  We consider periodic traveling wave solutions to the focusing nonlinear
Schrodinger equation (NLS) that have been shown to persist when the NLS is
perturbed to the complex Ginzburg-Landau equation (CGL). In particular, we show
that these periodic traveling waves are spectrally stable solutions of NLS with
respect to periodic perturbations. Furthermore, we use an argument based on the
Fredholm alternative to find an instability criterion for the persisting
solutions to CGL.
