Conformable Gross–Pitaevskii dynamical equation: Soliton solutions, energy balance, and nonlinear dynamics

This study explores the nonlinear conformable Gross–Pitaevskii equation (NL-CGPE), a fundamental model in Bose–Einstein condensate (BEC) theory that describes a single quantum state shared by ultra-cold bosonic particles. Beyond its quantum significance, the NL-CGPE also models optical soliton propagation with applications in data transfer, telecommunication networks, and long-distance optical fibers. By employing the complete discrimination system of polynomial method (CDSPM), we derive multiple families of exact analytical soliton solutions, including periodic, hyperbolic, rational, trigonometric, and Jacobi elliptic (JE) types. Furthermore, JE solutions are transformed into single-wave structures, enriching the solution landscape. To capture the system’s nonlinear dynamics, we conduct sensitivity analysis, Poincaré mapping, time-series profiling, and identify critical conditions leading to quasi-periodic behavior. An energy balance analysis (EBA) is also performed, confirming approximate periodic oscillations under energy conservation principles. In addition, machine learning regression techniques are applied to validate the equilibrium states of the model and to determine parameter thresholds consistent with dynamical analysis. Collectively, these results highlight both the diverse soliton behaviors supported by the NL-CGPE and the utility of integrating analytical, qualitative, and computational approaches to uncover its complex dynamics.