Stability analysis of (1+1)-dimensional cnoidal waves in media with cubic nonlinearity

In the present paper we perform stability analysis of stationary (1+1)-dimensional cnoidal waves of cn and dn types (anomalous group velocity dispersion) and sn type (normal group velocity dispersion). The mathematical model is based on the nonlinear Schrödinger equation. With this aim we developed a method that takes into consideration the properties of complex eigenvalues of Cauchy matrix for perturbation vectors. We show that cnoidal sn-wave is stable in the whole domain of its existence, whereas cn- and dn-waves are unstable. The instability of cn- and dn-waves is suppressed in the limiting case of strong localization when waves evolve into a set of well-separated fundamental bright solitons.