Exploring soliton solutions in nonlinear spatiotemporal fractional quantum mechanics equations: an analytical study

In this work, travelling wave solutions of a nonlinear system of fractional Schrödinger equations (FSEs) with conformable fractional derivatives are studied. We examine the fractional generalization of the Schrödinger equation, a topic of great importance in quantum physics, using the analytic approach known as the modified extended direct algebraic method. Our approach involves the use of a fractional complex transformation to produce nonlinear ordinary differential equations, which are then solved to reveal travelling wave solutions. The two- and three-dimensional graphs that provide visual representations of the system’s behaviour present a variety of wave profiles, including periodic, kink, anti-kink, shocks, lumps, and other soliton waves. The study sheds light on the dynamics of FSEs by revealing multiple families of travelling wave solutions and their complex relationships. These results provide insight into nonlinear fractional partial differential equations and a greater understanding of the dynamics of FSEs than previous attempts in the literature.