Damped-Driven KdV and Effective Equations for Long-Time Behaviour of its Solutions

Abstract

For the damped-driven KdV equation $$ \dot{u}-{\nu}u_{xx} + u_{xxx} - 6uu_{x} = \sqrt{\nu}\,\eta(t, x), x \in S^1, \int udx \equiv \int \eta dx \equiv 0, $$ with 0 < ν ≤ 1 and smooth in x white in t random force η, we study the limiting long-time behaviour of the KdV integrals of motions (I1, I2, . . . ), evaluated along a solution uν(t, x), as ν → 0. We prove that for \({0 \leq \tau := {\nu}t \lesssim 1}\) the vector Iν(τ) = (I1(uν(τ, ·)), I2(uν(τ, ·)), . . . ), converges in distribution to a limiting process \({I^{0}(\tau) = (I_{1}^{0}, I_{2}^{0}, \ldots )}\). The j-th component \({I_{j}^{0}}\) equals \({\frac{1}{2}(v_{j}(\tau)^2 + v_{-j}(\tau)^{2})}\), where the vector v(τ) = (v1(τ), v−1(τ), v2(τ), . . . ) is a solution of a system of effective equations for the damped-driven KdV. These new equations are a quasilinear stochastic heat equation with a non-local nonlinearity, written in the Fourier coefficients. They are well posed.