Traveling Waves for the Nonlinear Variational Wave Equation

TL;DR

This paper investigates traveling wave solutions of a nonlinear variational wave equation, demonstrating how to construct global weak solutions from classical ones and applying the approach to recover known solutions for the Camassa-Holm equation.

Contribution

It introduces a method to derive global weak traveling wave solutions with monotone and constant segments from local classical solutions, and applies it to the Camassa-Holm equation.

Findings

Construction of global bounded weak solutions from local classical solutions

Characterization of wave solutions with monotone and constant segments

Recovery of known traveling wave solutions for the Camassa-Holm equation

Abstract

We study traveling wave solutions of the nonlinear variational wave equation. In particular, we show how to obtain global, bounded, weak traveling wave solutions from local, classical ones. The resulting waves consist of monotone and constant segments, glued together at points where at least one one-sided derivative is unbounded. Applying the method of proof to the Camassa--Holm equation, we recover some well-known results on its traveling wave solutions.