Complex soliton wave patterns of Gross–Pitaevskii systems: application in quantum and optical engineering

The purpose of this work is to explore precise solutions, particularly soliton solutions, by fractionally analyzing the multicomponent Gross-Pitaevskii problem, a basic nonlinear Schrödinger equation. Soliton solutions are essential for comprehending the complex system dynamics, providing insight into superfluidity, superconductivity, and related nonlinear effects. The complex fractional Gross-Pitaevskii equation is solved by considering the β-derivative. Numerous optical solutions, including trigonometric, hyperbolic, rational function, and complex multiple soliton structures, are achieved by using elegant integration methods. All reported solutions are verified using the Wolfram Mathematica program by putting back-substitution into the governing equation. The novel solutions are acquired, capitalizing on the new extended hyperbolic function method (EHFM) and the unified method that holds significant implications across various scientific disciplines. Furthermore, we portray various wave profiles with the corresponding parametric values under the influence of β-derivative. This visualization style enhances our understanding of the acquired solutions and facilitates a thorough examination of their potential practical applications. By engaging the aforementioned cutting-edge methods, we obtain an effective framework for analyzing the complex nonlinear phenomena that emerge in several physical contexts.