Stationary breather modes of generalized nonlinear Klein-Gordon lattices

This paper studies the form of stationary breather modes in discrete generalized nonlinear Klein-Gordon equations, with symmetric and non-symmetric potential energy functions. In the case of static breathers, the discrete nature of the spatial dimension has a much more subtle effect on the breather than the moving breather mode. This effect is analysed using a variety of approximating partial differential equations, whose solutions are found by using an extended multiple-scales asymptotic approach to reduce the equation to a nonlinear Schrödinger equation at leading order and more complex equations at higher order, where secularity conditions are required to fully specify the solution. As well as the much studied discrete sine-Gordon equation, the methods are demonstrated on a discrete nonlinear Klein-Gordon equation with second-neighbour interactions and non-symmetric on-site potential. New partial differential equations which approximate these lattice systems are also proposed and analysed.