Bifurcations and exact solutions of generalized nonlinear Schrödinger equation

To find the exact explicit solutions of the generalized nonlinear Schrödinger equation, we first give the corresponding differential system for the amplitude component, which constitutes a planar dynamical system featuring a singular straight line. By analyzing its corresponding traveling wave system, we can derive the dynamical behavior of the amplitude component and give the corresponding phase portraits. Under different parameter conditions, we obtain exact explicit solitary wave solutions, periodic wave solutions, as well as peakons and periodic peakons. By comparing our results with previous studies on the generalized nonlinear Schrödinger equation, we correct the error regarding the first integral and present accurate solutions to the equation.