# Stability of Periodic, Traveling-Wave Solutions to the Capillary-Whitham Equation

## TL;DR

This paper investigates the stability of periodic traveling-wave solutions to the capillary-Whitham equation, which models shallow water surface waves, by computing solutions and analyzing how parameters affect their stability.

## Contribution

It provides the first detailed analysis of the stability of these solutions across various parameters, enhancing understanding of wave dynamics in shallow water models.

## Key findings

- Solutions vary with wavelength, wave speed, and surface tension.
- Certain parameter ranges lead to stable solutions.
- Surface tension significantly influences wave stability.

## Abstract

Recently, the Whitham and capillary-Whitham equations were shown to accurately model the evolution of surface waves on shallow water. In order to gain a deeper understanding of these equations, we compute periodic, traveling-wave solutions to both and study their stability. We present plots of a representative sampling of solutions for a range of wavelengths, wave speeds, wave heights, and surface tension values. Finally, we discuss the role these parameters play in the stability of the solutions.