Breathers are nontrivial time-periodic spatially localized solutions of nonlinear dispersive PDEs. They have been found for certain integrable PDEs, such as the 1D sine-Gordon equation, but are believed to be rare in non-integrable ones. 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 Stokes constant which depends on the nonlinearity analytically, but is independent of the frequency.This rigorously justifies the formal asymptotics by Kruskal and Segur (1987) in the analysis of small breathers. It is a joint work with O. Gomide, M. Guardia, and T. Seara.

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