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

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.

Figure 1

Amplitude profile and phase for $\delta =-0.012, \beta =0.3, \epsilon =0.2$, and $R_r=0.2$ (a) high-amplitude and (b) low-amplitude solutions.

Figure 2

Existence and stability of the high amplitude solutions on the $(\beta ,\epsilon )$ plane for $\delta =-0.012$ and $R_r=0.2$. The existence boundary from perturbation approach is shown as a dashed (green) curve.

Figure 3

Existence and stability of the high-amplitude solutions on the $(R_r,\epsilon )$ plane for $\delta =-0.012$ and $\beta =0.3$. The existence boundary from perturbation approach is shown as a dashed (green) curve.

Figure 4

Velocity, peak amplitude, and propagation constant dependence on $R_r$ (left column), for $\delta =-0.012, \beta =0.3$, and $\epsilon =0.2$, and on $\epsilon$ (right column), for $\delta =-0.012, \beta =0.3$, and $R_r=0.2$. The high-amplitude solutions are the upper curves and the low-amplitude solutions are the lower curves. Solid lines (black) are results from shooting and dashed lines (green) are results from perturbation.

Figure 5

Characteristics of the high-amplitude solution for nonzero $\xi , R_i, S_r$, and $S_i$ and $\delta =-0.012, \beta =0.3, \epsilon =0.2$, and $R_r=0.2$. The $x$ axis represent each of the referred parameters as in the curve legends.

Figure 6

(a) Trajectory of the discrete eigenvalue (the one with positive real part) as $\epsilon$ increases. (b) Imaginary part of the same eigenvalue as function of $\epsilon$. Both graphs correspond to high-amplitude solutions and $\delta =-0.012, \beta =0.3$, and $R_r=0.2$.

Figure 7

Propagation for $\delta =-0.012, \beta =0.3, \epsilon =0.2$, and $R_r=0.2$ (a) $\text{sech}\left(T\right)$ at the input (b) peak amplitude for $A\text{sech}\left(AT\right)$ at the input.

Figure 8

Propagation for $\delta =-0.012, \beta =0.3$, and $R_r=0.2$ with $\text{sech}\left(T\right)$ at the inputs (a) $\epsilon =0.15$ (no solution region) and (b) $\epsilon =0.28$ (unstable region).