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. The key to obtaining the spectral stability is that the periodic waves can be characterised by an alternative Hamiltonian structure, different from the standard formulation.

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 we will see 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.