Abstract:

Breathers are periodic in time spatially localized solutions of evolutionary PDEs. They are known to exist for the sine-Gordon equation but are believed to be rare in other Klein-Gordon equations. Exchanging the roles of time and position, breathers can be interpreted as homoclinic solutions to a steady solution.

In this talk, I will explain how to show that, under generic asumptions, the Klein-Gordon equation does not have breathers whose amplitude is smaller than a certain quantity. The key point is to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solution when the steady solution has weakly hyperbolic one dimensional stable and unstable manifolds. Their distance is exponentially small with respect to the amplitude of the breather and therefore classical perturbative techniques cannot be applied.

This is a joint work with O. Gomide (Universidade Federal de Goiás), M. Guardia (U. Politecnica de Catalunya) and C. Zeng (Georgiatech I.)