High frequency perturbation of cnoidal waves in KdV

TL;DR

This paper investigates how high frequency radiation interacts with cnoidal waves in the KdV equation, showing that the interaction remains weak over finite times and the radiation behaves approximately like an Airy wave.

Contribution

It provides a rigorous analysis of the weak interaction between low and high frequency components in the KdV equation, with a focus on finite time behavior.

Findings

High frequency radiation approximately satisfies the Airy equation.

Interaction between cnoidal waves and high frequency radiation is weak over finite times.

The study offers insights into wave stability and energy transfer in KdV dynamics.

Abstract

The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. The interaction of a periodic solitary wave (cnoidal wave) with high frequency radiation of finite energy ($L^2$-norm) is studied. It is proved that the interaction of low frequency component (cnoidal wave) and high frequency radiation is weak for finite time in the following sense: the radiation approximately satisfies Airy equation.