Stability of cnoidal waves in the parametrically driven nonlinear Schrödinger equation

Details Stability of cnoidal waves in the parametrically driven nonlinear Schrödinger equation Stability of cnoidal waves in the parametrically driven nonlinear Schrödinger equation

2011-12-31

arxiv.org/abs/1201.0263v1

Nonlinear Sciences

Pattern Formation and Solitons

14 pages, 6 figures

Scientific paper

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

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