Symmetric waves are traveling waves

TL;DR

The paper proves that symmetric water waves are necessarily traveling waves for the Euler equations and related models, providing a general principle and detailed analysis for specific equations like Camassa-Holm.

Contribution

It introduces a general principle linking symmetry to traveling wave solutions and applies it to various nonlinear PDEs including water wave models.

Findings

Symmetric water waves are traveling waves for Euler equations.

Detailed analysis of weak solutions of the Camassa-Holm equation.

Existence of nonsymmetric linear rotational waves for Euler equations.

Abstract

We show that horizontally symmetric water waves are traveling waves. The result is valid for the Euler equations, and is based on a general principle that applies to a large class of nonlinear partial differential equations, including some of the most famous model equations for water waves. A detailed analysis is given for weak solutions of the Camassa-Holm equation. In addition, we establish the existence of nonsymmetric linear rotational waves for the Euler equations.