Asymptotic analysis of breather modes in a two-dimensional mechanical lattice

Wattis, Jonathan; Alzaidi, Ahmed

Abstract

We consider a two-dimensional square lattice in which each node is restricted to the plane of the lattice, but is permitted to move in both directions of the lattice. We assume nodes are connected to nearest neighbours along the lattice directions with nonlinear springs, and to diagonal neighbours with linear springs. We consider a generalised Klein-Gordon system, that is, where there is an onsite potential at each node in addition to the (nonlinear) nearest-neighbour interactions. We derive the equations of motion for the displacements from the Hamiltonian. We use asymp-totic techniques to derive the form of small amplitude breather solutions, and find necessary conditions required for their existence. We find two types of mode, which we term 'optical' and 'acoustic', based on the analysis of other lattices which support dispersion relations with multiple branches. In addition to the usual inequality on the sign of the nonlinearity in order for the NLS to be of the focusing type, we obtain an additional ellipticity constraint, that is a restriction in the two-dimensional wavenumber space, required for the spatial differential operator to be elliptic. Highlights • we consider a 2D square lattice with in-plane motion of nodes • we use a weakly nonlinear asymptotic expansion to derive envelope equation • we find breather solutions of an associated 2D NLS • two conditions for breathers: usual focusing, additional ellipticity constraint

Citation

Wattis, J., & Alzaidi, A. (2020). Asymptotic analysis of breather modes in a two-dimensional mechanical lattice. Physica D: Nonlinear Phenomena, 401, Article 132207. https://doi.org/10.1016/j.physd.2019.132207

• Journal Article Type: Article
• Acceptance Date: Sep 17, 2019
• Online Publication Date: Sep 24, 2019
• Publication Date: Jan 1, 2020
• Deposit Date: Sep 20, 2019
• Publicly Available Date: Sep 25, 2020
• Journal: Physica D
• Print ISSN: 0167-2789
• Electronic ISSN: 1872-8022
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 401
• Article Number: 132207
• DOI: https://doi.org/10.1016/j.physd.2019.132207
• Keywords: breathers; nonlinear waves; discrete systems; asymptotic analysis
• Public URL: https://nottingham-repository.worktribe.com/output/2641437
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0167278919301782
• Additional Information: This article is maintained by: Elsevier; Article Title: Asymptotic analysis of breather modes in a two-dimensional mechanical lattice; Journal Title: Physica D: Nonlinear Phenomena; CrossRef DOI link to publisher maintained version: https://doi.org/10.1016/j.physd.2019.132207; Content Type: article; Copyright: © 2019 Elsevier B.V. All rights reserved.
• Contract Date: Sep 20, 2019