06-313 Sergei B. Kuksin and Andrey L. Piatnitski
Khasminskii--Whitham averaging for randomly perturbed KdV equation (442K, pdf) Nov 2, 06
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Abstract. We consider the damped-driven KdV equation $$ \dot u-\nu{u_{xx}}+u_{xxx}-6uu_x=\sqrt\nu\,\eta(t,x),\; x\in S^1,\ \int u\,dx\equiv \int\eta\,dx\equiv0\,, $$ where $0<\nu\le1$ 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,\dots) $ 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 $\nu\to0$ the vector $ I(t)=(I_1(u(t,\cdot)),I_2(u(t,\cdot)),\dots) $ satisfies the (Whitham) averaged equation.

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