On Longtime Dynamics of Perturbed KdV Equations

Abstract

Abstract. Consider a perturbed KdV equation:ut + uxxx − 6uux = ǫf (u(·)), x ∈ T = R/Z, 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 ǫ-quasiinvariant measure µ and satisfies some additional mild assumptions. Let u ǫ (t) be a solution. Then on time intervals of order ǫ −1 , as ǫ → 0, its actions I(u ǫ (t, ·)) can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is µ-typical. IntroductionThe KdV equation on the circle, perturbed by smoothing perturbations, was studied in [5]. There an averaging theorem that describes the long-time behavior for solutions of the perturbed KdV equation was proved. In this work, we suggest an abstract theorem which applies to a large class of ǫ-perturbed KdV equations which have ǫ-quasi-invariant measures; the latter notion is explained in the main text. We show that the systems considered in [5], satisfy this condition, and believe that it may be verified for many other perturbations of KdV. More exactly, we consider a perturbed KdV equation with zero mean-value periodic boundary condition:where ǫf is a nonlinear perturbation, specified below. For any p ∈ R we introduce the Sobolev space of real valued function on T with zero mean-value:Hereû k andû −k , k ∈ N are the Fourier coefficients of u with respect to the trigonometric base e k = √ 2 cos 2πkx, k > 0 and e k = √ 2 sin 2πkx, k < 0, i.e. u = k∈Nû k e k +û −k e −k . It is well known that KdV is integrable. It means that the function space H p admits analytic coordinates v = (v 1 , v 2 , . . . ) = Ψ(u(·)), where v j = (v j , v −j ) t ∈ R 2 , such that the quantities I j = 1 2 |v j | 2 and ϕ j = Arg v j , j 1, are action-angle variables for KdV. In the (I, ϕ)-varibles, KdV takes the integrable formİ = 0,φ = W (I), (0.2) 1