Figure 1

Soliton ($S=0$, black solid lines) and vortex-ring ($S=1$, red dashed lines) solutions of Eq. (6) for $\beta =1$ (a) and $-1$ (b). Examples of the density distribution $\rho =|{\chi }_{1,1}{|}^2$ for a bright soliton ($\beta =1$, $S=0$, $J_x=J_y=1$, $k_x=0$, $k_y=0.95\pi /2$, ${\Omega }_0=1.01$) and dark vortex ring ($\beta =-1$, $S=1$, $k_x=k_y=1.05\pi /2$, ${\Omega }_0=1.1$) are shown in the insets in panels (a) and (b), respectively. The dark vortex ring in the inset in panel (b) has been truncated to zero by the function $0.5[1-\tanh (r-r_0)]$ with $r_0=5$.Reuse & Permissions

Figure 2

Snapshots [amplitude (left column) and density (right column) plots] of a collision in the $mn$ plane between a bright soliton [plotted in inset in Fig. 1a] and dark vortex-ring (plotted in inset in Fig. 1b]. $t=0$ (a) and (b) the bright soliton at the coordinate (64,100) with velocity in the $3\pi /2$ direction (angle with respect to the positive $m$ axis) and a dark vortex-ring at (30,30) with velocity in the $\pi /4$ direction. $t=20$ (c and d) collision at (0,0). $t=40$ (e) and (f) bright soliton at (64,30) and dark vortex-ring at (110,110). Arrows indicate direction of motion. Other parameters are given in the caption of Fig. 1.Reuse & Permissions

Figure 3

(a) Speed $v=|{\mathbf{v}}_{1,1}|$ vs the quasimomentum components $k_x$ and $k_y$ in the first Brillouin zone. The meshed surface corresponds to the theoretical prediction, Eq. (5), and the dots are results from numerical simulations. (b) Amplitude coefficient $\eta$ [see Eq. (8)] vs the quasimomentum components $k_x$ and $k_y$ in the first Brillouin zone. $\{k_x,k_y\}=\{0,0\}$ (A), $\{0.95\pi /2,0\}$ (B), $\{1.75\pi /2,0\}$ (C), and $\{0.95\pi /2,0.95\pi /2\}$ (D). Other parameters: $J_x=J_y=U=1$ and $|{\Omega }_0-1|=0.01$.Reuse & Permissions

Figure 4

(a) Normalized maximum ${\rho }_{\max }$ vs $t$ for cases in Fig. 3b with $|{\Omega }_0-1|=0.01$. (b) For comparison, the case $|{\Omega }_0-1|=0.001$. (c) Soliton position $\{m,n\}$ vs $t$ for $k_x=0.95\pi /2$ and/or $k_y=0.95\pi /2$ and periodic boundary conditions.Reuse & Permissions