Existence and stability of solutions of the cubic complex Ginzburg-Landau equation with delayed Raman scattering

Abstract

We found two stationary solutions of the cubic complex Ginzburg-Landau equation (CGLE) with an additional term modeling the delayed Raman scattering. Both solutions propagate with nonzero velocity. The solution that has lower peak amplitude is the continuation of the chirped soliton of the cubic CGLE and is unstable in all the parameter space of existence. The other solution is stable for values of nonlinear gain below a certain threshold. The solutions were found using a shooting method to integrate the ordinary differential equation that results from the evolution equation through a change of variables, and their stability was studied using the Evans function method. Additional integration of the evolution equation revealed the basis of attraction of the stable solutions. Furthermore, we have investigated the existence and stability of the high amplitude branch of solutions in the presence of other higher order terms originating from complex Raman, self-steepening, and imaginary group velocity.