Homoclinic orbits in the near-integrable double discrete sine-Gordon equation

Abstract

We establish the existence of homoclinic orbits for the near-integrable double discrete sine-Gordon (dDSG) equation under periodic boundary conditions. The hyperbolic structure and homoclinic orbits are constructed through the Bäcklund transformation and Lax pair. A geometric perturbation method based on Mel'nikov analysis is used to establish necessary criteria for the persistent of temporally homoclinic orbits for the class of dDSG equations with dissipative perturbations.