Periodic traveling wave solutions of discrete nonlinear Klein-Gordon lattices. (English) Zbl 1534.37003
Summary: We prove the existence of periodic traveling 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 nontrivial 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 traveling wave solutions is facilitated by a variational approach based on the mountain pass theorem. Thresholds on the averaged kinetic energy for these solutions to exist are also derived.
© 2023 John Wiley & Sons, Ltd.
© 2023 John Wiley & Sons, Ltd.
MSC:
| 37K60 | Lattice dynamics; integrable lattice equations |
| 37L60 | Lattice dynamics and infinite-dimensional dissipative dynamical systems |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q51 | Soliton equations |
Keywords:
fixed-point theorems; mountain pass theorem; nonlinear lattices; periodic traveling waves; variational methodsReferences:
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