On Longtime Dynamics of Perturbed KdV Equations

Abstract

Consider a perturbed KdV equation:

$$u_t+u_{xxx}-6uu_x=\epsilon f(u)(x),\quad x\in\mathbb{T}=\mathbb{R}/\mathbb{Z},\;\int_{\mathbb{T}}u(x,t)dx=0, $$

where the nonlinear perturbation defines analytic operators u(⋅)↦f(u(⋅)) in sufficiently smooth Sobolev spaces. Assume that the equation has an 𝜖-quasi-invariant measure μ and satisfies some additional mild assumptions. Let u ^𝜖(t) be a solution. Then on time intervals of order 𝜖 ⁻¹, as 𝜖→0, its actions I(u ^𝜖(t,⋅)) can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is μ-typical.