###  V.4 Stable 2D two-component solitons with hidden vorticity

Proceeding to the consideration of two-component systems under the action of
the self-attractive nonlinearity and trapping potential, it is straightforward
to investigate stability of vortex solitons with equal wave functions of both
components (Yakimenko, Zaliznyak, and Lashkin, 2009). A more intriguing
possibility is to seek for stable two-component solitons with HV (hidden
vorticity), which are built of components carrying opposite winding numbers,
$S_{1,2}=\pm S$ (Brtka, Gammal, and Malomed, 2010).

In this case, the system of nonlinearly coupled GP equations for wave
functions $\psi_{1,2}$ of the two components is

$i\frac{\partial\psi_{1}}{\partial
t}=\left[-\frac{1}{2}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)-\left(|\psi_{1}|^{2}+\beta\left|\psi_{2}\right|^{2}\right)+\frac{1}{2}\left(x^{2}+y^{2}\right)\right]\psi_{1},$
(185a) $i\frac{\partial\psi_{2}}{\partial
t}=\left[-\frac{1}{2}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)-\left(|\psi_{2}|^{2}+\beta\left|\psi_{1}\right|^{2}\right)+\frac{1}{2}\left(x^{2}+y^{2}\right)\right]\psi_{2},$
(186)

The system is assumed to be fully symmetric, with strengths of the self-
attraction in each component and HO potential scaled to be $1$, like in the
single-component GP equation (5.1), while $\beta$, that may be both positive
and negative, is the relative strength of the interaction between the
components (attraction for $\beta>0$ and repulsion for $\beta<0$). In this
notation, the collapse threshold for the norm of each component of the
symmetric ($\psi_{1}=\psi_{2}$) soliton with $S_{1,2}=\pm 1$ follows from Eq.
(5.10):

$N_{\mathrm{TS}}^{(S=\pm 1)}\approx 24.1\left(1+\beta\right)^{-1}.$ (187)

HV states are stationary solutions of Eqs. (5.24), written in the polar
coordinates as

$\psi_{1}=\exp\left(-i\mu t+iS\theta\right)\phi(r),\psi_{2}=\exp\left(-i\mu
t-iS\theta\right)\phi(r),$ (188)

with $S\geq 1$ and real function $\phi(r)$ determined by the same radial
equation as in the case of the state with the explicit vorticity (i.e., with
identical phase terms $S\theta$ in both components, rather than $\pm S\theta$
in Eq. (5.26)):

$\mu\phi=-\frac{1}{2}\left(\frac{d^{2}\phi}{dr^{2}}+\frac{1}{r}\frac{d\phi}{dr}-\frac{S^{2}}{r^{2}}-r^{2}\right)-\left(1+\beta\right)\phi^{3}.$
(189)

While the density profile of the HV modes is the same as of the usual 2D
vortex solitons, up to obvious rescaling
$\phi\rightarrow\left(1+\beta\right)^{-1/2}\phi$ (provided that $\beta>-1$),
the stability is different. To address this point, a perturbed HV solution is
looked for as

$\psi_{1,2}(r,t)=[R(r)+u_{1,2}(r)\exp\left(\lambda
t+iL\theta\right)+v_{1,2}^{\ast}(r)\exp\left(\lambda^{\ast}t-iL\theta\right)]\exp\left(-i\mu
t\pm iS\theta\right),$ (190)

where $+S$ and $-S$ correspond to $\psi_{1}$ and $\psi_{2}$, respectively, and
$L$ is an integer azimuthal index, cf. Eq. (5.7). The substitution of this
expression in Eqs. (5.24) and the linearization with respect to small
perturbations $\left(u_{1,2},v_{1,2}\right)$ leads to the respective BdG
equations. Then, the numerical solution of the BdG equations produces the
stability chart for $S=1$, displayed in Fig. 14, in which the stability area
for the HV states is shown in the plane of $\left(N,\beta\right)$ ($N$ is
defined as the norm of one component). For $S\geq 2$, all the HV states are
completely unstable.

Figure 14: The stability chart for the HV (hidden-vorticity) 2D solitons with
$S_{1,2}=\pm 1$ (see Eq. (5.26)) in the plane of $\left(N,\beta\right)$,
produced by the numerical solution of Eqs. (5.24) and the respective BdG
equations for small perturbations. Here $N$ is the norm of one component of
the unperturbed state (the total norm is $2N$). In domains labeled by
different values of $L$ the HV states are unstable against perturbation modes
with the respective value of azimuthal index $L$, see Eq. (5.28). In the
“forbidden” area, bounded by curve $N\approx 24.1/\left(1+\beta\right)$ (see
Eq. (5.25)), vortex solitons with $S=1$ do not exist (source: Brtka, Gammal,
and Malomed 2010).

In other parts of the plane displayed in Fig. 14, instability domains
dominated by perturbation eigenmodes with different values of azimuthal index
$L$ are identified, as per the results of Brtka, Gammal, and Malomed (2010).
It is worthy to note an essential difference from the instability for the
usual vortex solitons which, as shown above in Fig. 9, is always dominated by
the perturbation eigenmode with $L=2$, hence the unstable vortex spontaneously
splits in two fragments (Fig. 10). Figure 14 demonstrates that the eigenmode
with $L=2$ remains dominant for the HV solitons at, roughly speaking,
$|\beta|<0.5$; on the other hand, $L=1$, $3$ and $4$ may be the azimuthal
indices of the leading perturbation eigenmode at $|\beta|>0.5$. When, in
particular, the leading unstable perturbation has $L=1$, it does not split the
HV soliton. Instead, the weak instability slowly deforms it into a rotating
crescent-shaped mode, as shown in Fig. 52 (eventually, this mode implodes
under the action of the intrinsic collapse).

Figure 15: The slowly developing instability of the HV soliton in the case
when the dominant perturbation eigenmode has azimuthal index $L=1$ (see Fig.
14). The top and bottom rows of panels display the distribution of the
densities in two components of the HV state at indicated moments of time, as
produced by simulations of Eqs. (5.24) with $\beta=0.5$. The unperturbed HV
soliton has the norm (in each component) $N=7.5$ and chemical potential
$\mu=0.83$ (source: Brtka, Gammal, and Malomed 2010).

On the other hand, the instability corresponding to the dominant eigenmodes
with $L=3$ or $4$ splits the HV soliton into three or four fragments,
respectively (later, the fragments are destroyed by the intrinsic collapse).
In particular, the instability dominated by $L=4$ is very strong, leading to
fast splitting of the HV soliton, as shown in Fig. 16.

Figure 16: The rapidly developing instability of the HV soliton in the case
when the dominant perturbation eigenmode has azimuthal index $L=4$ (see Fig.
14). The set of panels display identical distributions of the density in both
components at indicated moments of time. The results were produced by
simulations of Eq. (5.24) with $\beta=-0.5$. The unperturbed HV soliton has
the norm (in each component) $N=34.1$ and chemical potential $\mu=-0.2$
(source: Brtka, Gammal, and Malomed 2010).

##  VI Nonlinear trapping potentials

The trapping potentials considered in the previous chapter provide the
traditional method for maintaining and stabilizing localized states, which, by
itself, applies independently of the presence of nonlinearity in the system.
Nonlinear potentials, induced by spatial modulation of the local strength of
the cubic or other nonlinearity, offer a completely different method for the
creation of self-trapped states (quasi-solitons). An efficient implementation
of the latter method was proposed by Borovkova et al. (2011a,b), in the form
of the self-repulsive cubic term with the coefficient growing fast enough from
the center to periphery, as per Eqs. (4.28) and (4.31) or (4.32). This scheme
offers options for the creation of various localized states that would not
exist or would be unstable without the use of nonlinear potentials. Although
experimental realization of the scheme has not yet been reported, many
possibilities of its use have been explored theoretically. In particular, an
essential asset of the theoretical work in this direction is that, although it
is naturally based on numerical methods, many important results may be
obtained in an analytical form, approximately or exactly.

