Being stable and discrete

Many discrete lattice systems possess solutions that take the form of localized, stationary structures. In this communication we introduce the discrete version of the Evans function, an analytic function whose zeros correspond to the eigenvalues of the linear stability problem for a spatially localized equilibrium solution. This function provides a convenient and useful tool for investigating the linear eigenvalue spectrum. Notably, it allows us to construct sufficient stability conditions and detect “internal modes” (neutral oscillatory modes that correspond to localized oscillations about the static structure). We illustrate with the discrete sine-Gordon equation, also known as the Frenkel–Kontorova model. A complementary approach suitable for systems with nearest neighbour coupling and based upon techniques of linear algebra (the bisection method) is also described.