Symmetric waves are traveling waves for a shallow water equation for surface waves of moderate amplitude

TL;DR

This paper proves that for a shallow water wave equation modeling moderate amplitude surface waves, all symmetric wave solutions are necessarily traveling waves, confirming a general principle for this class of equations.

Contribution

It establishes that symmetry implies traveling wave solutions for a specific shallow water wave equation, extending the understanding of wave symmetry and dynamics.

Findings

All symmetric solutions are traveling waves.

Supports the general principle relating symmetry to traveling waves.

Applicable to equations modeling moderate amplitude shallow water waves.

Abstract

Following a general principle introduced by Ehrnstr\"{o}m et.al. we prove that for an equation modeling the free surface evolution of moderate amplitude waves in shallow water, all symmetric waves are traveling waves.