The analytic solutions of Schrödinger equation with Cubic-Quintic nonlinearities

This paper deals with the nonlinear Schrödinger equation with Cubic-Quintic nonlinearities. After transforming the non-autonomous nonlinear Schrödinger equation into a stationary equation by adopting the BPVK idea, we give the classification of the solutions to the Cubic-Quintic nonlinear Schrödinger equation by utilizing the complete discrimination system for polynomial method. As an auxiliary result, we use the homotopy renormalization method (HTR) to obtain the global solutions to the Ermakov-Pinney equation satisfied by the width function.