Oscillatory instability and stability of stationary solutions in the parametrically driven, damped nonlinear Schrödinger equation

We found two stationary solutions of the parametrically driven, damped nonlinear Schrödinger equation with a nonlinear term proportional to ${\left|\psi (x,t)\right|}^{2\kappa }\psi (x,t)$ for positive values of $\kappa$. By linearizing the equation around these exact solutions, we derived 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, the damping coefficient, and the 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 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 oscillatory stability.