Homoclinic intersections and Mel'nikov method for perturbed sine-Gordon equation

We describe and characterize rigorously the homoclinic structure of the perturbed sine-Gordon equation under periodic boundary conditions. The existence of invariant manifolds for a perturbed sine-Gordon equation is established. The Mel'nikov method, together with geometric analysis are used to assess the persistence of the homoclinic orbits under bounded and time-periodic perturbations.