  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 $H_\epsilon$ is destroyed giving rise to heteroclinic
connections between certain elements (at infinity) for exponentially small (in
$\epsilon$) energy levels. In this setting Melnikov theory does not apply
because there are exponentially small phenomena.
