Khasminskii--Whitham averaging for randomly perturbed KdV equation

TL;DR

This paper proves that solutions of a randomly perturbed damped-driven KdV equation follow an averaged Whitham equation for small viscosity and short times, linking stochastic PDE dynamics to integrable system invariants.

Contribution

It establishes the validity of the Khasminskii--Whitham averaging principle for the damped-driven KdV equation with stochastic forcing.

Findings

The KdV integrals of motion approximately satisfy the averaged Whitham equations.

The result holds for small viscosity and short time scales proportional to the inverse of viscosity.

Provides a rigorous connection between stochastic PDE solutions and integrable system averaging.

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,...) $ 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)),...) $ satisfies the (Whitham) averaged equation.