Stability analysis of spatiotemporal cnoidal waves in cubic nonlinear media

We analyze numerically the modulational instability of spatiotemporal cnoidal waves of cn, dn, and sn types that are periodic along a single space coordinate and are uniform in time. The band of possible increments is calculated for all three types of cnoidal waves as a function of parameter describing the degree of localization of the wave field energy. It is shown that this band transforms into a set of discrete values for waves of cn and dn types in the limit of strong spatial localization. Simulation of perturbed cnoidal-wave propagation revealed suppression of collapse and multiple-wave filamentation on the developed stage of instability. Different instability scenarios are considered in detail.