NONPERSISTENCE OF BREATHER FAMILIES FOR THE PERTURBED SINE-GORDON EQUATION

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 = epsilonDELTA(u) + O(epsilon2) where DELTA 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 A 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 epsilon.