Figure 1

(Color online) Curves of ${\alpha }_j\left(t\right)$, $g\left(t\right)$, and $\gamma \left(t\right)$ given by Eqs. (12, 13) vs $t$ for ${\rho }_0=6.0$, ${\gamma }_0=m=0.1$, and $b_j=n=1.0$ $\left(j=1,2,3\right)$.Reuse & Permissions

Figure 2

(Color online) Propagations (left) and contour plots (right) of density for the bright soliton [Eq. (14)] in $\left(t,\zeta \equiv \mathbf{b}\cdot \mathbf{r}\right)$ space for $r_1=0.5$, $s_1={\rho }_0=2.0$, and ${\gamma }_0=0.01$. The others are the same as Fig. 1. (a) The breathing bright soliton propagating in a zigzag trace for $m=0.1$. In the given sodium sample, the maximum density and width of the soliton are about $1.28\mu {\text{m}}^{-3}$ and $12.13\mu \text{m}$. The period is about 53.1 ms. (b) The bright soliton chain for $m=0.9$. The maximum density and width of the soliton are about $1.28\mu {\text{m}}^{-3}$ and $9.1\mu \text{m}$. The period is about 91.0 ms.Reuse & Permissions

Figure 3

(Color online) Propagations (left) and contour plots (right) for collisions between bright two solitons [Eq. (15)] in $\left(t,\zeta \equiv \mathbf{b}\cdot \mathbf{r}\right)$ space for ${\gamma }_0=0.01$, ${\delta }_{1,2}={\rho }_0=1.0$, and $m=0.6$. The others are the same as Fig. 1. (a) Two zigzag solitons without interaction for $r_1=1.0$, $r_2=1.2$, and $s_{1,2}=0$. In the given sodium sample, the maximum density and width of the left (right) soliton are about $1.8\mu {\text{m}}^{-3}$ and $7.6\mu \text{m}$ ($1.5\mu {\text{m}}^{-3}$ and $9.1\mu \text{m}$). (b) Two strong zigzag solitons with interaction for $r_{1,2}=s_1=-s_2=1.2$. The maximum density and width of the left (right) soliton are about $6.2\mu {\text{m}}^{-3}$ and $3.0\mu \text{m}$ ($17.5\mu {\text{m}}^{-3}$ and $2.4\mu \text{m}$). (c) Two strong-weak zigzag solitons with interaction for $r_1=1.2$, $r_2=1.5$, $s_1=1.05$, and $s_2=0$. The maximum density and width of the left (right) soliton are about $6.2\mu {\text{m}}^{-3}$ and $4.2\mu \text{m}$ ($13.1\mu {\text{m}}^{-3}$ and $3.8\mu \text{m}$). The period of the zigzag oscillation is about 26.5 ms in all panels.Reuse & Permissions

Figure 4

(Color online) Numerical simulations of bright soliton from Fig. 2 vs $\zeta$ at $t=0$, 10, and 20. The initial conditions are given by Eq. (14) with $s_1=0.1$, $m=0.1$ in panel (a) and $m=0.9$ in panel (b). The other parameters are the same as those used in Fig. 2.Reuse & Permissions