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The parametrically driven, damped nonlinear Schr\"odinger equation has twocn- and two dn-wave solutions. We show that one pair of the cn and dn solutionsis unstable for any combination of the driver's strength, dissipationcoefficient and spatial period of the wave; this instability is againstperiodic perturbations. The second dn-wave solution is shown to be unstableagainst antiperiodic perturbations --- in a certain region of the parameterspace. We also consider quasiperiodic perturbations with long modulationwavelength, in the limit where the driving strength is only weakly exceedingthe damping coefficient.

Stability of cnoidal waves in the parametrically driven nonlinear Schr\"odinger equation