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arXiv · 2105.00745

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

Dirk Hennig·Nikos I. Karachalios
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Abstract

The problem of showing the existence of localised modes in nonlinear lattices has attracted considerable efforts from the physical but also from the mathematical viewpoint where a rich variety of methods has 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 establish concisely not only the existence of localised structures but also a required rate of spatial localisation. As a case study we consider lattices of coupled particles with nonlinear nearest neighbour interaction and prove the existence of exponentially spatially localised breathers exhibiting either even-parity or odd-parity symmetry under necessary non-resonant conditions accompanied with the proof of energy bounds of the solutions.

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Dirk Hennig, Nikos I. Karachalios. 2021-12-07. Existence of exponentially spatially localised breather solutions for lattices of nonlinearly coupled particles: Schauder's fixed point theorem approach. https://doi.org/10.1063/5.0058381

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