Dynamical behavior and multiple optical solitons for the fractional Ginzburg–Landau equation with \(\beta \)-derivative in optical fibers

The main goal of the current work is to study dynamical behavior and dispersive optical solitons for the fractional Ginzburg–Landau equation in optical fibers. Starting with the traveling wave transformations, the fractional Ginzburg–Landau model is converted into an equivalent ordinary differential traveling wave system. Then, the Hamiltonian function and orbits phase portraits of this system are found. Here, we derived explicit fractional periodic wave solutions, bell-shaped solitary wave solutions and kink-shaped solitary wave solutions through the bifurcation theory of differential dynamical system. In addition to, some other traveling wave solutions are obtained by using the polynomial complete discriminant method and symbolic computation. Most notably, we give the classification of all single traveling wave solutions of fractional Ginzburg–Landau equation at the same time. The obtained optical soliton solutions in this work may substantially improve or complement the corresponding results in the known references. Finally, we give the comparison between our solutions and other’s results.