###  VI.1 The basic setting

The basic model providing self-trapping of various 2D and 3D modes in the
effective nonlinear trapping potential is (Borovkova et al. (2011a,b)),

$i\psi_{t}+\frac{1}{2}\nabla^{2}\psi-\sigma(r)|\psi|^{2}\psi=0,$ (191)

where $r$ is the radial coordinate in the 2D or 3D space in which Eq. (6.1) is
written, and the local self-defocusing coefficient, $\sigma(r)>0$, grows from
the center ($r=0$) to periphery ($r\rightarrow\infty$). This equation
conserves the total norm defined as per Eq. (4.13),

$N_{\mathrm{3D}}=\int\int\int
dxdydz\left|\psi\right|^{2},N_{\mathrm{2D}}=\int\int
dxdy\left|\psi\right|^{2},$ (192)

the Hamiltonian (cf. Eq. (4.15)),

$H_{\mathrm{3D}}=\frac{1}{2}\int\int\int
dxdydz\left[\left|\nabla\psi\right|^{2}+\sigma(r)|\psi|^{4}\right]$ (193)

(or its counterpart in the 2D case), and the vectorial angular momentum, given
above by Eq. (4.16). In the 2D case, the norm, Hamiltonian, and $z$-component
of the angular momentum are produced by obvious reduction of the 3D
expressions.

Stationary states with real chemical potential $\mu$ are sought for as
solutions to Eq. (6.1) in the form of

$\psi\left(r,t\right)=\exp\left(-i\mu t\right)w(\mathbf{r}),$ (194)

where $\mathbf{r}$ is the set of coordinates in the respective 2D or 3D space,
and function $w$ (generally, a complex one) satisfies the stationary equation,

$\mu w+\frac{1}{2}\nabla^{2}w-\sigma(\mathbf{r})|w|^{2}w=0.$ (195)

The fundamental condition necessary for the existence of physically relevant
self-trapped states produced by Eq. (6.5) is the _convergence_ at
$r\rightarrow\infty$ of the integral expression (6.2) which defines the norm
of the state. In particular, if $\sigma(r)$ grows asymptotically at
$r\rightarrow\infty$ as

$\sigma_{\mathrm{asympt}}(r)=\sigma_{0}r^{\alpha}$ (196)

with $\sigma_{0}>0$ and $\alpha>0$, it is easy to identify the convergence
condition by means of the TF approximation, which neglects term
$(1/2)\nabla^{2}w$ in Eq. (6.4) (a rigorous justification of the application
of the TF to models of the present type was developed by Malomed and
Pelinovsky (2015)). In this approximation, the solution is obvious,

$\left(|w|^{2}\right)_{\mathrm{TF}}=\mu/\sigma(r)$ (197)

(it exists for $\mu>0$), and the substitution of this in Eq. (6.2) immediately
shows that the integral converges under the condition

$\alpha>D.$ (198)

In fact, condition (6.8) secures the self-trapping for localized modes of all
types, i.e., not only fundamental ones, but also states with an intrinsic
structure, such as embedded vorticity (Borovkova et al. (2011a)). It is
relevant to mention that the same condition also secures the convergence of
the integral which defines Hamiltonian (6.3).

In the model with the asymptotic form (6.6) of the spatially modulated local
nonlinearity, the asymptotic solution including the term given by Eq. (6.7)
and the first post-TF correction, produced by term $(1/2)\nabla^{2}w$ at
$r\rightarrow\infty$, is

$w_{\mathrm{TF-
corrected}}(r)\approx\sqrt{\frac{\mu}{\sigma_{0}}}r^{-\alpha/2}+\frac{1}{4\sqrt{\mu\sigma_{0}}}\left[\frac{\alpha}{2}\left(\frac{\alpha}{2}-\left(D-2\right)\right)-m^{2}\right]r^{-\alpha/2-2}$
(199)

(the term $\sim m^{2}$ appears in the 2D case, representing the effect of the
vorticity with integer winding number $m$, see Eq. (6.11) below).

Expression (6.9) shows that the TF approximation yields the _asymptotically
exact_ expression for the soliton’s tail at $r\rightarrow\infty$, as the first
correction to it is negligibly small, hence the convergence condition, given
by Eq. (6.8), is always correct. Further, it is relevant to stress that
stationary equation (6.5) with the nonlinearity-modulation profile (6.6) is
_nonlinearizable_ : while the solution’s tail decays at $r\rightarrow\infty$,
its asymptotic form, determined by the TF approximation (6.7) and (6.9),
cannot be found from the linearization of Eq. (6.5), unlike the usual
situation for nonlinear equations in the free space.

###  VI.2 Typical examples of the self-trapped modes

Basic results illustrating the realization of the concept of the effective
nonlinear trapping potential were reported by Borovkova et al. (2011a), who
considered 2D version of Eq. (6.1) with

$\sigma(r)=1+r^{\alpha},$ (200)

whose asymptotic form complies with expression (6.6) (the 1D version of the
model was also considered in that work). Along with the fundamental 2D
solitons, their counterparts with embedded vorticity $m$ were considered too,
with Eqs. (6.4) and (6.5) replaced by

$\psi\left(r,t\right)=\exp\left(-i\mu t+im\theta\right)w(r),$ (201) $\mu
w+\frac{1}{2}\left(\frac{d^{2}w}{dr^{2}}+\frac{1}{r}\frac{dw}{dr}-\frac{m^{2}}{r^{2}}w\right)-\left(1+r^{\alpha}\right)w^{3}=0,$
(202)

where $\theta$ is the angular coordinate. In this case, the substitution of
the TF approximation (6.6), with $\sigma(r)$ taken as per Eq. (6.10), yields
the prediction for the total norm of the fundamental ($m=0$) solitons, which
is relevant for all dimensions, $D=1$, $2$, and $3$:

$N_{\mathrm{TF}}=\frac{2\pi^{D}\mu}{\alpha\sin\left(\pi D/\alpha\right)}.$
(203)

In particular, this expression provides the convergent norm precisely under
condition (6.8).

Typical examples of radial profiles of stationary states, produced by the
numerical solution of Eq. (6.12) with $m=0$, $1$, and $2$, are displayed in
Fig. 17, and families of the respective solution families, characterized by
dependences $N(\mu)$ and $N(\alpha$), are shown in Fig. 55\. The latter plot
includes the TF prediction (6.13) for $m=0$, showing that the TF approximation
is quite accurate at all values of $\alpha$. Note also that the linear
dependence of $N_{\mathrm{TF}}$ on $\mu$, predicted by Eq. (6.13), is in
perfect agreement with the linear line for $m=0$ in Fig. 18(a).

