Critical velocity in kink-defect interaction models: rigorous results

Abstract

In this work we study a model of interaction of kinks of the sine-Gordon equation with a weak defect. We obtain rigorous results concerning the so-called critical velocity derived in [7] by a geometric approach. More specifically, we prove that a heteroclinic orbit in the energy level 0 of a 2-dof Hamiltonian is destroyed giving rise to heteroclinic connections between certain elements (at infinity) for exponentially small (in e) energy levels. In this setting Melnikov theory does not apply because there are exponentially small phenomena.