Bilinearization of the generalized coupled nonlinear Schrödinger equation with variable coefficients and gain and dark-bright pair soliton solutions

We investigate coupled nonlinear Schrödinger equations (NLSEs) with variable coefficients and gain. The coupled NLSE is a model equation for optical soliton propagation and their interaction in a multimode fiber medium or in a fiber array. By using Hirota's bilinear method, we obtain the bright-bright, dark-bright combinations of a one-soliton solution (1SS) and two-soliton solutions (2SS) for an $n$-coupled NLSE with variable coefficients and gain. Crucial properties of two-soliton (dark-bright pair) interactions, such as elastic and inelastic interactions and the dynamics of soliton bound states, are studied using asymptotic analysis and graphical analysis. We show that a bright $2\text{-soliton}$, in addition to elastic interactions, also exhibits multiple inelastic interactions. A dark $2\text{-soliton}$, on the other hand, exhibits only elastic interactions. We also observe a breatherlike structure of a bright 2-soliton, a feature that become prominent with gain and disappears as the amplitude acquires a minimum value, and after that the solitons remain parallel. The dark $2\text{-soliton}$, however, remains parallel irrespective of the gain. The results found by us might be useful for applications in soliton control, a fiber amplifier, all optical switching, and optical computing.

Figure 1

Evolution of a bright soliton in two components, $|q_1|,|q_2|$, in (a) and (b) and a dark soliton in one component, $|q_3|$, in (c) with $\gamma =1$, ${\sigma }_1={\sigma }_2=-{\sigma }_3=1$, $\eta =1-i$, $\zeta =4$, $\chi =1+2\mathit{i}$, ${\alpha }_{11}=2+i$, ${\alpha }_{12}=1-\mathit{i}$ and the soliton management parameters $k=\frac{\pi }{4}$, $\sigma =0.005$, and $\Gamma \left(z\right)=0.0025$.

Figure 2

Elastic interactions between two bright solitons in (a) component $|q_1|$ and (b) component $|q_2|$ with $\gamma =1$, ${\sigma }_1={\sigma }_2={\sigma }_3=-{\sigma }_4=1$, ${\eta }_1=2-i$, ${\eta }_2=2+i$, ${\zeta }_3=1$, ${\zeta }_4=3$, ${\chi }_3=1+3i$, ${\chi }_4=1-3i$, and ${\alpha }_{11}={\alpha }_{12}={\alpha }_{21}={\alpha }_{22}=1$, and soliton management parameters $k=\frac{\pi }{18}$, $\sigma =0.005$, and $\Gamma \left(z\right)=0.0025$.

Figure 3

Elastic interactions between two dark solitons, component-wise, in (a) component $|q_3|$ and (b) component $|q_4|$ with $\gamma =1$, ${\sigma }_1={\sigma }_2={\sigma }_3=-{\sigma }_4=1$, ${\eta }_1=2-i$, ${\eta }_2=2+i$, ${\zeta }_3=1$, ${\zeta }_4=3$, ${\chi }_3=1+3i$, ${\chi }_4=1-3i$, and ${\alpha }_{11}={\alpha }_{12}={\alpha }_{21}={\alpha }_{22}=1$, and soliton management parameters $k=\frac{\pi }{18}$, $\sigma =0.005$, and $\Gamma \left(z\right)=0.0025$.

Figure 4

Contour plot of elastic soliton interactions, component-wise, in (a) $|q_1|$, (b) $|q_2|$, (c) $|q_3|$, and (d) $|q_4|$ with the parameters as given in the caption of Figs. 2 and 3.

Figure 5

Elastic interactions between two bright solitons [in (a) $|q_1|$ and (b) $|q_2|$] and between two dark solitons [in (c) $|q_3|$ and (d) $|q_4|$]: $\gamma =1$, ${\sigma }_1={\sigma }_2={\sigma }_3=-{\sigma }_4=1$, ${\eta }_1=4-0.5i$, ${\eta }_2=4.1+0.5i$, ${\zeta }_3=4$, ${\zeta }_4=3$, ${\chi }_3=1+3i$, ${\chi }_4=1-3i$, ${\alpha }_{11}={\alpha }_{22}=0$, ${\alpha }_{12}=2$, and ${\alpha }_{21}=1$, and soliton management parameters $k=\frac{\pi }{16}$, $\sigma =0.01$, and $\Gamma \left(z\right)=0.005$.

Figure 6

Contour plot of elastic soliton interactions, component-wise, in (a) $|q_1|$, (b) $|q_2|$, (c) $|q_3|$, and (d) $|q_4|$ with the parameters as given in the caption of Fig. 5.

Figure 7

Elastic interactions between two bright solitons [in (a) $|q_1|$ and (b) $|q_2|$] and between two dark solitons [in (c) $|q_3|$ and (d) $|q_4|$] with $\gamma =1$, ${\sigma }_1={\sigma }_2={\sigma }_3=-{\sigma }_4=1$, ${\eta }_1=4+i$, ${\eta }_2=4.1+i$, ${\zeta }_3=4$, ${\zeta }_4=3$, ${\chi }_3=1+3i$, ${\chi }_4=1-3i$, ${\alpha }_{12}$, ${\alpha }_{21}=0$, ${\alpha }_{11}=4$, and ${\alpha }_{22}=1$, and soliton management parameters $k=\frac{\pi }{18}$, $\sigma =0.01$, and $\Gamma \left(z\right)=0.005$.

