Critical velocity in kink solutions of the sine-Gordon equation

The goal of this work is to present a way to deduce the value of the critical velocity observed in soliton-like solutions of a perturbed version of the sine-Gordon equation. To do so, an ODE system is obtained from the perturbed sine-Gordon equation using a variational approach; the resulting Hamiltonian system is then studied. From that, a Melnikov integral formula for the critical velocity is deduced via an energy balance reasoning. Finally, the problem is approached from a geometrical point of view that allows for an interpretation of the previous results in terms of intersections of invariant manifolds of periodic orbits