Exact solutions of a coupled space-time fractional nonlinear Schrödinger type equation in quantum mechanics

In this paper, we consider a coupled space-time fractional nonlinear Schrödinger type equation which can be used for describing nonrelativistic quantum mechanical behavior. Under the help of the fractional complex transform and the conformable fractional derivative sense, we apply the extended auxiliary equation method, the modified He’s semi-inverse method and the improved Weierstrass elliptic function method to obtain many exact solutions. These exact solutions include the bright, dark, bright-dark, bright-like and kinky-bright optical soliton solutions as well as trigonometric function solutions, the Jacobi elliptic function solutions and the general solutions in form of Weierstrass elliptic function. The behaviors of some obtained solutions are displayed through the 3D graphs for different fractional orders by using Matlab. The obtained results show that these proposed methods are straightforward, efficient and can provide more forms of exact solutions, which are helpful to understand the complex physical meaning of space-time fractional coupled nonlinear Schrödinger equation.