Existence of exponentially and superexponentially spatially localized breather solutions for nonlinear klein–gordon lattices in ℤd, d ≥ 1. Issue 2 (16th May 2022)

• Record Type: Journal Article
• Authors: Hennig, Dirk; Karachalios, Nikos I.
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 65:Issue 2(2022)
• Page Start: 480
• Page End: 499
• Publication Date: 2022-05-16
• Subjects: discrete Klein–Gordon -- higher-dimensional lattices -- breathers -- exponential and superexponential localization -- Schauder fixed-point theorem; 37K40 -- 37K60 -- 47H10; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091522000189
• Languages: English
• ISSNs: 0013-0915

Abstract

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.