Stability of stationary gap solitary waves at periodically modulated surfaces

Nonlinear optical waveguides with periodically modulated surfaces or interf aces can support stationary localized waves, often called gap solitons, wit h frequencies lying in the stop gaps of the spectrum of linear excitations. They are solutions of evolution equations that have been derived for insta ntaneous Kerr-type, thermal (diffusive) as well as instantaneous resonant a nd nonresonant second-order nonlinearity. A numerical linear stability anal ysis is carried out for some examples of these gap solitary wave solutions based on discretization of the spatial coordinate. In addition to numerical instabilities, which are a consequence of discretization and which pose a problem to numerical integration schemes, weak physical instabilities have been found, which correspond to radiation away from the solitary wave. The growth rates are strongly dependent on the boundary conditions imposed at t he edges of the spatial domain. Growth rates and radiation frequencies have also been computed for an infinite spatial domain. The influence of the di ffusion length on the instability has been investigated.