Figure 17: (a) Radial profiles of self-trapped solutions produced by the
numerical solution of Eq. (6.12) with $\alpha=5$ and a fixed chemical
potential, $\mu=10$, for vorticities $m=0$, $m=1$, and $m=2$. (b) Vortex
states for $\alpha=5$, $m=1$ and $\mu=5$, $10$, and $20$ (curves 1, 2, and 3,
respectively). The red curve shows the respective nonlinearity-modulation
profile, defined as per Eq. (6.10) (source: Borovkova et al., 2011a).  Figure
18: (a) The norm of 2D self-trapped states (denoted $U$, instead of $N$)
produced by the numerical solution of Eq. (6.12), vs. the chemical potential
(denoted $-b$, instead of $\mu$), for $\alpha=5$ and two values of the
vorticity, $m=0$ and $2$. (b) Dependences of the 2D norm on power $\alpha$
from Eq. (6.10) for a fixed chemical potential, $\mu=10$, and three values of
the vorticity: $m=0$, $1$, and $2$. Value $\alpha_{\mathrm{cr}}=2$ is
determined by Eq. (6.7), with the norm diverging at
$\alpha\geq\alpha_{\mathrm{cr}}$. The red curve shows the analytical
prediction for $m=0$ produced by the TF approximation in the form of Eq.
(6.13) (source: Borovkova et al., 2011a).

###  VI.3 Stability of the self-trapped states

Stability of the 2D self-trapped states produced by Eq. (6.12) against small
perturbations was investigated by Borovkova et al. (2011(a)) by means of the
numerical solution of the respective BdG equations, cf. Eqs. (5.8). As a
result, it was concluded that the entire family of fundamental solitons, with
$m=0$, is stable, in accordance with the fact that the respective $N(\mu)$
dependence in Fig. 18(a) obeys the anti-VK criterion, $dN/d\mu>0$, which is a
necessary stability condition for families of localized states supported by
the self-repulsive nonlinearity (on the contrary to the opposite VK criterion
per se, in the case of self-attraction), as shown by Sakaguchi and Malomed
(2010). In the notation adopted Fig. 6.2(a), the anti-VK criterion means
$dU/db<0$.

Unlike the fundamental self-trapped states, ones with embedded vorticity tend
to be vulnerable to instability, their stabilization being possible for
sufficiently steep nonlinearity-modulation profiles in Eq. (6.10); in
particular, the entire family of the vortices with $m=1$ was found by
Borovkova et al. (2011a) to be completely stable at $\alpha\geq 8$, being only
partly stable at smaller $\alpha$ (e.g., for $-\mu>30$ at $\alpha=5$).

Direct simulations demonstrate that instability of the vortex mode with $m=1$
expels the vortex’ pivot from the central position, and drives its drift
towards periphery. As shown in Fig. 19(a), the pivot eventually vanishes in
the peripheral area. The angular momentum of the original vortex state is also
ousted to the periphery where, in actual numerical simulations, it is
eliminated by a numerical absorber providing stability of the numerical
scheme. In the case when a double vortex with $m=2$ is unstable, small
perturbations may split it in two unitary vortices, one staying as a stable
one with $m=1$ at the center, while the other one spontaneously drifting to
the periphery, where it vanishes eventually. The latter scenario of the
instability development is shown in Fig. 19(b).

Figure 19: (a) The development of the drift instability of the vortex state
with $m=1$ and $\mu=3$, produced by simulations of Eq. (6.1) with the
nonlinearity-modulation profile (6.10), where $\alpha=3.5$. Colors denote
values of the local density, $\left|\psi\left(x,y,t\right)\right|^{2}$, with
variable $t$ replaced by $\xi$. (b) The same for the splitting-drift
instability of the double vortex, with $m=2$, $\mu=20$, and $\alpha=4$
(source: Borovkova et al., 2011a).

###  VI.4 Exact, approximate, and numerical results for the anti-Gaussian
spatial-modulation profile

####  VI.4.1 The formulation and analytical findings

Stronger results for self-trapped states supported by the spatial modulation
of the cubic repulsive term were reported by Borovkova et al. (2011b) in the
framework of Eq. (6.1) with a very steep (anti-Gaussian) modulation profile
(cf. the algebraic one in Eq. (6.10)):

$\sigma(r)=\exp\left(\alpha r^{2}\right).$ (204)

Note that the scaling invariance of Eq. (6.1) with $\sigma(r)$ taken as per
Eq. (6.14) implies that the solutions actually depend not separately on
$\alpha$ and $\mu$, but solely on the ratio, $\mu/\alpha$.

A remarkable finding is that the 2D version of Eqs. (6.1) and (6.5) with
$\sigma(r)$ taken as per Eq. (6.14) admits a particular _exact solution_ for
the self-trapped mode with embedded vorticity $m=1$:

$\psi=\frac{\alpha}{\sqrt{2}}r\exp\left(-i\mu
t+i\theta-\frac{\alpha}{2}r^{2}\right),\mu=2\alpha$ (205)

(the exact solution is available for the single value of $\mu$, as written in
Eq. (6.15)). As concerns the asymptotic form of the solution’s tail at
$r\rightarrow\infty$, it is determined by the balance of the cubic and
gradient terms in Eq. (6.5):

$w(r)=\left[\frac{\alpha}{\sqrt{2}}r+\frac{2\mu-\alpha\left(D+2\right)}{2\sqrt{2}\alpha
r}+\mathcal{O}\left(\frac{1}{r^{3}}\right)\right]\exp\left(-\alpha
r^{2}/2\right).$ (206)

This asymptotic solution is a _universal_ one, as, apart from the correction
$\mathcal{O}\left(1/r^{3}\right)$, it does not depend on $\mu$, nor on the
presence of embedded vorticity $m$, and only the correction $\sim 1/r$ depends
on the spatial dimension, $D$. Vanishing of the latter term in the case of
$D=2$, $\mu=2\alpha~{}$agrees with the existence of the exact solution given
by Eq. (6.15).

Similar to the case of modulation profile (6.10), the one represented by Eq.
(6.14) makes equation (6.5) nonlinearizable, as the tail solution (6.16),
although it vanishes at $r\rightarrow\infty$, cannot be found in the correct
form, unless the nonlinear term is kept in the analysis. On the other hand,
the approximation leading to Eq. (6.16) is different from the TF
approximation, which was used above to derive expression (6.9), as the
gradient term is negligible in the latter case.

In any spatial dimension $D$, fundamental solitons (with $m=0$) produced by
Eqs. (6.5) and (6.14) can be very accurately approximated by the variational
method. To this end, one notes that the 2D ($D=2$) or 3D ($D=3$) Lagrangian of
Eq. (6.5) with real field $w(r)$ is

$L_{w}=\frac{1}{2}\int_{0}^{\infty}\left[\mu
w^{2}-\frac{1}{2}\left(\frac{dw}{dr}\right)^{2}-\frac{1}{2}\sigma(r)w^{4}\right]r^{D-1}dr.$
(207)

A natural ansatz approximating the fundamental soliton is

$w(r)=A\exp\left(-\frac{\alpha}{2}r^{2}\right),$ (208)

with the norm

$N=\left(\pi/\alpha\right)^{D/2}A^{2},$ (209)

where amplitude $A$ is a free variational parameter.

The substitution of ansatz (6.18) in Lagrangian (6.29) yields the following
effective Lagrangian, in which $A^{2}$ is expressed in terms of $N$, by means
of relation (6.19):

$L_{\mathrm{2D}}=\frac{1}{4\pi}\left(\mu-\frac{\alpha}{2}\right)N-\frac{\alpha}{8\pi^{2}}N^{2},$
(210)
$L_{\mathrm{3D}}=\frac{1}{8\pi}\left(\mu-\frac{3\alpha}{4}\right)N-\frac{\alpha^{3/2}}{16\pi^{5/2}}N^{2}.$
(211)

Finally, the Euler-Lagrange equation, $\partial L/\partial N=0$, predicts the
following relation between the norm and chemical potential:

$N=\left(\pi/\alpha\right)^{D/2}\left(\mu-D\alpha/4\right).$ (212)

Numerically found dependences $N(\alpha)$ for the fundamental solitons are
indistinguishable from the VA prediction (6.21), in both cases $D=2$ and $3$
(Borovkova et al. 2011b).

