Jochen Denzler: Second order nonpersistence of the sine Gordon breather under an exceptional perturbation. Annales de l’Institut Henri Poincaré, Analyse non linéaire, 12(1995), 201-239

We consider the only non-trvial perturbation of the sine Gordon equation of the type

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

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 ruled out 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.