Existence of exponentially spatially localized breather solutions for lattices of nonlinearly coupled particles: Schauder’s fixed point theorem approach

The problem of showing the existence of localized modes in nonlinear lattices has attracted considerable efforts not only from the physical but also from the mathematical viewpoint where a rich variety of methods have been employed. In this paper, we prove that a fixed point theory approach based on the celebrated Schauder’s fixed point theorem may provide a general method to concisely establish not only the existence of localized structures but also a required rate of spatial localization. As a case study, we consider lattices of coupled particles with a nonlinear nearest neighbor interaction and prove the existence of exponentially spatially localized breathers exhibiting either even-parity or odd-parity symmetry under necessary non-resonant conditions accompanied with the proof of energy bounds of solutions.