Necessary conditions for breathers on continuous media to approximate breathers on discrete lattices

Smith, Warren; Wattis, Jonathan

Abstract

We start by considering the sine-Gordon partial differential equation (PDE with an arbitrary perturbation. Using the method of Kuzmak-Luke, we investigate those conditions the perturbation must satisfy in order for a breather solution to be a valid leading-order asymptotic approximation to the perturbed problem. We analyse the cases of both stationary and moving breathers. As examples, we consider perturbing terms which include typical linear damping, periodic sinusoidal driving, and dispersion caused by higher order spatial derivatives. The motivation for this study is that the mathematical modelling of physical systems, often leads to the discrete sine-Gordon system of ODEs which are then approximated in the long wavelength limit by the continuous sine-Gordon PDE. Such limits typically produce fourth-order spatial derivatives as higher order correction terms. The new results show that the stationary breather solution is a consistent solution of both the quasi-continuum SG equation and the forced/damped SG system. However, the moving breather is only a consistent solution of the quasi- continuum SG equation and not the damped SG system.

Citation

Smith, W., & Wattis, J. (2015). Necessary conditions for breathers on continuous media to approximate breathers on discrete lattices. European Journal of Applied Mathematics, 27(1), https://doi.org/10.1017/S0956792515000273

• Journal Article Type: Article
• Acceptance Date: May 29, 2015
• Publication Date: Jun 23, 2015
• Deposit Date: Jul 6, 2017
• Publicly Available Date: Jul 6, 2017
• Journal: European Journal of Applied Mathematics
• Print ISSN: 0956-7925
• Electronic ISSN: 1469-4425
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 27
• Issue: 1
• DOI: https://doi.org/10.1017/S0956792515000273
• Keywords: singular perturbations, breathers, sine-Gordon equation
• Public URL: https://nottingham-repository.worktribe.com/output/753896
• Publisher URL: https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/article/necessary-conditions-for-breathers-on-continuous-media-to-approximate-breathers-on-discrete-lattices/7F081C2CA5B5527DC222D7BC065EECB2
• Contract Date: Jul 6, 2017