Impact of high-order effects on soliton explosions in the complex cubic-quintic Ginzburg-Landau equation

We investigate the impact of higher-order nonlinear and dispersive effects on the dynamics of a single soliton solution in the complex cubic-quintic Ginzburg-Landau equation. Operating in the regime of soliton explosions, we show how the splitting of explosion modes is affected by the interplay of the high-order effects (HOEs) resulting in the controllable selection of right- or left-side periodic explosions. In addition, we demonstrate that HOEs induce a series of pulsating instabilities, significantly reducing the stability region of the single soliton solution.

Figure 1

(a) A maximal intensity of a single LS of Eq. (1) as a function of the loss parameter $\delta$ for ${\beta }_3=s={\tau }_r=0$. A LS is stable between an AH point $H_0$ and the threshold of explosions, given by a double AH point $H_L,H_R$ at $\delta =-0.5537$. (b) Space-time representation of the time evolution of a LS from a DNS for $\delta =-0.3$. (c),(d) Real parts of the critical eigenfunctions (red) at $\delta =-0.6$, corresponding to asymmetrical and symmetrical explosions, respectively. Cyan lines represent the $\mathrm{Re}\left(A\right)$ field, whereas the black line corresponds to the intensity profile. All quantities are dimensionless. Parameters are $\epsilon =1.0188, \beta =0.125, \mu =-0.1$, and $\nu =-0.6$.

Figure 2

(a),(b) Single LS branch of Eq. (1) as a function $\delta$ for ${\beta }_3=0.016, {\tau }_R=0.032, s=0.009$ where the splitting of AH points $H_R,H_L$ can be seen. (a) Maximal intensity and (b) drift velocity. A LS is stable between an AH point $H_0$ and the threshold of right-side explosions $H_R$. (c),(d) Real parts of the critical eigenfunctions (red), $\mathrm{Re}\left(A\right)$ (cyan), and $I={\left|A\right|}^2$ (black) at $\delta =-0.6$ for the right- and left-side explosions, respectively. All quantities are dimensionless.

Figure 3

(a),(b) Bifurcation diagram of a single LS of Eq. (1) in the $(\delta ,{\beta }_3)$ plane showing the splitting of the explosions modes for (a) ${\tau }_R=0, s=0$ and (b) ${\tau }_R=0.032, s=0.009$. Magenta solid lines $F$ correspond to folds, blue dashed lines $H_0$ are symmetrical pulsation thresholds, whereas $H_L$ and $H_R$ are lines corresponding to the left- and right-side explosion thresholds, respectively. (c),(d) Space-time plots showing the (c) left- and (d) right-side explosions calculated within DNSs at $(\delta ,{\beta }_3)=(-0.45,0.001)$ and $(-0.5,0.01)$, respectively. All quantities are dimensionless.

Figure 4

(a) Intensity profile (black dashed line) of a single LS of Eq. (1) calculated at ${\beta }_3=0.068$ and $\delta =-1$ together with the real part of the critical eigenfunction $H_1$ (red) and $\mathrm{Re}\left(A\right)$ (cyan). (b) Drift velocity $v$ as a function of the TOD coefficient ${\beta }_3$. The points $H_{1-4}$ correspond to the TOD-induced AH bifurcations, whereas $H_0$ corresponds to a symmetric pulsation (cf. Fig. 3). (c),(d) Time evolution of the LS calculated from DNS for ${\beta }_3=0.07$ and ${\beta }_3=0.08$, respectively. All quantities are dimensionless.