Seminario de Matemática aplicada ICMAT-UAM

Ponente: Tere M-Seara (Universitat Politècnica de Catalunya)

Título: BREAKDOWN OF SMALL AMPLITUDE BREATHERS FOR THE REVERSIBLE
KLEIN-GORDON EQUATION

DATE: Tuesday, January 21, 2020 - 15:00
VENUE: Aula Naranja, ICMAT

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.    Breathers    can    be
interpreted    as    homoclinic solutions to a steady solution in an
infinite dimensional space. In this talk, we prove 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.This
formula  allows  to  say  that  for  a  wide  set  of  Klein-Gordon
equations breathers do not exist.The distance is exponentially small
with respect to the amplitude of the breather and therefore classical
perturbative techniques cannot be applied.