About the non-existence of small breathers in Klein-Gordon equations

Speaker. Tere M-Seara
Date. 27.09.22 at 3:30 pm

Abstract. Breathers are solutions of evolutionary PDEs, which are periodic in time and spatially localized. They are known to exist for the sine-Gordon equation but are believed to be rare in other Klein-Gordon equations.

When the spatial dimension is equal to one, working in the spatial dynamics framework (which consists in exchanging the roles of time and space variables), breathers can be seen as homoclinic solutions to steady states in an infinite dimensional phase space consisting of periodic in time solutions, which arise from the intersections of the stable and unstable manifolds of the steady states.

In this talk, taking the temporal frequency as a parameter, we shall study small breathers of the non-linear Klein-Gordon equation generated in an unfolding bifurcation as a pair of eigenvalues collide at the origin.

Due to the presence of the oscillatory modes, generally the finite dimensional stable and unstable manifolds do not intersect in the infinite dimensional phase space, but with an exponentially small splitting (relative to the amplitude of the breather) in this singular perturbation problem of multiple time scales.

When the steady solution has weakly hyperbolic one dimensional stable and unstable manifolds we will prove an asymptotic formula for their distance.

This formula allows to say that for a wide set of Klein-Gordon equations breathers do not exist.