Nearly linear dynamics of nonlinear dispersive waves

TL;DR

This paper demonstrates that solutions to the KdV equation with periodic boundary conditions behave nearly linearly at high frequencies, supported by numerical simulations, and applies this to shallow water wave dynamics.

Contribution

It introduces a novel analysis showing high frequency solutions of the KdV equation are nearly linear, combining dispersive averaging with numerical validation and applications to water waves.

Findings

High frequency solutions of KdV are nearly linear

Numerical simulations confirm the approximation accuracy

Application to shallow water wave dynamics in the KdV limit

Abstract

Dispersive averaging effects are used to show that KdV equation with periodic boundary conditions possesses high frequency solutions which behave nearly linearly. Numerical simulations are presented which indicate high accuracy of this approximation. Furthermore, this result is applied to shallow water wave dynamics in the limit of KdV approximation, which is obtained by asymptotic analysis in combination with numerical simulations of KdV.