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

2010/02/05 by Sergei B. Kuksin, Kuksin, Sergei B. · 1 citation
Mathematics · Physics and Astronomy · #35R60 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP #msc:35R60

paper · pdf · doi:10.48550/arxiv.1002.1294

arxiv created 2010/02/05 · arxiv updated 2010/02/26

Abstract

For the damped-driven KdV equation u-νu_xx+u_xxx-6uu_x=√νη(t,x), x∈ S¹, ∫ u dx≡ ∫ηdx≡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 u^ν(t,x), as ν→0. We prove that %if u=u^ν(t,x) is a solution of the equation above, for 0≤τ:= νt \lesssim1 the vector I^ν(τ)=(I₁(u^ν(τ,⋅)),I₂(u^ν(τ,⋅)),...), converges in distribution to a limiting process I⁰(τ)=(I⁰₁,I⁰₂,...). The j-th component I_j⁰ equals \12(v_j(τ)²+v_-j(τ)²), where v(τ)=(v₁(τ), v_-1(τ),v₂(τ),...) 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.

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