Stability of waves in discrete systems

Abstract

The Evans function is a complex analytic function whose zeros correspondto eigenvalues of an associated linear operator. Over the last decade, it hasbeen shown to be a very powerful tool for examining eigenvalue problemsof linearized partial differential equations. In this work the Evansfunction is constructed for differential-difference, i.e. lattice,equations. It is then used to study the stability of waves inHamiltonian lattice models. In particular, the stability ofnonlinear waves for the discrete nonlinear Schrödingerequation, as well as for the discrete sine-Gordon equation, isestablished. The linear spectrum for each problem is identified,and terms which are exponentially small in the discretenessparameter are quantified. The analytic results are supplementedwith detailed numerical simulations.