Bifurcation analysis, chaos, sensitivity, and diverse soliton solutions with propagation insights a nonlinear spatiotemporal fractional quantum mechanics system

In this study, we amalgamate the solitary wave solutions for the nonlinear fractional Schrödinger equation by using the modified Sardar sub-equation method. The significance of data pertaining to the nonlinear fractional Schrödinger equation in nonrelativistic quantum mechanics stems from these equations’ capacity to depict intricate physical phenomena beyond the scope of traditional models. By leveraging the above-mentioned computational approach, we manifested the novel soliton solutions in the form of dark, bright-dark, dark-bright, periodic, singular, rational, and exponential forms, which are not reported in previously. Compared with other recent analytical approaches, the modified Sardar sub-equation scheme employed here is more versatile, yielding a richer spectrum of soliton structures with reduced computational complexity. This comparative advantage underscores the novelty of the present work and highlights its effectiveness for nonlinear fractional models. Subsequently, the dynamical characteristics of the model are analyzed from various perspectives, such as bifurcation analysis, chaos behavior, and sensitivity. Bifurcation occurs at critical points in the dynamical system when an external force is applied, revealing the onset of chaotic behavior. This chaotic behavior is illustrated through two-dimensional plots, time series, multistability analysis, and Lyapunov exponents. Additionally, the sensitive analysis of the model is examined using the Runge–Kutta method. We show that our proposed techniques are effective in evaluating the nonlinear fractional Schrödinger equation with analytical tools and numerical simulations, providing new insights into their behavior and solutions. With ramifications for numerous areas of physics and applied mathematics, our findings contribute to the development of mathematical tools for studying nonlinear partial differential equations and offer fresh perspectives on the dynamics of nonlinear fractional Schrödinger equations.