Seminário de Sistemas Dinâmicos

Resumo: Breathers are nontrivial time-periodic and spatially localized solutions of evolutionary Partial Differential Equations (PDE's). It is known that the sine-Gordon equation (a special case of Klein-Gordon equation) admits an explicit family of breathers. Nevertheless this kind of solution is expected to be rare in other Klein-Gordon equations. In this work, we discuss the non-existence of small amplitude breathers for reversible Klein-Gordon equations (RKG) through a rigorous analysis. Roughly speaking, we look at the RKG as an evolutionary PDE with respect to the spatial variable in such a way that a breather becomes a homoclinic orbit to a critical point (origin). We obtain an asymptotic formula for the distance between the stable and unstable manifolds of such critical point which happens to be exponentially small with respect to the amplitude of the breather and therefore classical Melnikov Theory cannot be used. This is a joint work with M. Guardia, T. Seara and C. Zeng.