Dissipative solitons for generalizations of the cubic complex Ginzburg-Landau equation

We found stable soliton solutions for two generalizations of the cubic complex Ginzburg-Landau equation, namely, one that includes the term that, in optics, represents a delayed response of the nonlinear gain and the other including the self-steepening term, also in the optical context. These solutions do not require the presence of the delayed response of the nonlinear refractive index, such that, they exist regardless of the term previously considered essential for stabilization. The existence of these solitons was predicted by a perturbation approach, and then confirmed by solving the ordinary differential equations, resulting from a similarity reduction, and also by applying a linear stability analysis. We found that these solitons exist for a large region of the parameter space and possess very asymmetric amplitude profiles as well as a complicated chirp characteristic.

Figure 1

Solutions for $R=0.026$ and $S=0$. Region of existence and stability (a): the lines associated with ${\epsilon }_m$ and ${\epsilon }_c$ correspond to the existence and stability boundaries, respectively, obtained with the perturbation approaches. Peak amplitude of the perturbation and ODE solutions and of the CGLE soliton as a function of $\epsilon$ for $\beta =0.4$ (b). Comparison of (c) amplitude and (d) phase derivative profiles obtained with the perturbation approach and with the ODEs (10) for $\beta =0.4$ and $\epsilon =0.23$.

Figure 2

Solutions for $R=0.1i$ and $S=0$. Region of existence and stability (a). Peak amplitude of the perturbation and ODE solutions and of the CGLE soliton as a function of $\epsilon$ for $\beta =0.3$ (b). Comparison of (c) amplitude and (d) phase derivative profiles obtained with the perturbation approach and with the ODEs (10) for $\beta =0.3$ and $\epsilon =0.2$.

Figure 3

Solutions for $R=0$ and $S=0.05$. Region of existence and stability (a). Peak amplitude of the perturbation and ODE solutions and of the CGLE soliton as a function of $\epsilon$ for (b) $\beta =0.045$ and (c) $\beta =0.25$. Comparison of (d) amplitude and (e) phase derivative profiles obtained with the perturbation approach and with the ODEs (10) for $\beta =0.25$ and $\epsilon =0.15$.

Figure 4

Evolution to stable pulses of (a) $15\text{sech}\left(15T\right)$ for $\beta =0.3$, $\epsilon =0.2$, $R=0.1i$, and $S=0$ and of (b) $25\text{sech}\left(25T\right)$ for $\beta =0.25$, $\epsilon =0.15$, $R=0$, and $S=0.05$. The insets show the input amplitude profile (bottom) and the output amplitude profile (top).