New Exact Wave Solutions on the Complex Ginzburg–Landau Equation with Extended Rational Sin–Cos and Sinh–Cosh Method

Abstract In this paper, a new complex waves representing solutions of the complex Ginzburg–Landau equation with Kerr law nonlinearity is investigated. we used the extended rational sin–cos and sinh–cosh function methods construct precise solutions to the nonlinear equation. Bright periodic solution, periodic solution, dark wave soliton solution of phenomena that occur in nonlinear optics or in plasma physics are obtained. The physical meaning of the geometrical structures for some solutions is discussed for different choices of the free parameters. The proposed method provides an important and effective mathematical tool to construct exact solutions according to different complex equations. The results present the dynamics properties of the different waves with different the 3D and contour plots.