Stability of parametrically driven, damped nonlinear Dirac solitons

TL;DR

This paper analyzes the linear stability of exact stationary solutions in a parametrically driven, damped nonlinear Dirac equation, revealing conditions for stability and instability through eigenvalue analysis and simulations.

Contribution

It provides the first detailed stability diagram for these solutions, showing how dissipation and driven frequency influence stability, supported by a novel numerical algorithm.

Findings

One solution is always unstable, confirming previous variational results.

Large dissipation stabilizes the second solution.

Low-frequency solitons are stable across all parameters.

Abstract

The linear stability of two exact stationary solutions of the parametrically driven, damped nonlinear Dirac equation is investigated. Stability is ascertained through the resolution of the eigenvalue problem, which stems from the linearization of this equation around the exact solutions. On the one hand, it is proven that one of these solutions is always unstable, which confirms previous analysis based on a variational method. On the other hand, it is shown that sufficiently large dissipation guarantees the stability of the second solution. Specifically, we determine the stability curve that separates stable and unstable regions in the parameter space. The dependence of the stability diagram on the driven frequency is also studied, and it is shown that low-frequency solitons are stable across the entire parameter space. These results have been corroborated with extensive simulations of the parametrically driven and damped nonlinear Dirac equation by employing a novel and recently proposed numerical algorithm that minimizes discretization errors.