Figure 8

Contour plot of elastic soliton interactions, component-wise, in (a) $|q_1|$, (b) $|q_2|$, (c) $|q_3|$, and (d) $|q_4|$ with the parameters as given in the caption of Fig. 7.

Figure 9

Inelastic interactions between two bright solitons in two components, $|q_1|$ in (a) and $|q_2|$ in (b), with $\gamma =1$, ${\sigma }_1={\sigma }_2=-{\sigma }_3={\sigma }_4=1$, ${\eta }_1=4+i$, ${\eta }_2=4.1+i$, ${\zeta }_3=4$, ${\zeta }_4=3$, ${\chi }_3=1+3i$, ${\chi }_4=1-3i$, ${\alpha }_{11}={\alpha }_{12}={\alpha }_{21}=1$, and ${\alpha }_{22}=8$, and soliton management parameters $k=\frac{\pi }{18}$, $\sigma =0.005$, and $\Gamma \left(z\right)=0.0025$.

Figure 10

Elastic interactions between two dark solitons in two components, $|q_3|$ in (a) and $|q_4|$ in (b), with the same values of the parameters as given in the caption of Fig. 9.

Figure 11

Contour plot of component-wise inelastic bright soliton interactions in (a) $|q_1|$ and (b) $|q_2|$, and elastic dark soliton interaction in (c) $|q_3|$ and (d) $|q_4|$, with the parameters as given in the captions of Figs. 9 and 10.

Figure 12

Soliton complexes of bright $2\text{-solitons}$ [in (a) $|q_1|$ and (b) $|q_2|$] and of dark $2\text{-solitons}$ [in (c) $|q_3|$ and (d) $|q_4|$] with $\gamma =1$, ${\sigma }_1={\sigma }_2={\sigma }_3={\sigma }_4=1$, ${\eta }_1=4$, ${\eta }_2=4.1$, ${\zeta }_3=1$, ${\zeta }_4=3$, ${\chi }_3=1+3i$, ${\chi }_4=1-2i$, and ${\alpha }_{11}={\alpha }_{22}={\alpha }_{12}={\alpha }_{21}=1$, and soliton management parameters $k=\frac{\pi }{18}$, $\sigma =0.05$, and $\Gamma \left(z\right)=0.025$.

Figure 13

Contour plot of $2\text{-soliton}$ complexes, component-wise, in (a) $|q_1|$, (b) $|q_2|$, (c) $|q_3|$, and (d) $|q_4|$ with the parameters as given in the caption of Fig. 12.

Figure 14

Soliton complexes of bright $2\text{-solitons}$ [in (a) $|q_1|$ and (b) $|q_2|$] and of dark $2\text{-soliton}$ [in (c) $|q_3|$ and (d) $|q_4|$], with $\gamma =1$, ${\sigma }_1={\sigma }_2={\sigma }_3={\sigma }_4=1$, ${\eta }_1=4$, ${\eta }_2=4.1$, ${\zeta }_3=1$, ${\zeta }_4=3$, ${\chi }_3=1+3i$, ${\chi }_4=1-2i$, ${\alpha }_{11}=1,{\alpha }_{22}=2$, and ${\alpha }_{12}={\alpha }_{21}=0$, and soliton management parameters $k=\frac{\pi }{18}$, $\sigma =0.05$, and $\Gamma \left(z\right)=0.025$.

Figure 15

Soliton complexes of a bright $2\text{-soliton}$ [in (a) $|q_1|$ and (b) $|q_2|$] and of dark $2\text{-solitons}$ [in (c) $|q_3|$ and (d) $|q_4|$] with $\gamma =1$, ${\sigma }_1={\sigma }_2={\sigma }_3=-{\sigma }_4=1$, $\gamma =1$, ${\eta }_1=4$, ${\eta }_2=4.1$, ${\zeta }_3=1$, ${\zeta }_4=3$, ${\chi }_3=1+3i$, ${\chi }_4=1-2i$, ${\alpha }_{12}=1,{\alpha }_{21}=2$, and ${\alpha }_{11}={\alpha }_{22}=0$, and soliton management parameters $k=\frac{\pi }{18}$, $\sigma =0.05$, and $\Gamma \left(z\right)=0.025$.

Figure 16

Breatherlike bright $2\text{-soliton}$ [in (a) $|q_1|$ and (b) $|q_2|$] and of normal dark $2\text{-solitons}$ [in (c) $|q_3|$ and in (d) $|q_4|$] with $\gamma =1$, ${\sigma }_1={\sigma }_2={\sigma }_3=-{\sigma }_4=1$, ${\eta }_1=1$, ${\eta }_2=1.6$, ${\zeta }_3=1$, ${\zeta }_4=3$, ${\chi }_3=1+i$, ${\chi }_4=1-i$, ${\alpha }_{21}=-1$, and ${\alpha }_{11}={\alpha }_{22}={\alpha }_{12}=1$, and soliton management parameters $k=\frac{\pi }{12}$, $\sigma =0.005$, and $\Gamma \left(z\right)=0.0025$.

Figure 17

Contour plot of $2\text{-soliton}$ complexes, component-wise, in (a) $|q_1|$, (b) $|q_2|$, (c) $|q_3|$, and (d) $|q_4|$ with the parameters as given in the caption of Fig. 16.