Cubic nonlinear fractional Schrödinger equation with conformable derivative and its new travelling wave solutions

In the present paper, the fractional-order cubic nonlinear Schrödinger equation is considered. The Schrödinger equation with time and space fractional derivative is studied at the same time. Different types of travelling wave solutions including the kink solution, soliton solution, periodic solution, and singular solution for the mentioned equation are obtained by using the Jacobi elliptic functions expansion method. It is shown that the solutions turn into the exact solutions when the fractional orders go to 1. This method can be relied on gaining the solutions to time or space fractional order partial differential equations as well as ordinary ones. Throughout this work, the fractional derivative is given in a conformable sense.