On small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting (2107.14566v2)

Abstract: Families of breathers have been found for certain integrable PDEs but are believed to be rare in non-integrable ones such as nonlinear Klein-Gordon equations. In this paper 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 {\it small} breather of {\it 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 \cite{KS87} in the analysis of small breathers. We rely on the spatial dynamics approach where breathers can be seen as homoclinic orbits. The birth of such small homoclinics is analyzed via a singular perturbation setting where a Bogdanov-Takens type bifurcation is coupled to infinitely many rapidly oscillatory directions. The leading order term of the exponentially small splitting between the stable/unstable invariant manifolds is obtained through a careful analysis of the analytic continuation of their parameterizations. This requires the study of another limit equation in the complexified evolution variable, the so-called \emph{inner equation}.