Sergei B. Kuksin and Andrey L. Piatnitski
Khasminskii--Whitham averaging for randomly 
perturbed KdV equation
(442K, pdf)

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.