A two-component extension of Eqs. (6.1), in the form of

$i\left(\psi_{1,2}\right)_{t}+\frac{1}{2}\nabla^{2}\psi_{1,2}-\sigma(r)\left(|\psi_{1,2}|^{2}+C\left|\psi_{2,1}^{2}\right|\right)\psi=0,$
(213)

with an XPM coefficient $C>0$, was elaborated by Kartashov et al. (2011).

####  VI.4.2 Vortices and rotary structures

Shapes of fundamental and vortical self-trapped modes created by the anti-
Gaussian modulation profile (6.14) are quite similar to those shown above in
Fig. 17 for profile (6.10). However, results for the stability of vortices are
different: unlike the above model, where vortices even with the lowest winding
number, $m=1$, tend to be unstable, see Fig. 56(a), all the 2D vortex states
supported by the anti-Gaussian profile (6.14) are _completely stable_ , as
well as all the fundamental modes in all dimensions (Borovkova et al., 2011b).
Alternation of stability and instability commences from $m=2$, as shown in
Fig. 20. In case the vortex modes with $m=3$ and $3$ are unstable, they do not
start drift to $r\rightarrow\infty$, unlike what is shown above in Fig. 19(a);
instead, they spontaneously split in robust rotating sets of two or three
unitary vortices, as seen in Fig. 21.

Figure 20: Self-trapped vortex modes with $m=2$, generated by Eqs. (6.1) and
(6.14), are stable and unstable in white and shaded sectors of the parametric
plane $\left(\alpha,b\equiv b\equiv-\mu\right)$. Note that the scaling
invariance of the underlying equation implies that all boundaries of the
stability sectors are strict straight lines, $\mu/\alpha=\mathrm{const}$, as
seen in the figure (source: Borovkova et al., 2011b).  Figure 21: (a)
Relaxation to the stationary state of a perturbed stable vortex solution of
Eqs. (6.1) and (6.17) with $m=2$ and $\mu=17$. (b) Spontaneous splitting of an
unstable vortex with $m=2$ and $\mu=11$ into a steadily rotating pair of
unitary vortices. (c) The splitting of the unstable vortex with $m=3$ and
$\mu=9$ into a rotating triplet of unitary vortices. In this figure $\xi\equiv
t$ (source: Borovkova et al., 2011b).

The same 2D model, based on Eq. (6.1) with the modulation profile given by Eq.
(6.14), also supports stable rotational motion of a single vortex with the
pivot shifted off the center, as well of clusters composed of several vortices
(Kartashov et al., 2017). The respective GP equation, written in the reference
frame rotating with angular velocity $\omega$, is written as

$i\psi_{t}+\frac{1}{2}\nabla^{2}\psi+i\omega\left(x\frac{\partial}{\partial
y}-y\frac{\partial}{\partial x}\right)\psi-\sigma(r)|\psi|^{2}\psi=0.$ (214)

($\omega>0$ corresponds to the counter-clockwise rotation).

A snapshot of the density distribution in a stable vortex moving along a
circular trajectory is displayed in Fig. 22(a). Further, separated vortex and
antivortex, with winding numbers $\pm 1$, may form a non-rotating dipole, as
shown in Fig. 22(b), and a non-rotating quadrupole built of alternating
vortices and antivortices is shown in Fig. 22(d) (generally speaking, vortex
dipoles and quadrupoles are subject to weak instability in the absence of
rotation (Möttönen et al., 2005)). In the usual confined BEC, vortex-
antivortex dipoles were experimentally created by Freilich et al. (2010).

When the vortex dipole is unstable, its evolution may expel one vortex to
periphery, where it eventually disappears, while the remaining antivortex
falls onto the center, thus converting the unstable dipole into a stable
stationary vortex with the unitary topological charge. This instability-
development scenario resembles the one displayed above in Fig. 19(b). The
application of the rotation ($\omega\neq 0$ in Eq. (6.23)) helps to stabilize
vortex-antivortex clusters. Simultaneously, Figs. 22(c), (e), and (f)
demonstrate that dipoles, quadrupoles, and sextupoles (a set of three vortices
alternating with three antivortices), which are symmetric complexes in the
absence of the rotation, are deformed by the rotation. The deformation is
caused by an effective Coriolis force acting with opposite signs on individual
vortices and antivortices. In particular, Fig. 22(c) shows that this force
pushes the vortex and antivortex, belonging to the dipole, towards $r=0$ and
$r\rightarrow\infty$, respectively, and Fig. 22(f) shows that the sextupole,
which has a symmetric hexagonal shape at $\omega=0$, is deformed into a
triangular structure. Finally, if $\omega$ is too large, the fast rotation
destroys the clusters.

Figure 22: (a): A snapshot of the density distribution in a 2D vortex with
$m=1$ and norm $N=58.4$, which performs orbiting motion in the framework of
Eqs. (6.23) and (6.14) with $\alpha=0.5$ and $\omega=1.04$. Panels (b) and
(d): the density distribution in stationary spatially symmetric dipole and
quadrupole vortex-antivortex clusters, produced by Eqs. Eqs. (6.23) and (6.14)
with $\alpha=0.5$, $\omega=0$, and $\mu=10$. (c) and (e): The same clusters
stabilized and simultaneously deformed by the rotation. (f) The density
distribution in a stable rotating sextupole with $\alpha=0.5$, $\mu=20$, built
as a cluster of three vortices alternating with three antivortices. Each panel
shows an area in the $\left(x,y\right)$ plane of size $\left(-5,+5\right)$ in
each direction (source: Kartashov et al., 2017).

Additional results for 2D and 3D fundamental and vortex solitons were produced
by means of numerical methods, in similar models with more sophisticated
(specially devised) profiles of the spatial modulation of the strength of the
self-repulsive cubic nonlinearity (Tian et al. 2012; Wu et al. 2013). Another
direction of the work was the analysis of a similar model with the local
quintic self-repulsive nonlinearity (Zeng and Malomed, 2012).

####  VI.4.3 A discrete 2D model with the exponential nonlinearity-modulation
profile

It is relevant to mention that a discrete version of the models considered
here was elaborated by Kevrekidis et al. (2015). The corresponding lattice
(discrete) NLS equation, with the 2D discrete coordinates $\left(m,n\right)$
is

$i\frac{d\psi_{mn}}{dt}=-\varepsilon\left(\psi_{m+1,n}+\psi_{m-1.n}+\psi_{m,n+1}+\psi_{m,n-1}-4\psi_{m,n}\right)+\sigma\left(m,n\right)\left|\psi_{m,n}\right|^{2}\psi_{m,n}~{},$
(215)

cf. Eqs. (2.163) and (6.1), with the exponential profile of the spatial
modulation of local nonlinearity,

$\sigma\left(m,n\right)=\exp\left(2\left(|m|+|n|\right)\right),$ (216)

cf. Eq. (6.14). Various species of 2D discrete solitons supported by Eqs.
(6.24) and (6.25) have been constructed, and their stability has been
investigated. Among them were two types of discrete vortex solitons, viz., OS-
centered vortex crosses, whose shape is similar to that displayed above in
Fig. 24, and IS-centered vortex squares).

###  VI.5 Gyroscope solitons (vortex tori) in 3D

Full consideration of the structure and dynamics of fundamental and vortical
solitons in the 3D version of Eq. (6.1) with the modulation profile defined by
Eq. (6.14), where $\alpha=1/2$ is fixed by means of rescaling, was presented
by Driben et al. (2014a). The respective stationary equation (6.5) for modes
with embedded vorticity $S$, introduced as per Eq. (6.11) with real function
$w\left(r,z\right)$, takes the following form, in cylindrical coordinates
$\left(r,z\right)$:

$\mu w=-\frac{1}{2}\left(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}-\frac{S^{2}}{r^{2}}\right)w+\exp\left(\frac{1}{2}\left(r^{2}+z^{2}\right)\right)w^{3}.$
(217)

As shown above, the VA based on Eqs. (6.18), (6.19), and (6.21) with $D=3$
produces a very accurate prediction for the family of fundamental solitons,
with $S=0$. For 3D solitons with embedded vorticity $m\geq 1$, the VA is less
accurate. However, in this case quite relevant is the TF approximation.
Indeed, neglecting the derivative terms in Eq. (6.26), one obtains

$w_{\mathrm{TF}}^{2}=\begin{array}[]{c}0,~{}\mathrm{at}~{}r^{2}<m^{2}/\left(2\mu\right),\\\
\left[\mu-S^{2}/\left(2r^{2}\right)\right]\exp\left(-\frac{1}{2}\left(r^{2}+z^{2}\right)\right),~{}\mathrm{at}~{}r^{2}>S^{2}/\left(2\mu\right).\end{array}$
(218)

The TF approximation (6.27) produces the norm of the vortex state in the form
of

$N_{\mathrm{TF}}(\mu)=4\left(2\pi\right)^{3/2}\mu\exp\left(-\frac{S^{2}}{4\mu}\right)\int_{0}^{\infty}\frac{e^{-R}RdR}{4R+S^{2}/\mu}$
(219)

(here, the integral factor should be computed numerically).

Numerically found shapes of typical _stable_ 3D solitons with vorticities
$S=0$, $1$, and $2$ are displayed in Fig. 23 (because of their shape, the
solitons with $S\neq 0$ are often called vortex tori, alias 3D vortex rings).
It was found that all the states with $S=0$ and $S=1$ are stable, while ones
with $S=2$ and $3$ feature alternating stability and instability zones.
Generic examples of the dynamics of stable and unstable solitons are presented
in Fig. 24. It is seen that a stable soliton cleans itself of a strong initial
perturbation, while unstable ones with $S=2$ and $3$ spontaneously split in
rotating clusters of two and three unitary vortices, respectively.

Figure 23: Typical examples of stable 3D self-trapped modes produced by the
numerical solution of Eq. (6.26) with $\mu=16$ and $S=0$, norm $N=246.1$
(left), $S=1$, $N=225.2$ (central), and $S=2$, $N=191.7$. The modes are
displayed by means of isosuraces, $\left|w\left(x,y,z\right)\right|^{2}=3$
(source: Driben et al., 2014a).  Figure 24: Generic examples of the perturbed
dynamics of 3D solitons with embedded vorticity $S=1$ in the top row (a stable
vortex torus with $\mu=16$; the plots are shown at $t=0$, $t=230$, $t=310$);
$S=2$ in the middle row (an unstable vortex torus with $\mu=13$, shown at
$t=0$, $t=230$, $t=310$); and $S=3$ (an unstable vortex torus with $\mu=25$,
shown at $t=0$, $t=165$, $t=180$). $N=191.7$. The top and middle rows display
density isosurfaces $\left|\psi\left(x,y,z\right)\right|=2.5$, and bottom one
displays $\left|\psi\left(x,y,z\right)\right|=3$ (source: Driben et al.,
2014a).

Families of the completely stable solitons with $S=0$ and $1$ are
characterized by dependences $N(\mu)$ shown in Fig. 25. It is clearly seen
that the VA for the fundamental solitons with $S=0$ and the TF approximation
for ones with $S=1$ produce the $N(\mu)$ lines which are indistinguishable
from their numerically generated counterparts.

Figure 25: Black lines show the norm of the fundamental solitons (a) and ones
with $S=1$ (b) vs. their chemical potential. Chains of green dots represent
the VA prediction for the fundamental solitons, given by Eq. (6.21) with
$D=3$, and the TF prediction for $S=1$, given by Eq. (6.28). The empty circle
in (a) and bold red dot in (b) correspond, respectively, to the left plot in
Fig. 60, and the top row in Fig. 24 (source: Driben et al., 2014a).

The stable 3D solitons with embedded vorticity demonstrate macroscopic
dynamics resembling that of mechanical gyroscopes. An example if displayed in
Fig. 63. To generate it, Eqs. (6.1) and (6.14) with $\alpha=1/2$ was
simulated, with the input taken as the stable soliton with $S=1$, shown in
Figs. 60 and 61, to which a torque is applied at $t=0$, multiplying the
soliton’s wave function by the phase factor

$T=\exp\left(i\beta z\tanh\left(x/x_{0}\right)\right),$ (220)

with real $\beta$. This factor adds a nonzero component $M_{y}$ of the angular
momentum (see Eq. (4.16)). In particular, if the unperturbed vortex is taken
in the form defined by the TF approximation, as per Eqs. (6.11), (6.27), and
(6.28), the momentum produced by torque (6.29) is

$M_{y}=\frac{\beta}{2x_{0}}\left[N-\left(2\pi\right)^{3/2}\left(\mu-\frac{S^{2}}{4}\right)\right],$
(221)

while the the momentum of the unperturbed vortex soliton is

$M_{z}=SN.$ (222)

Then, as it may be expected, the application of the torque to the gyroscope
sets it in the state of precession, as shown in Fig. 63. This dynamical regime
corresponds to the total angular momentum with components (6.31) and (6.30).

Figure 26: The gyroscopic precession of the 3D vortex soliton (torus) with
$S=1$ and $\mu=16$, initiated by the application of torque (6.29) with
$\beta=4$ and $x_{0}=5$. The plots are produced by simulations of Eqs. (61.)
and (6.14) with $\alpha=1/2$. Three images in the top row correspond to times
$t=90$, $90.8$, and $91.5$. Time $\Delta t=1.5$ between the first and third
images is equal to the precession period. Green arrows indicate the
corresponding orientations of the axle of the precessing gyroscope. The bottom
plot shows a set of instantaneous orientations of the axle, the green ones
corresponding to those displayed in the top row (source: Driben et al.,
2014a).

###  VI.6 Hopfions: 3D solitons with two independent topological charges

As shown above in Fig. 23, the density distribution in 3D solitons with
embedded vorticity $m$ is shaped as a torus (“donut”). An additional
possibility is to impose intrinsic twist, characterized by integer number $s$,
onto the torus, in an attempts to create self-trapped modes carrying two
independent topological charges. States of such a type were originally
predicted in the Faddeev-Skyrme model (Faddeev, 1976; Faddeev and Niemi,
1997), based on a set of three real scalar fields. Soliton-like states in that
system realize the Hopf map, $R^{3}\rightarrow S^{2}$, therefore they are
called hopfions.

In usual settings, the creation of hopfions requires, at the minimum level,
two complex components (Sutcliffe and Manton, 2004; Sutcliffe, 2007). A unique
possibility offered by the 3D equation (6.1) with the modulation profile
(6.14) is to predict single-component stable hopfions (Kartashov et al.,
2014). The respective stationary solutions were looked for, in the cylindrical
coordinates $\left(r,\theta,z\right)$, as

$\psi=\exp\left(-i\mu t+im\theta\right)\,w\left(r,z\right),$ (223)

with complex field $w$ satisfying equation

$\mu w=-\frac{1}{2}\left(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}-\frac{m^{2}}{r^{2}}\right)w+\exp\left(\frac{1}{2}\left(r^{2}+z^{2}\right)\right)|w|^{2}w.$
(224)

Examples of hopfions with sets of topological numbers $\left(s=1,m=0\right)$,
$\left(s=1,m=1\right),$ $\left(s=1,m=1\right)$, and $\left(s=1,m=2\right)$,
found as numerical stationary solutions of Eq. (6.1), are shown, from the top
to bottom, in the left images of Fig. 27. Solutions with $s>1$ do not exist
(note that the solution with $S=0$ and $s=1$, i.e., zero overall vorticity,
exists, as shown in the top row of Fig. 27). Further, the evolution of the
hopfions, displayed in Fig. 27, demonstrates that all the hopfions with
$\left(s=1,m=0\right)$ are stable, ones with $\left(s=m=1\right)$ may be
stable or unstable, depending on their chemical potential (or norm), and all
the states with $\left(s=1,m\geq 2\right)$ are completely unstable against
spontaneous breakup (the results for the hopfions with $s=-1$ are the same).

Figure 27: Four rows display, from left to right, the distribution of the
density in hopfion stationary solutions of Eq. (6.33), and their evolution,
governed by the underlying equations (6.1) and (6.14) (with $\alpha=1/2$).
Parameters (twist number $s$ of the intrinsic torus, overall vorticity $m$,
and chemical potential $\mu$) are, from the top to bottom:
$\left(s=1,m=0,\mu=15\right)$, $\left(s=1,m=1,\mu=16\right)$,
$\left(s=1,m=1,\mu=22\right)$, and $\left(s=1,m=2,\mu=22\right)$. Snapshots of
the evolving hopfions are taken at moments of time indicated in the figure. In
the second row, the density isosurface is drawn as
$\left|\psi\left(x,y,z\right)\right|^{2}=0.2$, and in all others is it
$\left|\psi\left(x,y,z\right)\right|^{2}=1$ (source: Kartashov et al., 2014).

Shapes of the hopfions displayed in Fig. 27 suggest that the respective
solution to Eq. (6.26) may be approximated by the following ansatz, proposed
by Kartashov et al. (2014) (assuming $s\geq 0$ and $m\geq 0$):

$w=A\left[\left(r-R\right)+iz\right]^{s}r^{m}\exp\left[-\frac{1}{2}\left(r^{2}+z^{2}\right)\right].$
(225)

Here variational parameter $R$ is the radius of the coiled core of the
hopfion, at which the field must vanish, in the midplane drawn through $z=0$,
due to the presence of the vorticity with winding number $s$ in the cross
section of the torus. The norm of this ansatz is

$N=\left(2\pi\right)^{3/2}A^{2}\left(R^{2}-\sqrt{2\pi}R+3\right).$ (226)

The Lagrangian of Eq. (6.33) is (cf. Lagrangian (6.17) for the spherically
symmetric real field):

$L_{w}=\int_{0}^{\infty}rdr\int_{-\infty}^{+\infty}dz\left[\left(\mu-\frac{m^{2}}{2r^{2}}\right)|w|^{2}-\frac{1}{2}\left(\left|\frac{\partial
w}{\partial r}\right|^{2}+\left|\frac{\partial w}{\partial
z}\right|^{2}\right)-\frac{1}{2}\exp\left(\frac{1}{2}\left(r^{2}+z^{2}\right)\right)|w|^{4}\right].$
(227)

The substitution of ansatz (6.33) in expression (6.36) yields the respective
effective Lagrangian, in which $A^{2}$ is replaced by $N$ according to Eq.
(6.35). The ensuing Euler-Lagrange equation,
$\partial\left(L_{w}\right)_{\mathrm{eff}}/\partial R=0$, leads to a
cumbersome but usable equation which determines $R$ as a function of $N$. It
takes a rather simple form for “heavy” hopfions, in the limit of
$N\rightarrow\infty$ (which may be considered as the TF limit):

$2\left(4-\pi\right)R_{\infty}^{3}-3\sqrt{2\pi}R_{\infty}^{2}+4(2\pi-3)R_{\infty}=3\sqrt{2\pi}.$
(228)

A relevant root of Eq. (6.37) is

$R_{\infty}\approx 1.06.$ (229)

Numerical and analytical findings for hopfion families are presented in Fig.
28 by means of $R(N)$ dependences, which represent the most relevant property
of the states. The figure includes results for the stability, produced by
means of both numerical solution of the respective BgG equations and direct
simulations of the perturbed evolution in the framework of Eq. (6.1) (in
particular, some results of the simulations are displayed in Fig. 28). It is
seen that the VA prediction for $R(N)$ becomes very close to its numerical
counterpart at $N\gtrsim 100$. Actually, Eq. (6.38) yields an _asymptotically
exact_ value of the radius of the coiled core of the hopfion in the limit of
$N\rightarrow\infty$. An additional prediction of the VA, which is also very
close to the numerical findings, is that the radius of the cross section of
the hopfion’s toroidal core shrinks with the increase of $N$ (unlike the
asymptotically constant overall radius of the core, given by Eq. (6.38)):

$\rho_{\mathrm{core}}\approx\left(2\pi^{3}\right)^{1/4}/\sqrt{N}.$ (230)

Figure 28: Continuous black and red lines represent dependences of radius $R$
of the coiled axis of the hopfion, along which the wave function vanishes, on
its norm, as obtained form the numerical solution of Eq. (6.33). Black and red
segments correspond, respectively, to stable and unstable hopfions (the
stability boundary splits the solution branch with $s=m=1$ as per Eq. (6.40)).
The chain of green dots represents the VA prediction based on ansatz (6.34).
The lower blue curve shows the $R(N)$ dependence for a stable hopfion family
with $s=1$ and $m=0$, produced by Eq. (6.1) with a smoother spherically
isotropic modulation profile, $\sigma(r)=1+r^{6}$, instead of one defined by
Eq. (6.14) (source: Kartashov et al., 2014).

As concerns the stability, Fig. 28 demonstrates, as mentioned above, that the
hopfions with $s=1,m=0$ are completely stable, the ones with $s=m=1$ are
stable at

$\mu\geq\mu_{\min}\approx 11.2,$ (231)

and the modes with $s=1,m\geq 2$ are completely unstable.

Finally, the application of torque (6.30), with the axis belonging to the
plane of the coiled axis of the hopfion, may readily set the stable hopfion in
mechanical rotation around the torque’s axis. An example is displayed in Fig.
29 for the hopfion with $s=1,m=0$.

Figure 29: Stable rotation of the hopfion with $s=1,m=0$, initiated by
multiplying the stationary wave function by torque factor (6.30) with
$\beta=x_{0}=5$. Densities isosurfaces,
$\left|\psi\left(x,y,z\right)\right|^{2}=0.2$, are plotted at times $t=141$,
$156$, and $177,$ the rotation period being $T\approx 48$ (source: Kartashov
et al., 2014).

