Subsonic periodic traveling waves in Fermi–Pasta–Ulam type systems with nonlocal interaction on 2d-lattice

Abstract

The paper is devoted to Fermi--Pasta--Ulam type system that describe an infinite system of nonlinearly coupled particles with nonlocal interaction on a two dimensional integer-valued lattice. It is assumed that each particle interacts nonlinearly with several neighbors horizontally and vertically on both sides. 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 traveling waves solutions with periodic velocity profiles. Note that the profiles of such waves are not necessarily periodic. The problem of the existence of such solutions is reduced to a variational problem for the action functionals. We obtain sufficient conditions for the existence of such solutions with the aid of the critical point method and the Linking Theorem for functionals satisfying the Palais--Smale condition and possessing linking geometry. We prove that under natural assumptions there exist subsonic traveling waves. While in our previous paper, the existence of supersonic periodic traveling waves in this system was established using variational techniques and a corresponding version of the Mountain Pass Theorem for action functionals that satisfy the Cerami condition instead of the Palais--Smale condition.