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Existence of exponentially and superexponentially spatially localized breather solutions for nonlinear klein–gordon lattices in ℤd, d ≥ 1

Part of: Infinite-dimensional Hamiltonian systems Nonlinear operators and their properties

Published online by Cambridge University Press:  16 June 2022

Dirk Hennig  and
Nikos I. Karachalios Open the ORCID record for Nikos I. Karachalios [Opens in a new window]
Dirk Hennig
Affiliation:
Department of Mathematics, University of Thessaly, Lamia GR35100, Greece ([email protected], [email protected])
Nikos I. Karachalios
Affiliation:
Department of Mathematics, University of Thessaly, Lamia GR35100, Greece ([email protected], [email protected])
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Abstract

We prove the existence of exponentially and superexponentially localized breather solutions for discrete nonlinear Klein–Gordon systems. Our approach considers $d$-dimensional infinite lattice models with general on-site potentials and interaction potentials being bounded by an arbitrary power law, as well as, systems with purely anharmonic forces, cases which are much less studied particularly in a higher-dimensional set-up. The existence problem is formulated in terms of a fixed-point equation considered in weighted sequence spaces, which is solved by means of Schauder's Fixed-Point Theorem. The proofs provide energy bounds for the solutions depending on the lattice parameters and its dimension under physically relevant non-resonance conditions.

Keywords

discrete Klein–Gordonhigher-dimensional latticesbreathersexponential and superexponential localizationSchauder fixed-point theorem

MSC classification

Primary: 37K40: Soliton theory, asymptotic behavior of solutions
Secondary: 37K60: Lattice dynamics 47H10: Fixed-point theorems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 2 , May 2022 , pp. 480 - 499
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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