Bright-dark solitons and their collisions in mixed $N$-coupled nonlinear Schrödinger equations

Mixed-type (bright-dark) soliton solutions of the integrable $N$-coupled nonlinear Schrödinger (CNLS) equations with mixed signs of focusing- and defocusing-type nonlinearity coefficients are obtained by using Hirota’s bilinearization method. Generally, for the mixed $N$-CNLS equations the bright and dark solitons can be split up in $\left(N-1\right)$ ways. By analyzing the collision dynamics of these coupled bright and dark solitons systematically we point out that for $N>2$, if the bright solitons appear in at least two components, nontrivial effects, such as onset of intensity redistribution, amplitude-dependent phase shift, and change in relative separation distance take place in the bright solitons during collision. However their counterparts, the dark solitons, undergo elastic collision but experience the same amplitude-dependent phase shift as that of bright solitons. Thus, in the mixed CNLS system, there is a coexisting shape-changing collision of bright solitons and elastic collision of dark solitons with amplitude-dependent phase shift, thereby influencing each other mutually in an intricate way.

Figure 1

Intensity plots of one-soliton of the mixed CNLS equations with $N=2$ for different values of the background parameter $c_1$ for a fixed value of $z$. Note that the intensity of the bright soliton increases as the depth of the dark soliton decreases. The other parameters are chosen as $k_1=1+i,{\alpha }_1^{\left(1\right)}=1,b_1=0.2$.

Figure 2

Intensity plots of one-soliton of the mixed CNLS equations with $N=3$ for different values of the $\alpha$ parameters for a fixed value of $z$. (a) ${\alpha }_1^{\left(1\right)}=0.2+0.01i,{\alpha }_1^{\left(2\right)}=0.5+0.05i$, (b) ${\alpha }_1^{\left(1\right)}=1+i,{\alpha }_1^{\left(2\right)}=2+i$, and (c) ${\alpha }_1^{\left(1\right)}=13-13i,{\alpha }_1^{\left(2\right)}=17+0.3i$. The role of $\alpha$ parameters are shown in both the intensity and phase of the bright soliton while it affects the phase (central position) of the dark soliton. The parameters $c_1$, $b_1$, and $k_1$ are chosen as $\left|c_1\right|=0.56,b_1=0.2,k_1=1+i$.

Figure 3

Elastic collision of (bright-dark) two solitons in the mixed CNLS system for the $N=2$ case. The parameters are chosen as given in the text.

Figure 4

Intensity profiles of two colliding bright solitons of the mixed CNLS equations with $N=2$, before $\left(z=-4\right)$ and after $\left(z=4\right)$ collision: (a) in the absence of dark component [$c_1=0$ in Eq. (11)]; (b) in the presence of dark component [$c_1\ne 0$ in Eq. (11)]. The figure is plotted for a special choice of parameters (as given in the text) with $\left|c_1\right|=0.56$. Note the elastic nature of the collision.

Figure 5

Shape-changing collision of two solitons in the mixed CNLS system for the $N=3$ case. The parameters are as given in the text.

Figure 6

Intensity profiles showing the collision scenario of two bright solitons of the mixed CNLS equations with $N=3$ in the absence of dark component, $\left(c_1=0\right)$ with ${\alpha }_1^{\left(1\right)}/{\alpha }_2^{\left(1\right)}\ne {\alpha }_1^{\left(2\right)}/{\alpha }_2^{\left(2\right)}$. (a) $z=-4$ and (b) $z=4$, given by the specific choice of parameters as given in the text.

Figure 7

Intensity profiles showing the collision scenario of two bright solitons of the mixed CNLS equations with $N=3$ in the presence of dark component $\left(c_1\ne 0\right)$ with ${\alpha }_1^{\left(1\right)}/{\alpha }_2^{\left(1\right)}\ne {\alpha }_1^{\left(2\right)}/{\alpha }_2^{\left(2\right)}$. (a) $z=-4$ and (b) $z=4$, given by the specific choice of parameters as given in the text.

Figure 8

Intensity profiles showing the collision scenario of two bright solitons of the mixed CNLS equations with $N=3$ in the absence of dark component $\left(c_1=0\right)$, for ${\alpha }_1^{\left(1\right)}/{\alpha }_2^{\left(1\right)}={\alpha }_1^{\left(2\right)}/{\alpha }_2^{\left(2\right)}$. (a) $z=-4$ and (b) $z=4$. The parameters are chosen as given in the text.

Figure 9

Intensity profiles showing the collision scenario of two bright solitons of the mixed CNLS equations with $N=3$ in the presence of dark component $\left(c_1\ne 0\right)$, for ${\alpha }_1^{\left(1\right)}/{\alpha }_2^{\left(1\right)}={\alpha }_1^{\left(2\right)}/{\alpha }_2^{\left(2\right)}$. (a) $z=-4$ and (b) $z=4$. The parameters are chosen as in the text.

Figure 10

Intensity profiles showing the propagation of two dark solitons in the mixed CNLS system, before $\left(z=-5\right)$ and after $\left(z=5\right)$ collision: (a) for the special choice of parameters (as given in the text) with ${\alpha }_1^{\left(1\right)}/{\alpha }_2^{\left(1\right)}\ne {\alpha }_1^{\left(2\right)}/{\alpha }_2^{\left(2\right)}$; (b) for the special choice of parameters (as given in the text) with ${\alpha }_1^{\left(1\right)}/{\alpha }_2^{\left(1\right)}={\alpha }_1^{\left(2\right)}/{\alpha }_2^{\left(2\right)}$.