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It is shown that for an interesting class of perturbation functions, at most one of the continuum of sine-Gordon breathers can persist for the perturbed equation. This question is much more subtle than the question of persistence of large portions of the family, because analytic continuation arguments in the amplitude parameter are no longer available. Instead, an asymptotic analysis of the obstructions to persistence for large Fourier orders is made, and it is connected to the asymptotic behaviour of the Taylor coecients of the perturbation function by means of an inverse Laplace transform and an integral transform whose kernel involves hypergeo- metric functions in a way that is degenerate in that asymptotic analysis in- volves a splitting monkey saddle. Only rst order perturbation theory enters into the argument. The reason- ing can in principle be carried over to other perturbation functions than the ones considered here.

A hypergeometric function approach to the persistence problem of single sine-Gordon breathers