Damped-driven KdV and effective equation for long-time behaviour of its solutions

TL;DR

This paper investigates the long-time behavior of solutions to the damped-driven KdV equation with stochastic forcing, showing convergence of integrals of motion to a process described by an effective stochastic heat equation.

Contribution

It introduces a new effective quasilinear stochastic heat equation in Fourier space that accurately describes the long-time dynamics of the damped-driven KdV solutions as viscosity vanishes.

Findings

Convergence of KdV integrals of motion to a limiting process.

Derivation of a well-posed effective stochastic heat equation.

Identification of the limiting behavior as viscosity tends to zero.

Abstract

For the damped-driven KdV equation $$ \dot u-\nu{u_{xx}}+u_{xxx}-6uu_x=\sqrt\nu \eta(t,x), x\in S^1, \int u dx\equiv \int\eta dx\equiv0, $$ with $0<\nu\le1$ and smooth in $x$ white in $t$ random force $\eta$, we study the limiting long-time behaviour of the KdV integrals of motions $(I_1,I_2,...)$, evaluated along a solution $u^\nu(t,x)$, as $\nu\to0$. We prove that %if $u=u^\nu(t,x)$ is a solution of the equation above, for $0\le\tau:= \nu t \lesssim1$ the vector $ I^\nu(\tau)=(I_1(u^\nu(\tau,\cdot)),I_2(u^\nu(\tau,\cdot)),...), $ converges in distribution to a limiting process $I^0(\tau)=(I^0_1,I^0_2,...)$. The $j$-th component $I_j^0$ equals $\12(v_j(\tau)^2+v_{-j}(\tau)^2)$, where $v(\tau)=(v_1(\tau), v_{-1}(\tau),v_2(\tau),...)$ is the vector of Fourier coefficients of a solution of an {\it effective equation} for the dam-ped-driven KdV. This new equation is a quasilinear stochastic heat equation with a non-local nonlinearity, written in the Fourier coefficients. It is well posed.