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

Abstract

The paper deals with the Fermi--Pasta--Ulam type systems that describe an infinite systems of nonlinearly  coupled particles with nonlocal interaction on a two dimensional lattice. It is assumed that each particle interacts nonlinearly with several neighbors horizontally and vertically on both sides. The main result concerns the existence of traveling waves solutions with periodic relative displacement profiles. We obtain sufficient conditions for the existence of such solutions with the aid of critical point method and a suitable version of the Mountain Pass Theorem for functionals satisfying the Cerami condition instead of the Palais--Smale condition. We prove that under natural assumptions there exist monotone traveling waves.