Abstract

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.

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