Perturbation theory for kinds

In this paper we prove the validity of formal asymptotic results on perturbation theory for kink solutions of the sine-Gordon equation, originally obtained by McLaughlin and Scott. We prove that for appropriate perturbations, of size β in an appropriate norm, slowly varying in time in the rest frame of the kink, the shape of the kink is unaltered in the L°° norm to 0(e) for a time of 0 ί - 1 . The kink parameters, which represent its velocity and centre, evolve slowly in time in the way predicted by the asymptotics. The method of proof uses an orthogonal decomposition of the solution into an oscillatory part and a one-dimensiona l "zero-mode" term. The slow evolution of the kink parameters is chosen so as to suppress secular evolution of the zero-mode.