Chaos-driven stability and decay of dark solitons in gain-assisted nonlinear systems

We present a detailed stability analysis of dark solitons in a gain-assisted nonlinear quantum system, with particular relevance to Bose–Einstein condensates. By systematically varying the dark-soliton phase parameter from cos(ϕ=π/5) to cos(ϕ=π/11), we investigate the instability growth rate, Im[χ], across normalized time and frequency domains. Three-dimensional surface plots, density maps, and contour representations reveal a pronounced periodic washboard structure, identifying resonant parameter windows where dark solitons undergo dynamical instability. The results demonstrate strong tunability of soliton robustness, with instability growth rates ranging from near-zero values (∼0.01%, indicating long-lived localization) to extreme resonant amplification (∼100%, corresponding to complete soliton breakdown). Complementary spherical visualizations capture the spatial manifestation of the snake instability, illustrating how initially planar dark solitons deform and evolve into higher-dimensional coherent structures such as vortex rings. These findings establish phase and field control as effective mechanisms for steering dark solitons between stable localization and controlled decay, providing valuable insight into nonlinear wave dynamics in driven-dissipative quantum systems.