###  VI.7 Vortex-antivortex hybrids in the peanut-shaped modulation profile

####  VI.7.1 Composite modes with opposite vorticities

Another species of unusual 3D self-trapped states is supported by Eq. (6.1) in
which the anti-Gaussian isotropic modulation profile (6.14) is replaced by a
“peanut”-shaped one, which is stretched along axis $z$ by distance $d$. It was
introduced by Driben et al. (1914b) in terms of the cylindrical coordinates:

$\sigma\left(r,z\right)=\exp\left[\frac{1}{2}\left(r^{2}+\left(|z|-\frac{d}{2}\right)^{2}\right)\right].$
(232)

If $d$ is large enough, this modulation profile makes it possible to create
“hybrid states”, alias composite ones, in the form of vortex tori with
_opposite winding numbers_ , $S=\pm 1$, trapped in the top and bottom lobes of
the “peanut”. Examples of both the usual vortex mode, with winding numbers
$S=1$ in both lobes, i.e., with

$\psi=e^{iS\theta}w\left(r,z)\right),$ (233)

and hybrids, with vorticities $\pm 1$ and $\pm 2$, are presented in Fig. 30.

Figure 30: Density profiles of 3D vortex modes supported by Eq. (6.1) with the
“peanut”-shaped modulation profile of the local nonlinearity strength, defined
as per Eq. (6.41). The profiles are shown by means of isosurfaces
$\left|\psi\left(x,y,z\right)\right|^{2}=0.2$. The left panel displays a usual
stable vortex state, taken in the form of Eq. (6.42), with equal winding
numbers $S=1$ in both lobes of the peanut. The central and right panels
represent “hybrids” (composite modes), with opposite vorticities, $S/-S$, in
the top and bottom lobes (note the strong breaking of the axial symmetry
exhibited by the hybrids). The left and right panels correspond to $d=3$ in
Eq. (6.41) and chemical potential $\mu=10$. The central panel corresponds to
$d=5$ and $\mu=7$. The usual and hybrid vortex modes with vorticities $1$ and
$\pm 1$, shown in the left and central panels, are stable. The hybrid with
vorticities $\pm 2$, shown in the right panel, is unstable (source: Driben et
al., 2014b).

The hybrid states may be represented by an approximate stationary solution in
the following form:

$\psi=e^{-i\mu
t}\left[w_{+}\left(r,z\right)e^{iS\theta}+w_{-}\left(r,z\right)e^{-iS\theta}\right].$
(234)

The substitution of this ansatz in Eq. (6.1) leads, in the approximation which
neglects angular harmonics $\exp\left(\pm 3iS\theta\right)$, to a system of
two coupled stationary equations for real functions $w_{\pm}$:

$\left[\mu+\frac{1}{2}\left(\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}-\frac{S^{2}}{r^{2}}+\frac{\partial^{2}}{\partial
z^{2}}\right)\right]\phi_{\pm}=\sigma\left(r,z\right)\left[2\left(\phi_{\mp}\right)^{2}+\left(\phi_{\pm}\right)^{2}\right]\phi_{\pm},$
(235)

where $\sigma\left(r,z\right)$ is taken as per Eq. (6.41). Numerical solution
of Eqs. (6.44) predicts shapes of the hybrids which are quite close to those
produced, as stationary solutions, by full equation (6.1) with the modulation
profile (6.41).

The hybrid modes, combining opposite azimuthal harmonics, break the axial
symmetry of the resulting states. This can be easily seen looking at ansatz
(6.42) in the midplane, $z=0$, where $\phi_{\pm}(r,z=0)$ amount to the single
real function, $\phi_{0}(r)$:

$\left|\psi\left(r,\theta,z=0\right)\right|^{2}=4\phi_{0}^{2}(r)\cos^{2}\left(S\theta\right).$
(236)

The strong azimuthal pattern exhibited by Eq. (6.45) (it resembles the so-
called azimuthon states known in 2D systems (Desyatnikov, Sukhorukov, and
Kivshar, 2005)) models the actual structure seen in the middle panel of Fig.
67.

The study of stability of the hybrids was performed by Driben et al. (2014b)
by solving the respective BdG equations for small perturbations, and also by
means of direct simulations of the perturbed evolution. As a result, it was
found that the hybrids with vorticities $\pm 1$ may be stable provided that
the stretching parameter $d$ in Eq. (6.41) exceeds a minimum value,

$d>\left(d_{\min}\right)_{S=+1,-1}\approx 4.$ (237)

In particular, the hybrids are stable at $\mu<15.8$ (alias $N<394.9$) at
$d=4.5$, and at $\mu<13.5$ ($N<329.3$) at $d=5$. The instability of the
hybrids at larger values of $\mu$ is explained by the fact that (as shown, in
particular, by the TF approximation – see Eq. (6.37)) the increase of $\mu$
makes the localized modes broader, hence the stretching parameter $d$ becomes
relatively smaller, failing to produce the stabilization. In direct
simulations, those hybrids with $S=\pm 1$ which are unstable develop the
instability in the form of corrugation of the mode’s central axis. All hybrids
with vorticities $\pm S$ are unstable for $S\geq 2$.

####  VI.7.2 Fundamental-vortex complexes

The same model, based on Eqs. (6.1) and (6.41), admits dynamical composite
modes, in the form of fundamental and vortex states juxtaposed in the bottom
and top lobes of the peanut-shaped trapping profile. As shown in Fig. 31, this
complex may readily form a state in which the vortex component with $S_{1}=1$
performs completely stable periodic precession on top of the fundamental
“base”, with $S_{2}=0$. A broad stability area for the precessing composite
states was found provided that $d$ exceeds a minimum value,

$d>\left(d_{\min}\right)_{S_{1,2}=1,0}\approx 4.8,$ (238)

cf. Eq. (6.46).

Figure 31: An example of robust periodic precession of a vortex mode with
$S=1$ on top of the immobile zero-vorticity one, which are trapped,
respectively, in the top and bottom lobes of the “peanut-shaped” structure
(6.41) with $d=5$. The initial state was built taking halves of the respective
stationary states with $\mu=15$. The motion is displayed by means of density
isosurfaces $\left|\psi\left(x,y,z\right)\right|^{2}=1$, produced by
simulations of Eq. (6.1). In the present case, the period of the precession is
$\approx 20$ (source: Driben et al., 2014b).

###  VI.8 Weakly localized dark vortices

As mentioned above, the fundamental condition which secures the convergence of
the norm of localized modes produced by Eqs. (6.1) and (6.6) is given by Eq.
(6.8). If this condition does not hold, Eq. (6) gives rise to weakly localized
states with the divergent norm, which may be considered as localized dark
solitons (Zeng and Malomed, 2017), as they share the property of the
divergence of the norm with dark solitons in usual (free-space) models.

Localized dark solitons were considered by Zeng and Malomed (2017) in the 2D
version of Eq. (6.1) with the modulation profile

$\sigma(r)=r^{\alpha},~{}\alpha\leq 2,$ (239)

cf. Eq. (6.10). The weakly localized solutions to Eq. (6.1) with vorticity $S$
were looked for, in the polar coordinates, as

$\psi=\exp\left(-i\mu t+iS\theta\right)w(r),$ (240)

(essentially the same as Eq. (6.11)), where real function $w(r)$ satisfies the
equation

$\mu
w+\frac{1}{2}\left(\frac{d^{2}w}{dr^{2}}+\frac{1}{r}\frac{dw}{dr}-\frac{S^{2}}{r^{2}}w\right)-r^{\alpha}w^{3}=0,$
(241)

cf. Eq. (6.12). The asymptotic form of the solution at $r\rightarrow\infty$ is

$w=\sqrt{\mu}r^{-\alpha/2}+\frac{1}{4\sqrt{\mu}}\left(\frac{\alpha^{2}}{4}-S^{2}\right)r^{-\alpha/2-2}+\mathcal{O}\left(r^{-\alpha/2-4}\right),$
(242)

where the first term is tantamount to the TF approximation, cf. Eq. (6.9).
Solutions (6.49) with $S=0$ and $S\neq 0$ may be called, respectively,
localized CW states and localized dark vortices.

