Exploring optical solitons in the conformable nonlinear Gross–Pitaevskii equation: applications in telecommunications and Bose–Einstein condensates

The nonlinear Gross–Pitaevskii equation, in the sense of the conformable derivative, is typically derived within the framework of the second quantization formalism, which often goes beyond typical undergraduate curricula. It is a nonlinear Schrödinger equation with cubic nonlinearity and has various physical applications, such as in water waves and condensed matter physics. This equation provides an excellent description of the static and dynamic properties of a pure Bose–Einstein condensate composed of ultracold atoms. A Bose–Einstein condensate is a gas of bosons in the same quantum state, corresponding to the same wave function. The modified Sardar sub-equation method is employed to obtain a variety of solutions in the form of bright solitons, dark solitons, combo dark-bright, singular solitons, and periodic solutions. Additionally, we utilized the extended simple equation method to obtain dark, singular, and dark-singular soliton solutions. Optical solitons, fundamental building blocks of the telecommunication industry, are also modeled. Graphical visualizations of the results are illustarted by using suitable parametric values which demonstrating that the fractional order parameter controls the behavior of propagating solitary waves and provides a comparison between fractional operators and the classical derivative. Furthermore, 3-dimensional, 2-dimensional, and contour plots by using different values of constants are used to depict dynamic phenomena and interpret the physical meaning of the solutions.