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

Smith, Warren; Wattis, Jonathan

Authors

Warren Smith



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.

Journal Article Type Article
Publication Date Jun 23, 2015
Journal European Journal of Applied Mathematics
Print ISSN 0956-7925
Electronic ISSN 1469-4425
Publisher Cambridge University Press (CUP)
Peer Reviewed Peer Reviewed
Volume 27
Issue 1
APA6 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
DOI https://doi.org/10.1017/S0956792515000273
Keywords singular perturbations, breathers, sine-Gordon equation
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
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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