Jochen Denzler: Nonpersistence of breather families for the perturbed sine Gordon equation. Comm. Math. Phys., 158(1993), 397-430.

We show that, up to one exception and as a consequence of first order perturbation theory only, it is impossible that a large portion of the well-known family of breather solutions to the sine Gordon equation could persist under any nontrivial perturbation of the form

u_{tt} − u_{xx} + sin u = εΔ(u) + O(ε^{2}) ,

where Δ is an analytic function in an arbitrarily small neighbourhood of u = 0. Improving known results, we analyze and overcome the particular difficulties that arise when one allows the domain of analyticity of Δ to be small. The single exception is a one-dimensional linear space of perturbation functions under which the full family of breathers does persist up to first order in ε