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

TL;DR

This paper studies the parametrically driven, damped nonlinear Schrödinger equation with a general nonlinearity $|\psi|^{2\kappa}\psi$ ($\kappa>0$), demonstrating the existence of two exact stationary solutions $\Psi_{\pm}$ whose stability is governed by a Sturm–Liouville problem. By introducing an $\varepsilon$-parametrization, the authors derive a stability curve $r(\rho)$ that separates stable and unstable regions for each $\kappa$. They show that $\Psi_{-}$ is always unstable, while $\Psi_{+}$ can be stabilized via sufficient parametric forcing, with oscillatory instability for $\kappa<2$ and oscillatory stability for $\kappa\ge 2$ when $\varepsilon$ crosses a critical value. Numerical simulations reveal a two-branch structure of stability curves for $\kappa\ge 2$, where the soliton becomes stable between branches, indicating that the parametric drive can stabilize high-nonlinearity solitons but breaks Galilean invariance. These results advance understanding of how damping and parametric forcing control soliton stability in nonintegrable nonlinear wave equations, with potential implications for breather-like states in related Klein–Gordon systems.

Abstract

We found two stationary solutions of the parametrically driven, damped nonlinear Schrödinger equation with nonlinear term proportional to $|ψ(x,t)|^{2 κ} ψ(x,t)$ for positive values of $κ$. 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 $κ$. 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 $κ<2$ , an {\it oscillatory instability} is predicted analytically and confirmed numerically. Our principal result establishes that for $κ\ge 2$, there exists a critical value of $\varepsilon$ beyond which the unstable soliton becomes stable, exhibiting {\it oscillatory stability}.

Figures (5)

• Figure 1: Critical value of $\tilde{\varepsilon}_c$ as a function of $\kappa$ is depicted with a red dashed line. In the region above this curve, the necessary condition for stability $|\Lambda_i|>|\Lambda_r|$ holds. Black solid line (overimposing with red dashed line for $\kappa<2$) represents the values of $\tilde{\varepsilon}$ versus $\kappa$ for which a quadruplet emerges. The dotted region is stable since $\Lambda_r=0$. In the shadow region the condition $|\Lambda_i|<|\Lambda_r|$ is satisfied. In the dot-dashed region the necessary condition for stability \ref{['stability_condition']} is not satisfied and the solution is unstable. Parameters: $L=50$ and $N=501$.
• Figure 2: Crucial eigenvalues illustrating the emergence of complex quadruplet as $\varepsilon$ increases. Left-hand side panel: $\kappa=0.5$ ($\tilde{\varepsilon}_c=0.273$). Right-hand side panel: $\kappa=1.5$ ($\tilde{\varepsilon}_c=0.0268$). Circles: several eigenvalues of the continuous spectrum. Squares: discrete eigenvalues. The filled squares represent two very close eigenvalues near zero. Parameters: $L=50$ and $N=1001$.
• Figure 3: Upper panel: The stability curves, traveled from the lower to the upper boundaries as $\varepsilon$ increases ($\tilde{\varepsilon}_c \le \varepsilon \le 1$), are depicted for several values of $\kappa<2$: a red dotted line for $\kappa=0.5$, a black solid line for $\kappa=1$, a blue dashed line for $\kappa=1.5$, and a dot-dashed black line for $\kappa=1.75$. These curves are bounded below by $r=\rho$ and above by $r= \sqrt{1+\rho^2}$, both represented by black solid lines. For each value of $\kappa$, the soliton is stable when the point $(\rho,r)$ lie on the region to the right-hand side of the corresponding curve. Otherwise, it is deemed unstable. Lower panel: For $\kappa=1.5$ and $\kappa=1.75$, it is represented $r(\rho)-\rho$ versus $\rho$, where $r(\rho)$ is the stability curve. Parameters: $L=50$ and $N=501$.
• Figure 4: For $\kappa=2.5$, the real and imaginary parts of the eigenvalues which determine the soliton stability are represented as a function of $\varepsilon$. Parameters: $L=50$ and $N=501$.
• Figure 5: The stability curves are depicted with blue dashed line for $\kappa=2$ (upper panel) and $\kappa=2.5$ (lower panel). The curves are bounded below by $r=\rho$ and above by $r= \sqrt{1+\rho^2}$, both represented by black solid lines. The arrows indicate the direction of movement as $\epsilon$ increases: from A to C through B in the upper panel, and from D to F through E in the lower panel. For each value of $\kappa$, the soliton is stable when the point $(\rho,r)$ lies in the region to the right-hand side of the corresponding traveled curve. Otherwise, it is deemed unstable. Parameters: $L=50$ and $N=501$.