Being stable and discrete

Many discrete lattice systems possess solutions that take the form of local ized, stationary structures. In this communication we introduce the discret e version of the Evans function, an analytic function whose zeros correspon d to the eigenvalues of the linear stability problem for a spatially locali zed equilibrium solution. This function provides a convenient and useful to ol for investigating the linear eigenvalue spectrum, Notably, it allow's us to construct sufficient stability conditions and detect "internal modes" ( neutral oscillatory modes that correspond to localized oscillations about t he static structure). We illustrate with the discrete sine-Gordon equation, also known as the Frenkel-Kontorova model. A complementary approach suitab le for systems with nearest neighbour coupling and based upon techniques of linear algebra (the bisection method) is also described. (C)2000 Elsevier Science B.V. All rights reserved.