On the wave length of smooth periodic traveling waves of the Camassa-Holm equation

TL;DR

This paper investigates how the wavelength of smooth periodic traveling waves in the Camassa-Holm equation depends on wave height, revealing conditions for monotonicity and unimodality through bifurcation analysis.

Contribution

It establishes the monotonicity properties of the wavelength as a function of wave height and identifies bifurcation values that determine qualitative behavior.

Findings

Derived explicit bifurcation values for wave length behavior.

Linked wavelength properties to period functions of a planar differential system.

Identified conditions for monotonicity and unimodality of wave length.

Abstract

This paper is concerned with the wave length $\lambda$ of smooth periodic traveling wave solutions of the Camassa-Holm equation. The set of these solutions can be parametrized using the wave height $a$ (or "peak-to-peak amplitude"). Our main result establishes monotonicity properties of the map $a\longmapsto \lambda(a)$, i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated to the equation, which distinguish between the two possible qualitative behaviours of $\lambda(a)$, namely monotonicity and unimodality. The key point is to relate $\lambda(a)$ to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems.