Localized dark solitons and vortices in defocusing media with spatially inhomogeneous nonlinearity

Recent studies have demonstrated that defocusing cubic nonlinearity with local strength growing from the center to the periphery faster than $r^D$, in space of dimension $D$ with radial coordinate $r$, supports a vast variety of robust bright solitons. In the framework of the same model, but with a weaker spatial-growth rate $\sim r^{\alpha }$ with $\alpha \le D$, we test here the possibility to create stable localized continuous waves (LCWs) in one-dimensional (1D) and 2D geometries, localized dark solitons (LDSs) in one dimension, and localized dark vortices (LDVs) in two dimensions, which are all realized as loosely confined states with a divergent norm. Asymptotic tails of the solutions, which determine the divergence of the norm, are constructed in a universal analytical form by means of the Thomas-Fermi approximation (TFA). Global approximations for the LCWs, LDSs, and LDVs are constructed on the basis of interpolations between analytical approximations available far from (TFA) and close to the center. In particular, the interpolations for the 1D LDS, as well as for the 2D LDVs, are based on a deformed-tanh expression, which is suggested by the usual 1D dark-soliton solution. The analytical interpolations produce very accurate results, in comparison with numerical findings, for the 1D and 2D LCWs, 1D LDSs, and 2D LDVs with vorticity $S=1$. In addition to the 1D fundamental LDSs with the single notch and 2D vortices with $S=1$, higher-order LDSs with multiple notches are found too, as well as double LDVs, with $S=2$. Stability regions for the modes under consideration are identified by means of systematic simulations, the LCWs being completely stable in one and two dimensions, as they are ground states in the corresponding settings. Basic evolution scenarios are identified for those vortices that are unstable. The settings considered in this work may be implemented in nonlinear optics and in Bose-Einstein condensates.

Figure 1

Comparison of the analytical interpolation for 1D LCW states, given by Eq. (15), with their numerically found counterparts, for (a) $\alpha =0.2$ and $k=-1.35$, (b) $\alpha =0.65$ and $k=-3.4$, (c) $\alpha =1$ and $k=-2.1$, and (d) $\alpha =2$ and $k=-2.1$ (the latter case, with $\alpha >D=1$, corresponds to a bright soliton). (e) Numerically found dependence $N\left(k\right)$ for the 1D LCW state at different values of $\alpha$, with norm $N$ computed in a finite domain $\left|x\right|\le 20$. The dashed lines show the analytical counterpart, given by Eq. (10) with $D=1$ and $R=20$. This figure and all others pertain to $g_0=1$ in Eq. (8), fixed by scaling.

Figure 2

(a) Shapes of 1D fundamental (single-notch) LDSs with different values of parameters ($\alpha =0$ and $k=-0.5, \alpha =0.12$ and $k=-1.1$, and $\alpha =0.27$ and $k=-2.1$). The comparison of these numerically found profiles with their approximate analytical counterparts is displayed in Fig. 3. Included in (a), for the sake of comparison, is the commonly known exact dark soliton for $\alpha =0$. (b) and (c) Examples of higher-order LDSs with different numbers of notches: (b) two notches for $\alpha =0.25$ and $k=-2.2$ and (c) three notches for $\alpha =0.3$ and $k=-1$. (d) Norm $N$ vs the propagation constant $k$ for 1D fundamental LDSs in the truncated system with different values of $\alpha$. Dashed straight lines show the respective analytical approximations given by Eq. (10).

Figure 3

Comparison of numerically computed solutions for the 1D fundamental localized dark solitons with the analytical prediction based on the deformed-tanh interpolation (labeled D-tanh in the figure), given by Eq. (19) for (a) $\alpha =0.12$ and $k=-1.1$, with $\lambda =1.1$, and (b) $\alpha =0.27$ and $k=-2.1$, with $\lambda =1.5$.

Figure 4

Stability border for (a) the 1D fundamental LDSs and (b) the 2D LDVs with charge $S=1$. The dark solitons and vortices are stable above the respective borders. While the shape of the stability borders is relatively complex, the results meet the natural condition that all the modes are stable in the limit of $\alpha \to 0$, in compliance with the commonly known fact that dark 1D solitons and 2D vortices are completely stable in the case of the spatially uniform self-defocusing.

Figure 5

Evolution of (a) stable and (b) unstable 1D fundamental LDSs for (a) $\alpha =0.3$ and $k=-1.9$ and (b) $\alpha =0.35$ and $k=-2.4$. Also shown are the oscillations of weakly unstable perturbed 1D higher-order LDSs with different numbers of notches: (c) two, for $\alpha =0.25$ and $k=-2.2$, and (d) three, for $\alpha =0.3$ and $k=-1$.

Figure 6

Comparison of the analytical interpolation, given by Eq. (15), for 2D LCW states, with their numerically found counterparts, for (a) $\alpha =0.25$ and $k=-1.2$, (b) $\alpha =1$ and $k=-4.2$, and (c) $\alpha =2$ and $k=-2.2$. (d) Numerically found dependence $N\left(k\right)$ for the 2D LCW states in the truncated model, at different values of $\alpha$. Dashed lines display the analytical approximation for $N\left(k\right)$, as produced by Eq. (17).

Figure 7

(a) and (b) Numerically found profiles of 2D LDVs with topological charges, severally, $S=1$ and 2, propagation constant $k=-1.5$, and different values of $\alpha$. The profiles of the usual vortices in the uniform space, corresponding to $\alpha =0$, are shown for comparison. (c) Profiles of vortices with different values of $k$, for $\alpha =0.6$ and $S=2$. (d) Norm of the vortex in the truncated system, with $r\le R=15$, vs $k$, for the LDVs with $S=1,2$ and $\alpha =0.4$.

Figure 8

Comparison of numerically found profiles of the 2D LDVs with the fit provided by the deformed-tanh interpolation (19) (labeled D-tanh): (a) $S=1$ and $\lambda =1.1$ and (b) $S=2$ and $\lambda =1.5$. The propagation constant is $k=-4$ for $\alpha =0.6$ in (b), while $k=-1.5$ for the other vortices shown here.

Figure 9

Snapshots of the perturbed evolution of unstable 2D LDVs with different topological charges $S$: (a) a vortex with $S=1, \alpha =0.51$, and $k=-2.7$ turns into the eccentric vortex orbiting the center; (b) an unstable double vortex with $S=2, \alpha =1.2$, and $k=-1.8$ splits into a rotating pair of unitary vortices; (c) the transformation of an unstable double vortex with $S=2, \alpha =0.45$, and $k=-2.5$ into a broader, apparently stable, double vortex. All the plots are produced in the domain size $|x,y|<20$, as marked in (a).