Diverse optical solitons solutions of the fractional complex Ginzburg-Landau equation via two altered methods

This work evaluates the fractional complex Ginzburg-Landau equation in the sense of truncated M- fractional derivative and analyzes its soliton solutions and other new solutions in the appearance of a detuning factor in non-linear optics. The multiple, bright, and bright-dark soliton solutions of this equation are obtained using the modified $\left(G^{\prime }/G^2\right)$ and $\left(1/G^{\prime }\right)-$expansion methods. The equation is evaluated with Kerr law, quadratic –cubic law and parabolic law non-linear fibers. To shed light on the behavior of solitons, the graphical illustrations in the form of 2D and 3D of the obtained solutions are represented for different values of various parameters. All of the solutions have been verified by substitution into their corresponding equations with the aid of a symbolic software package. The various forms of solutions to the aforementioned nonlinear equation that arises in fluid dynamics and nonlinear processes are presented. Moreover, we guarantee that all the solutions are new and an excellent contribution in the existing literature of solitary wave theory.