Second order nonpersistence of the sine Gordon breather under an exceptional perturbation

We consider the only nontrivial perturbation of the sine Gordon equation of the type utt − uxx + sin u = eΔ(u) + O(e2) under which persistence of the unperturbed breather family cannot be ruled out by first order perturbation theory. We show that in this case, nonpersistence can be proved by second order perturbation theory. A resonant interaction of the second order perturbation function with the first order perturbation of the breathers is responsible for this phenomenon. Number theoretic techniques make the final analysis manageable.