Plain and oscillatory solitons of the cubic complex Ginzburg-Landau equation with nonlinear gradient terms

In this work, we present parameter regions for the existence of stable plain solitons of the cubic complex Ginzburg-Landau equation (CGLE) with higher-order terms associated with a fourth-order expansion. Using a perturbation approach around the nonlinear Schrödinger equation soliton and a full numerical analysis that solves an ordinary differential equation for the soliton profiles and using the Evans method in the search for unstable eigenvalues, we have found that the minimum equation allowing these stable solitons is the cubic CGLE plus a term known in optics as Raman-delayed response, which is responsible for the redshift of the spectrum. The other favorable term for the occurrence of stable solitons is a term that represents the increase of nonlinear gain with higher frequencies. At the stability boundary, a bifurcation occurs giving rise to stable oscillatory solitons for higher values of the nonlinear gain. These oscillations can have very high amplitudes, with the pulse energy changing more than two orders of magnitude in a period, and they can even exhibit more complex dynamics such as period-doubling.

Figure 1

Peak amplitude (a) and velocities (b) of the two branches of solutions for ${\delta }^{\prime }=-0.022, \beta =0.4$, and $R^{\prime }=0.177$, and other nonzero parameters as indicated in the curve labels.

Figure 2

Existence and stability for ${\delta }^{\prime }=-0.022, \xi =0, R^{\prime }=0.177$, and $S^{\prime }=0$. The three different regions are regions of absence of solitons, stable solitons, and unstable solitons, as indicated in the figure. The two curves are the threshold for existence and stability as predicted by perturbation.

Figure 3

Existence and stability for ${\delta }^{\prime }=-0.022, R^{\prime }=0.177$ and (a) $S^{\prime }=-0.1$, (b) $S^{\prime }=0.1$, (c) $R_i^{\prime }=0.14$, (d) ${\xi }^{\prime }=\pm 0.1$, (e) $S^{\prime }=-0.1i$, and (f) $S^{\prime }=0.1i$. The parameters that are not indicated are zero. The three different regions are regions of absence of solitons, stable solitons, and unstable solitons, as indicated in the figures. The two curves are the threshold for existence and stability as predicted by perturbation.

Figure 4

Profiles of amplitude ($F$) and derivative of phase (${\theta }^{\prime }$) for ${\delta }^{\prime }=-0.022, R_r^{\prime }=0.177, \beta =0.4, \epsilon =0.24$, and $S^{\prime }, R_i^{\prime }$, and ${\xi }^{\prime }$ as indicated in the labels.

Figure 5

Peak amplitude evolution for ${\delta }^{\prime }=-0.012, \beta =0.3, {\xi }^{\prime }=0, R^{\prime }=0.2, S^{\prime }=0$, and $\epsilon$ as in the inset.

Figure 6

Peak amplitude evolution for ${\delta }^{\prime }=-0.022, \beta =0.9, {\xi }^{\prime }=0, R^{\prime }=0.177, S^{\prime }=-0.1$, and $\epsilon$ as in the inset.

Figure 7

Pulse evolution for the case $\epsilon =0.42$ of Fig. 6.

Figure 8

Pulse energy vs peak amplitude for three of the cases represented in Fig. 6 as indicated in the inset.

Figure 9

(a) Peak amplitude evolution for ${\delta }^{\prime }=-0.022, \beta =0.1, {\xi }^{\prime }=0, R^{\prime }=0.177, S^{\prime }=-0.1i$, and $\epsilon$ as in the inset. (b) Type of solutions across the $\epsilon$ axis for the same parameters.

Figure 10

Pulse energy vs peak amplitude for ${\delta }^{\prime }=-0.022, \beta =0.1, \epsilon =0.76, {\xi }^{\prime }=0, R^{\prime }=0.177$, and $S^{\prime }=-0.1i$, and (a) $\epsilon =0.68$ and (b) $\epsilon =0.76$.