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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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