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

The sine-Gordon partial differential equation (PDE) with an arbitrary perturbation is initially considered. Using the method of Kuzmak-Luke, we investigate the conditions, which the perturbation must satisfy, 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. The motivation for this study is that the mathematical modelling of physical systems often leads to the discrete sine-Gordon system of ordinary differential equations, which are then approximated in the long wavelength limit by the continuous sine-Gordon PDE. Such limits typically produce fourth-order spatial derivatives as correction terms. The new results show that the stationary breather solution {\em is} a consistent solution of both the quasi-continuum sine-Gordon equation and the forced/damped sine-Gordon system. However, the moving breather is only a consistent solution of the quasi-continuum sine-Gordon equation and not the damped sine-Gordon system.