title: On the wave length of smooth periodic traveling waves of the Camassa-Holm equation.
abstract: This paper is concerned with the wave lengthλof smooth periodic traveling wave solutions of the Camassa-Holm equation. The set of these solutions can be parametrized using the wave heighta(or "peak-to-peak amplitude"). Our main result establishes monotonicity properties of the map [Formula: see text], i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated with the equation, which distinguish between the two possible qualitative behaviors of [Formula: see text], namely monotonicity and unimodality. The key point is to relate [Formula: see text] 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.
authors.affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, Spain.
authors.forename: A
authors.initials: A
authors.lastname: Geyer
authors.affiliation: Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Tarragona, Spain.
authors.forename: J
authors.initials: J
authors.lastname: Villadelprat
keywords: 34C25; 35Q35; Camassa–Holm equation; Center; Critical period; Traveling wave solution; Wave height; Wave length