Necessary conditions for breathers on continuous media to approximate breathers on discrete latticesTools Smith, Warren and Wattis, Jonathan (2015) Necessary conditions for breathers on continuous media to approximate breathers on discrete lattices. European Journal of Applied Mathematics, 27 (1). pp. 2341. ISSN 14694425
AbstractWe start by considering the sineGordon partial differential equation (PDE with an arbitrary perturbation. Using the method of KuzmakLuke, we investigate those conditions the perturbation must satisfy in order for a breather solution to be a valid leadingorder 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 sineGordon system of ODEs which are then approximated in the long wavelength limit by the continuous sineGordon PDE. Such limits typically produce fourthorder spatial derivatives as higher order correction terms. The new results show that the stationary breather solution is a consistent solution of both the quasicontinuum 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.
Actions (Archive Staff Only)
