The Solutions of the Space-Time Fractional Cubic Nonlinear Schrödinger Equation by Using the Unified Method

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Representing physical processes by introducing fractional derivatives in partial differential equations provides more realistic and flexible mathematical models. The solutions of nonlinear partial differential equations (NPDEs) can be derived from the solutions of the fractional nonlinear partial differential equations (FNPDEs) when the fractional derivatives go to 1 because FNPDEs are a generalization of NPDEs. Most of the exact solution methods for NPDEs based on the ansatz method can be extended easily to solve FNPDEs. In this study, we employ the unified method to obtain exact solutions in a more general form for the space-time fractional cubic nonlinear Schr¨odinger equation (stFCSE). Compared to other methods, this method not only gives more general solution forms with free parameters for the stFCSE, but also provides many novel solutions including hyperbolic, trigonometric, and rational function solutions. The solutions of the stFCSE approach the solutions of the cubic nonlinear Schr¨odinger equation when the fractional orders go to 1 for time and space. Moreover, three-dimensional graphs of some selected solutions with specific values of the parameters are presented to visualize the behavior and physical structures of the stFCSE.