Existence of exponentially spatially localised breather solutions for lattices of nonlinearly coupled particles: Schauder's fixed point theorem approach
arXiv:2105.00745 · doi:10.1063/5.0058381
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.
arXiv comment12 pages. To appear in Journal of Mathematical Physics
References in corpus (14)
- Thermal conduction in classical low-dimensional lattices
- The Fermi-Pasta-Ulam problem: 50 years of progress
- Excitation Thresholds for Nonlinear Localized Modes on Lattices
- Localization and Equipartition of Energy in the beta-FPU Chain : Chaotic Breathers
- Fermi, Pasta, Ulam and a mysterious lady
- Gap Solitons in Periodic Discrete Nonlinear Schroedinger Equations
- The Fermi-Pasta-Ulam "numerical experiment": history and pedagogical perspectives
- On the stability of multibreathers in Klein-Gordon chains
- Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations
- Asymptotic stability of breathers in some Hamiltonian networks of weakly coupled oscillators
- Existence and continuous approximation of small amplitude breathers in 1D and 2D Klein--Gordon lattices
- Breathers for the Discrete Nonlinear Schrödinger equation with nonlinear hopping
- Traveling wave solutions for the FPU chain: a constructive approach
- Lower and upper estimates on the excitation threshold for breathers in DNLS lattices