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GDMSeminar: Abstracts of the Talks (2021/2)


Tere M-Seara (UPC Barcelona)
Non existence of small amplitude  breathers for Klein-Gordon equations

Tuesday, March 2, 2021, 16:00 GMT

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 be interpreted as homoclinic solutions to a steady solution if one exchanges the roles of time and position.

In this talk, I will explain how to show that, under generic assumptions, 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 it has weakly hyperbolic one dimensional stable and unstable manifolds. This formula shows that, generically, these manifolds do not coincide.
We also show the existence of generalized breathers with exponentially small tails, which correspond intersections between the cener-stable and the center-unstable manifolds.
This is a joint work with O. Gomide (Universidade Federal de Goiás), M. Guardia (U. Politecnica de Catalunya) and C. Zeng (Georgiatech I.)