Anna Geyer: Stability and persistence of periodic traveling waves

Submitted by Jorge Cisneros on

Speaker: Anna Geyer, Delft University of Technology

Date: May 20, 2021

Title: Stability and persistence of periodic traveling waves

Abstract: In the first part of my talk, I will present a result on the stability of smooth periodic traveling waves of the Camassa-Holm equation. This equation models the propagation of shallow water waves and has been studied extensively. The problem of spectral stability of periodic waves however was still open. I will present a novel method to study the spectral and orbital stability, which is potentially applicable to other peakon-bearing equations. The key to obtaining this result is that the periodic waves can be characterized by an alternative Hamiltonian structure, different from the standard formulation common to the Korteweg–de Vries equation, which has the disadvantage that the period function is not monotone and the quadratic energy form may have two rather than one negative eigenvalues. By  exploring the nonstandard formulation one can prove that the period function is monotone in this case, and that the quadratic energy form has only one simple negative eigenvalue. Finally, a precise condition for the spectral and orbital stability of the smooth periodic waves is deduced.

In the second part of my talk, I will focus on the problem of persistence of periodic traveling waves in Hamiltonian PDE (for instance, the Camassa-Holm equation) under perturbations. I will show that the number of  traveling waves that persist are controlled by the zeros of certain Abelian integrals. Moreover I will show that one can design the perturbations precisely so that any prescribed number of traveling waves persists. 

The first part is  joint work with Dmitry Pelinovsky and Fabio Natali, the second part with Armengol Gasull and Víctor Mañosa.