Oscillatory instability and stability of stationary solutions in the parametrically driven, damped nonlinear Schr\"odinger equation

TL;DR

This paper analyzes the stability of stationary solutions in a parametrically driven, damped nonlinear Schrödinger equation, revealing conditions for oscillatory instability and stability depending on parameters like nonlinearity and forcing amplitude.

Contribution

It introduces a detailed stability analysis of stationary solutions in the nonlinear Schrödinger equation with parametric driving and damping, including the derivation of a stability diagram and identification of oscillatory stability regimes.

Findings

One stationary solution is unstable, the other's stability depends on parameters.

For < ext{kappa}<2, oscillatory instability occurs.

For   stability can be achieved beyond a critical parameter value.

Abstract

We found two stationary solutions of the parametrically driven, damped nonlinear Schr\"odinger equation with nonlinear term proportional to $|\psi(x,t)|^{2 \kappa} \psi(x,t)$ for positive values of $\kappa$. By linearizing the equation around these exact solutions, we derive the corresponding Sturm-Liouville problem. Our analysis reveals that one of the stationary solutions is unstable, while the stability of the other solution depends on the amplitude of the parametric force, damping coefficient, and nonlinearity parameter $\kappa$. An exceptional change of variables facilitates the computation of the stability diagram through numerical solutions of the eigenvalue problem as a specific parameter $\varepsilon$ varies within a bounded interval. For $\kappa <2$ , an {\it oscillatory instability} is predicted analytically and confirmed numerically. Our principal result establishes that for $\kappa \ge 2$, there exists a critical value of $\varepsilon$ beyond which the unstable soliton becomes stable, exhibiting {\it oscillatory stability}.