation
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We consider the damped-driven KdV equation $$ \dot u-ν{u_{xx}}+u_{xxx}-6uu_x=\sqrtνη(t,x), x\in S^1, \int u dx\equiv \intηdx\equiv0, $$ where $0<ν\le1$ and the random process $η$ 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 $0\le t\lesssimν^{-1}$ and $ν\to0$ the vector $ I(t)=(I_1(u(t,\cdot)),I_2(u(t,\cdot)),...) $ satisfies the (Whitham) averaged equation.