Wave Dynamics, Stability, and Integrability of a Generalized (3+1)‐Dimensional Breaking Soliton Equation

ABSTRACT This research investigates the complex behavior of soliton structures governed by the generalized ‐dimensional breaking soliton (gBS) equation, a critical framework for modeling nonlinear wave phenomena in fluid dynamics, plasma physics, and optical communications. Despite the importance of these higher‐dimensional models in capturing real‐world wave breaking and folded patterns, obtaining analytical solutions remains a significant challenge. To address this, we first establish the system's mathematical tractability and integrability through the Painlevé test, confirming its suitability for multi‐soliton solutions. A linear stability analysis near the trivial solution is conducted to derive a dispersion relation, revealing that stability regions are governed by an intricate interplay between system parameters and wave vectors. We then employ the generalized Riccati equation mapping method to derive new exact trigonometric, rational, and solitary wave solutions, providing a more diverse set of results than previously reported lump‐type or rogue wave studies. Furthermore, phase plane analysis is used to categorize equilibrium points and distinguish between periodic and unstable trajectories. The study's novelty is highlighted by the introduction of a perturbation‐based approach that uncovers the transition from stable periodic orbits to quasi‐periodic and chaotic dynamics, quantified by the largest Lyapunov exponent. These findings provide essential new insights into the role of nonlinearities in driving system complexity, offering a more comprehensive understanding of wave propagation in higher‐dimensional dispersive systems.