Stability of two-dimensional spatial solitons in nonlocal nonlinear media

We discuss the existence and stability of two-dimensional solitons in media with spatially nonlocal nonlinear response. We show that such systems, which include thermal nonlinearity and dipolar Bose-Einstein condensates, may support a variety of stationary localized structures, including rotating dipole solitons. We also demonstrate that the stability of these structures critically depends on the spatial profile of the nonlocal response function.

Figure 1

Nonlocal response functions for thermal nonlinearities $\left({\mathfrak{K}}_0\right)$ and dipolar BEC’s $\left(\mathfrak{R}\right)$. For comparison, a Gaussian response function is shown, too.

Figure 2

Dynamics of the vortex states in the thermal model: (a) unstable propagation at $P_0=200$, profiles are shown in an $xy$ box of about $5\times 5$; (b) unstable propagation at $P_0=500$, profiles are shown in an $xy$ box of about $3\times 3$; (c) stable propagation at $P_0=2000$, profiles are shown in an $xy$ box of about $1.5\times 1.5$.

Figure 3

Dynamics of azimuthons in thermal model of nonlinearity. Modulation depth $n=0.5$: (a) stable propagation at $P_0=500$, rotating with $\Omega \approx 4.9$, profiles are shown in an $xy$ box of about $3\times 3$; (b) stable propagation at $P_0=2000$, rotating with $\Omega \approx 11$, profiles are shown in an $xy$ box of about $1.5\times 1.5$.

Figure 4

Dynamics of localized structures in a BEC nonlinear model [Eq. (5)]. Power $P_0=1000$, $\alpha =0$: (a) unstable vortex state; (b) stable azimuthon, rotating with $\Omega \approx 5.1$, modulation depth $n=0.5$. All profiles are shown in an $xy$ box of about $4\times 4$.

Figure 5

Stable vortex states in the BEC model: (a) stable propagation at $P_0=5000$, $\alpha =0$, profiles are shown in an $xy$ box of about $2\times 2$; (b) stable propagation at $P_0=1000$, $\alpha =-0.05$, profiles are shown in an $xy$ box of about $5\times 5$.

Figure 6

Dipole states of the BEC nonlocal model at $P_0=1000$: (a) unstable propagation, $\alpha =0$, profiles are shown in an $xy$ box of about $4\times 4$; (b) stable propagation, $\alpha =-0.03$, profiles are shown in an $xy$ box of about $5\times 5$; (c) unstable propagation, $\alpha =-0.05$, profiles are shown in an $xy$ box of about $5\times 5$.

Figure 7

Dynamics of double-charge vortex states in a BEC nonlocal model with $P_0=2000$: (a) unstable propagation, $\alpha =0$, profiles are shown in an $xy$ box of about $4\times 4$; (b) stable propagation, $\alpha =-0.03$, profiles are shown in an $xy$ box of about $4\times 4$.

Figure 8

Approximate soliton solutions of Eqs. (1, 2) for $P_0=1000$: (a) dipole state ${\tilde{U}}_0$ computed from ${\tilde{\theta }}_0=P_0\exp (-r^2∕2)∕2\pi$, (b) dipole state ${\tilde{U}}_6$ after six iterations computed from ${\tilde{\theta }}_6$, (c) quadrupole state ${\tilde{U}}_0$ computed from ${\tilde{\theta }}_0=P_0\exp (-r^2∕2)∕2\pi$, and (d) quadrupole state ${\tilde{U}}_4$ after four iterations computed from ${\tilde{\theta }}_4$. The insets show the maximum intensity $I_{\mathit{max}}={\max }_{x,y}{\mid \psi \mid }^2$ upon propagation, employing the approximate solution as an initial condition.

Figure 9

(a) ${\tilde{\theta }}_0=P_0\exp (-r^2∕2)∕2\pi$ used to compute the approximate dipole and quadrupole states in Figs. 8a, 8c, (b) ${\tilde{\theta }}_6$ used to compute the approximate dipole state in Fig. 8b, and (c) ${\tilde{\theta }}_4$ used to compute the approximate quadrupole state in Fig. 8d.