The purpose of this study is to construct diverse forms of exact soliton solutions and conduct a comprehensive qualitative analysis. For this aim, we use the Gross–Pitaevskii system, which belongs to the family of nonlinear Schrödinger equations. This model is considered to be iconic and significant because it has potential applications in applied sciences, such as in physics, where it is used to exemplify quantum systems like Bose–Einstein condensates and illustrate the propagation of waves in optical fibers. Employing analytical techniques, the modified sine–cosine/sinh–cosh and extended rational sinh–Gordon expansion methods, we extract several waves from solutions in the shape of trigonometric, hyperbolic, and rational forms. To further deepen our insights related to the system’s behavior, we execute a detailed dynamical analysis, including sensitivity, bifurcation, and chaos, using the corresponding Hamiltonian structure. We also derive the instability modulation using linear stability theory. Using Mathematica, we systematically simulate and verify all constructed results and present some solutions for appropriate parameter values using 2D, 3D, and contour plots. The outcomes provide fruitful insights relevant to multiple scientific domains, including optical fiber technology, plasma, and condensed matter physics. This work contributes to the ongoing study of nonlinear models by applying novel solution techniques and offering a broader perspective on the complex behavior of such systems. The novelty of this study lies in the fact that the proposed model has not been previously explored using the aforementioned advanced methods and comprehensive dynamical analyses.
Aldwoah et al. (Wed,) studied this question.