An averaging theorem for a perturbed KdV equation

Abstract

We consider a perturbed KdV equation:For any periodic function u(x), let be the vector, formed by the KdV integrals of motion, calculated for the potential u(x). Assuming that the perturbation ϵf(x, u(x)) defines a smoothing mapping u(x) ↦ f(x, u(x)) (e.g. it is a smooth function ϵ f(x), independent from u), and that solutions of the perturbed equation satisfy some mild a priori assumptions, we prove that for solutions u(t, x) with typical initial data and for 0 ⩽ t ≲ ϵ−1, the vector I(u (t)) may be well approximated by a solution of the averaged equation.