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

TL;DR

This paper investigates the stability properties of cnoidal and dnoidal wave solutions in the parametrically driven nonlinear Schrödinger equation, revealing conditions under which these solutions are unstable against various perturbations.

Contribution

It provides a comprehensive stability analysis of cnoidal and dnoidal solutions, including the effects of parametric driving, damping, and perturbation types, which was not previously detailed.

Findings

One pair of cn and dn solutions is always unstable against periodic perturbations.

The second dn-wave solution is unstable against antiperiodic perturbations in certain parameter regions.

Quasiperiodic perturbations with long wavelengths are also considered, especially near weak driving conditions.

Abstract

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.