Stability of plane-wave solutions of a dissipative generalization of the nonlinear Schrِdinger equation

The modulational instability of perturbed plane-wave solutions of the cubic nonlinear Schrِdinger (NLS) equation is examined in the presence of three forms of dissipation. We present two families of decreasing-in-magnitude plane-wave solutions to this dissipative NLS equation. We establish that all such solutions that have no spatial dependence are linearly stable, though some perturbations may grow a finite amount. Further, we establish that all such solutions that have spatial dependence are linearly unstable if a certain form of dissipation is present.