Symmetry and decay of traveling wave solutions to the Whitham equation

TL;DR

This paper investigates the decay and symmetry characteristics of solitary wave solutions to a nonlocal shallow water wave model, demonstrating that supercritical solutions are symmetric and monotone, and establishing a link between symmetry and traveling waves.

Contribution

It proves symmetry and decay properties of solitary waves in the Whitham equation using a priori estimates and the method of moving planes, revealing new insights into their structure.

Findings

Supercritical solitary waves are symmetric and monotone.

A priori decay estimates are established for solutions.

A relation between symmetric and traveling wave solutions is demonstrated.

Abstract

This paper is concerned with decay and symmetry properties of solitary wave solutions to a nonlocal shallow water wave model. It is shown that all supercritical solitary wave solutions are symmetric and monotone on either side of the crest. The proof is based on a priori decay estimates and the method of moving planes. Furthermore, a close relation between symmetric and traveling wave solutions is established.