Title: Small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting
Abstract: Breathers are nontrivial time-periodic and spatially localized solutions of nonlinear dispersive
partial differential equations (PDEs). Families of breathers have been found for certain integrable PDEs, such as the sine-Gordon equation in 1-dim, but are believed to be rare in non-integrable ones such as nonlinear Klein-Gordon equations. In this talk we consider small breathers for semilinear Klein-Gordon equations with analytic odd nonlinearities. A breather with small amplitude exists only when its temporal frequency is close to be resonant with the linear Klein-Gordon dispersion relation. Our main result is that, for such frequencies, we rigorously identify the leading order term in the exponentially small (with respect to the small amplitude) obstruction to the existence of small breathers in terms of the so-called Stokes constant which depends on the nonlinearity analytically, but is independent of the frequency. As a corollary it proves that, for generic analytic odd nonlinearities, there does not exist any small breather of any temporal frequency, even though this had been intuitive for any single given frequency due to the dimension counting. In particular, this gives a rigorous justification of a formal asymptotic argument by Kruskal and Segur (1987) in the analysis of small breathers. The proof is carried out in a singular perturbation setting in the spatial dynamics framework. The leading order term in the exponentially small splitting is obtained through a careful analysis of the analytic continuation of the parameterizations of two specials small solutions which decay weakly at the spatial infinities. This is a joint work with O. Gomide, M. Guardia, and T. Seara.
