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

Date:
18.05.15   
Times:
15:30
Place:
UAB - Dept. Matemàtiques (C1/-128)
Speaker:
Anna Geyer
University:
UAB

Abstract:

We are 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.

The talk is based on a joint paper with Jordi Villadelprat (URV).