Application of the first integral method for solving (1 + 1) dimensional cubic-quintic complex Ginzburg-Landau equation

The main object of this paper is to construct the new exact solutions of a nonlinear evolution equation that appears in mathematical physics, specifically complex cubic-quintic Ginzburg-Landau equation by the first integral method. The periodic wave solutions and solitary wave solutions of the equation are obtained. The proposed technique analyzes the possibility of existence of limit cycles with specific parameters. This method is manageable, straightforward and a powerful tool to find the exact solutions of nonlinear partial differential equations (PDEs).