Small amplitude traveling waves in the full-dispersion Whitham equation

TL;DR

This paper introduces a new Fourier analysis-based method to construct and analyze small amplitude traveling waves in the full-dispersion Whitham equation, including stability considerations.

Contribution

It provides an alternative approach to construct small amplitude waves and proves their rigorous stability, extending understanding of Whitham type equations.

Findings

Successful construction of small amplitude traveling waves

Rigorous proof of wave stability

Applicable to both periodic and whole line cases

Abstract

In this article, we provide an alternative way to construct small amplitude traveling waves for general Whitham type equations, in both periodic and whole line contexts. More specifically, Fourier analysis techniques allow us to reformulate the problem to the study of waves that are small and regular perturbations of well-understood ODE's. In addition, rigorous stability of these waves is established.