An Averaging Theorem for Perturbed KdV Equation

TL;DR

This paper proves an averaging theorem for the perturbed KdV equation, showing that the integrals of motion are well approximated by an averaged system over long times under certain conditions.

Contribution

It establishes an averaging principle for the perturbed KdV equation with smoothing perturbations, extending the understanding of long-term behavior of solutions.

Findings

The integrals of motion are approximated by an averaged equation for times up to order 1/epsilon.

The theorem applies to typical initial data and smoothing perturbations.

Provides a rigorous foundation for averaging in infinite-dimensional Hamiltonian systems.

Abstract

We consider a perturbed KdV equation: [\dot{u}+u_{xxx} - 6uu_x = \epsilon f(x,u(\cdot)), \quad x\in \mathbb{T}, \quad\int_\mathbb{T} u dx=0.] For any periodic function $u(x)$, let $I(u)=(I_1(u),I_2(u),...)\in\mathbb{R}_+^{\infty}$ be the vector, formed by the KdV integrals of motion, calculated for the potential $u(x)$. Assuming that the perturbation $\epsilon f(x,u(\cdot))$ is a smoothing mapping (e.g. it is a smooth function $\epsilon f(x)$, 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 $0\leqslant t\lesssim \epsilon^{-1}$, the vector $I(u(t))$ may be well approximated by a solution of the averaged equation.