Study of the complex Ginzburg–Landau equation with parabolic law nonlinearity by the complete discrimination system for polynomial method

Abstract

In this paper, we use the complete discrimination system for polynomial method to study the complex Ginzburg–Landau equation with parabolic law nonlinearity. We firstly conduct the qualitative analysis to the equation, and show the existences of the soliton and the periodic solutions by the bifurcation method. In order to verify our results, we adjust concrete parameters to give concrete examples. Besides, we also construct other solutions such as rational type solution and the elliptic function double periodic solutions, which makes our results more integrated. To the best of our knowledge, some of the results in the present paper are initially given.