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Khasminskii-Whitham averaging for randomly perturbed KdV equation. (English) Zbl 1148.35077
Summary: We consider the damped-driven KdV equation:
$\dot u-\nu u_{xx}+u_{xxx}-6uu_x=\sqrt\nu \eta(t,x),\quad x\in S^1,\quad \int u\,dx\equiv\int \eta\,dx\equiv 0,$
where $$0<\nu\leq 1$$ and the random process $$\eta$$ is smooth in $$x$$ and white in $$t$$. For any periodic function $$u(x)$$ let $$I=(I_1,I_2,\dots)$$ 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 $$0\leq t\leq \nu^{-1}$$ and $$\nu\to 0$$ the vector $$I(t)=(I_1(u(t,\cdot)),I_2(u(t,\cdot)),\dots)$$ satisfies the (Whitham) average equation.

MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35R60 PDEs with randomness, stochastic partial differential equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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References:
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