Optical soliton dynamics of the conformable nonlinear evolution equation in Bose–Einstein condensates

In this study, we examine the dynamic behavior of optical solitons in the nonlinear conformable Gross–Pitaevskii equation within Bose–Einstein condensates, which refers to the phenomenon of many ultra-cold bosonic particles occupying a single quantum state. It is akin to the Ginzburg–Landau equation and resembles the nonlinear Schrödinger equation. We also discuss how fractional-order parameters affect the solution dynamics. In fiber optics, this model describes how light propagates in optical fibers when ultra-short pulses are generated. We characterize the model using the conformable time-fractional derivative operator to provide a more detailed description of the physical component. To extract exact solutions of the model, we employ the unified solver method, new Kudryashov method and \(\left( \frac{1}{\varphi (\varrho )},\frac{\varphi ^{\prime }(\varrho )}{\varphi (\varrho )}\right)\) $\left(\frac{1}{\phi (\varrho )},,,\frac{{\phi }^{\prime }(\varrho )}{\phi (\varrho )}\right)$ method. These methods for resolving nonlinear partial differential equations that arise in the natural sciences are intuitive, sturdy, and robust. Through consideration of the arising constraints over the parameters of the methods and nonlinear fractional Gross–Pitaevskii equation, a novel entire family of optical solitons are determined in the form of hyperbolic, trigonometric and rational function solutions. These solutions comprehend dark, bright, singular, and periodic-singular solitons. Visual representations of the derived solutions are provided through two-dimensional (2D) and three-dimensional (3D) graphs.