Periodic and compacton travelling wave solutions of discrete nonlinear
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.