Figure 1

(Color online) Example of dark soliton solution obtained numerically from Eqs. (3, 4) for $v=0.6$ and ${\sigma }^2=5$ (blue solid line), compared with the corresponding solution of the local case ${\sigma }^2=0$ (red dotted line): (a) modulus of the field; (b) phase of the field; and (c) refractive index change $N\left(X\right)$ across the soliton.Reuse & Permissions

Figure 2

(Color online) Snake instability gain $g$ computed numerically vs transverse wave number $q$ for different values of nonlocality ${\sigma }^2$: (a) stationary cases $v=0$ and (b) $v=0.4$.Reuse & Permissions

Figure 3

(Color online) Overall reduction of the snake instability: peak gain $g_{max}$ and cutoff (maximal) wave number $q_c$ vs ${\sigma }^2$ for three different velocities.Reuse & Permissions

Figure 4

(Color online) Results of numerical simulations based on Eqs. (1, 2): intensity distribution over the output transverse plane at propagation distance $Z=10$ for different values of $\sigma$ and $v$. Top row, local case $\sigma =0$: (a) $v=0$; (b) $v=0.2$; and (c) $v=0.4$. Bottom row, nonlocal case $\sigma =1$: (d) $v=0$; (e) $v=0.2$; and (f) $v=0.4$.Reuse & Permissions

Figure 5

(Color online) Radial intensity profiles (red solid line) of nonlocal vortex solutions and relative index profile (green dotted-dashed line) for different topological charges $m=1,2,3$ and $\sigma =1$. For comparison we report also the local intensity profile (blue dashed line).Reuse & Permissions

Figure 6

(Color online) Results of numerical simulations based on Eqs. (1, 2) of long term evolution of an input stationary dark soliton stripe. (a) Intensity distribution over the output transverse plane at $Z=80$. (b) and (c) Insets showing a portion of the transverse plain including two adjacent vortex solitons: (b) intensity distribution and (c) phase distribution, showing opposite helicity. (d) Noisy intensity profile along $x$ (blue solid line) of the upper vortex in (b) as seen at fixed $y=5$ compared with the theoretical profile (red dashed line) of the vortex mode as obtained from Eqs. (10, 11). Here $\sigma =1$.Reuse & Permissions

Figure 7

(Color online) DSW ruled by the local $1+2\text{D}$ NLS equation, contrasting the ideal noiseless case (a)–(c) with the evolution in the presence of noise (d)–(f). (a)–(d) Shock fan at $y=0$; (b)–(e) output transverse plane; and (c)–(f) output intensity profile (solid line) at $y=0$ superimposed to the input (dashed line). Here $\epsilon =0.05$ and the soliton stripe input is $U\left(x\right)=\text{tanh}\left(x\right)$.Reuse & Permissions

Figure 8

(Color online) As in Fig. 5, showing stabilization of a DSW against transverse instability in the nonlocal regime. Results from numerical integration of Eqs. (12, 13) with $\sigma =1$ and $\epsilon =0.05$. (a) Shock fan at $y=0$; (b) output $\left(z=8\right)$ transverse plane; (c) output intensity profile (blue thin solid line); and refractive index change $n\left(x\right)$ (red thick solid line) at $y=0$ superimposed to the input (dashed line).Reuse & Permissions

Figure 9

(Color online) DSW dynamics in a local medium $\left(\sigma =0\right)$ arising from a gray soliton shape $u_0=\sqrt{1-v^2}\text{tanh}\left(\sqrt{1-v^2}x\right)+iv$, with $v=0.5$. (a) and (b) Results from numerical integration of Eqs. (12, 13) in $1+1\text{D}$, with $\epsilon =0.05$: (a) the twin shock fan in the $x\text{−}z$ plane. The dashed line indicates the natural velocity of the input in the limit $\epsilon =1$ for which the input itself is a one soliton; (b) output intensity in the soliton trains (solid line) compared with the input intensity (dotted-dashed line). (c) and (d) Comparison of the results from Eqs. (12, 13) (solid line; blue) with those obtained in the hydrodynamic limit from Eqs. (14, 15) (thick dashed line; red): (c) equivalent fluid density $h={\left|\psi \right|}^2$ vs $x$; (d) equivalent fluid velocity $u={\phi }_x$ vs $x$.Reuse & Permissions

Figure 10

(Color online) As in Fig. 9 in $1+2\text{D}$, showing the onset of the instability growing from noise. (a) Shock fan at $y=0$; (b) output transverse plane; (c) and output intensity profile at $y=0$ (solid line) compared with the input (dashed line).Reuse & Permissions

Figure 11

(Color online) Stabilized DSW in the nonlocal case with $\sigma =1$. Results from numerical integration of Eqs. (12, 13) with $\epsilon =0.05$ and input shape characteristic of the nonlocal soliton $U\left(x\right)$ corresponding to $v=0.5$.Reuse & Permissions