Ground-state periodic traveling waves for Fermi-Pasta-Ulam-type systems on a two-dimensional lattice

Abstract

The paper is devoted to the Fermi-Pasta-Ulam-type system describing an infinite system of nonlinearly coupled particles on a two-dimensional integer-valued lattice. Each particle is assumed to interact nonlinearly with its four nearest neighbors. This system forms an infinite system of ordinary differential equations and is representative of a wide class of systems called lattice dynamical systems, which have been extensively studied in recent decades. Among the solutions of such systems, traveling waves deserve special attention. The main result concerns the existence of ground-state periodic traveling wave solutions with periodic velocity profiles. It is important to note that the profiles of such waves are not necessarily periodic. The existence problem is reduced to a variational problem for an associated action functional. Using variational method, including the Mountain Pass Theorem and the Nehari manifold approach, we establish sufficient conditions for the existence of nonconstant ground-state periodic traveling waves.