Periodic and compacton travelling wave solutions of discrete nonlinear Klein-Gordon lattices

We prove the existence of periodic travelling wave solutions for general discrete nonlinear Klein-Gordon systems, considering both cases of hard and soft on-site potentials. In the case of hard on-site potentials we implement a fixed point theory approach, combining Schauder’s fixed point theorem and the contraction mapping principle. This approach enables us to identify a ring in the energy space for non-trivial solutions to exist, energy (norm) thresholds for their existence and upper bounds on their velocity. In the case of soft on-site potentials, the proof of existence of periodic travelling wave solutions is facilitated by a variational approach based on the Mountain Pass Theorem. The proof of the existence of travelling wave solutions satisfying Dirichlet boundary conditions establishes rigorously the presence of compactons in discrete nonlinear Klein-Gordon chains. Thresholds on the averaged kinetic energy for these solutions to exist are also derived.