First, in the case of $\alpha=2$, which corresponds to the boundary between
localized dark and bright states, Eq. (6.50) with $S=1$ gives rise to simple
exact solutions, for all values $\mu>0$ (although these dark-vortex solutions
are unstable, see Fig. 33 below):

$w(r;\alpha=2;S=1)=\sqrt{\mu}r^{-1}.$ (243)

Obviously, the divergence of the integral norm of solutions to Eq. (6.50) is
logarithmic at $r\rightarrow\infty$.

The dark CW states may be quite accurately approximated by the TF
approximation (which corresponds to the first term on the right-hand side of
Eq. (6.51)) multiplied by a factor which accounts for vorticity $S\geq 1$:

$w(r)=\sqrt{\mu}r^{-\alpha/2}\left[\tanh(\lambda r)\right]^{S},$ (244)

where $\lambda$ is a fitting parameter. Typical radial profiles of the
localized dark vortices, along with approximation (6.53), are displayed, for
$S=1$ and $2$, in Fig. 32.

Figure 32: Blue lines show examples of localized dark vortices, produced by
the numerical solution of Eq. (6.50) with values of $\alpha$ indicated in the
figure, and vorticities $S=1$ (a) and $S=2$ (b). The chemical potential is
$\mu=4$ for $\alpha=0.6$ in (b), and $\mu=1.5$ for other solutions. Red dashed
lines show the analytical approximation given by Eq. (6.53), with fitting
factors $\lambda=1.1$ in (a), and $\lambda=1.5$ in (b). The vortices with
$S=1$ shown in (a) are stable, while the double vortices in (b) are weakly
unstable (source: Zeng and Malomed, 2017).

Stability of the localized CW states and dark vortices was identified by means
of systematic simulations of their perturbed evolution in the framework of the
Eq. (6.1). The conclusion is that the CW states are completely stable
(actually, they realize the system’s ground state), while all the localized
dark vortices with $S\geq 2$ are unstable. The analysis for vortices with
$S=1$ produces a nontrivial stability boundary, which is shown in Fig. 33. It
shows that the increase of both $\mu$ and $\alpha$ leads to destabilization of
the dark vortex. Actually, they may be stable only at

$\alpha<\alpha_{\max}\approx 0.55,$ (245)

i.e., if the the dark vortex is sufficiently stretched in the radial direction
(as seen from the approximate expression (6.53)).

Figure 33: The stability chart for the localized dark vortices with $S=1$,
produced by Eq. (6.50). In this plot, notation $k\equiv-\mu$ is used. The
vortices are completely unstable at $\alpha>0.55,$ see Eq. (6.54). All the
localized dark states with $S=0$ (CW) are completely stable, while all
vortices with $S\geq 2$ are unstable (source: Zeng and Malomed, 2017).

##  VII Conclusion

This chapter produces a systematic summary of theoretical results, both
numerical and analytical ones, obtained in numerous works aimed at the search
for physically relevant settings which allow one to create stable 2D and 3D
solitons, including topological modes with embedded vorticity, as well as
complex 3D modes in the form of hopfions, which carry two independent
topological charges. Stability is the main challenge in the studies of
multidimensional solitons because, on the contrary to diverse 1D settings, in
which solitons typically emerge as stable objects, they tend to be strongly
unstable in basic 2D and 3D situations. In particular, the ubiquitous NLS
(nonlinear Schrödinger) equation with the cubic self-focusing term
(essentially the same PDE is known as the GP (Gross-Pitaevskii) equation in
the BEC theory) creates only unstable solitons in 2D and 3D spaces alike,
because exactly the same equation gives rise to the destructive effects in the
form of the critical and supercritical wave collapse in 2D and 3D cases,
respectively.

The present chapter is focused on physically relevant multidimensional models
of the NLS/GP type. The chapter offers a relatively detailed review of two
basic topics. First, it is a general survey of various schemes which have been
elaborated to secure stability of 2D and 3D solitons, including those with
embedded vorticity. The main stabilization schemes outlined here are: (i)
competing (e.g., cubic-quintic) and saturable nonlinearities; (2) linear and
nonlinear trapping potentials; (3) the stabilizing effect provided by the LHY
(Lee-Huang-Yang) corrections to the mean-field BEC dynamics, leading to the
formation of stable QDs (quantum droplets), including ones with embedded
vorticity; (4) SOC (spin-orbit-coupling) effects in two- and three-dimensional
binary BEC; and (5) emulation of SOC in nonlinear optical waveguides,
including $\mathcal{PT}$-symmetric ones.

The second topic addressed in the chapter is a detailed summary of results
which demonstrate the creation of stable 2D and 3D solitons by means of
schemes based on the usual linear trapping potentials or effective nonlinear
ones. Nonlinear potentials may be induced by means of spatial modulation of
the local strength of the nonlinear term. The latter option is especially
promising, making it possible to use self-defocusing media, with the local
nonlinearity strength growing fast enough from the center to periphery, for
the creation of a great variety of multidimensional self-trapped modes. In
addition to fundamental quasi-solitons and vortex rings, the 3D modes may be
hopfions, i.e., twisted vortex rings. A remarkable fact is that many essential
results for the multidimensional solitons can be obtained, in such settings,
not only in a numerical form, but also by means of efficient analytical
methods, exact and approximate ones. In particular, basic properties of such
complex objects as hopfions are accurately predicted by the analytical
variational approximation.

The work on the vast topic surveyed in this chapter is far from being
completed. Among challenging open problems is, for instance, a possibility to
create stable complex 3D structures in the form of two linked rings, each one
being a vortex torus or, possibly, a hopfion. The greatest challenge is to
implement many theoretical predictions, outlined in the present review, in
real physical experiments – especially in BEC and nonlinear optics.

## Acknowledgments

I would like to thank my coauthors, in collaboration with whom many results
included in this review have been produced: Gregory Astrakharchik, Bakhtiyor
Baizakov, Milivoj Belić, Marijana Brtka, Olga Borovkova, George Boudebs,
Gennadiy Burlak, Cid de Araújo, Zhaopin Chen, Guangjiong Dong, Shenhe Fu,
Arnaldo Gammal, Yaroslav Kartashov, Vladimir Konotop, Hervé Leblond, Falk
Lederer, Ben Li, Yongyao Li, Valery Lobanov, Dumitru Mazilu, Torsten Meier,
Dumitru Mihalache, Michele Modugno, Wei Pang, Dmitry Pelinovsky, Han Pu, Jieli
Qin, Albert Reyna, Hidetsugu Sakaguchi, Mario Salerno, Elad Shamriz, Evgeny
Sherman, Yasha Shnir, Leticia Tarruell, Lluis Torner, Viktor Vysloukh, Li
Wang, Zhenya Yan, and Jianhua Zeng.

My thanks are to Prof. Weizhu Bao who has invited me to produce this chapter.
My work in this research area was supported, in part, by the Israel Science
Foundation through grant No. 1286/17.

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