Figure 13 – uploaded by Boris Malomed

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FIG. 13. Density and phase patterns of a stable discrete vortex soliton with S = 1, uy = —0.32 and C' = 0.05, generated b Eqs. (56) and (57), as per Ref. [269]. — Figure 13 Density and phase patterns of a stable discrete vortex soliton with S = 1, uy = —0.32 and C' = 0.05, generated b Eqs. (56) and (57), as per Ref. [269].

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FIG. 1. Spontaneous splitting of an unstable 3D vortex torus, with winding number S' = 2 and propagation constant k = 0.0 in the framework of the NLSE (10) with the CQ nonlinearity, as per Ref. [62]. Panels (a) and (b) display, respectively, th intensity isosurface of the stationary solution (the input at z = 0), and the result of its evolution at z = 250. In this figure, th coordinates are related to those in Eq. (10) by (X,Y,T) = V2 (a, y,t).

FIG. 2. Panels (al)-(a4) display density patterns of QDs (quantum droplets) with embedded vorticities S = 1,2,3,4 and fixec norm N = 1000, as per Ref. [161]. The first three QDs are stable, while the one corresponding to S = 4 is subject to the splitting instability. (b) Cross-sections of the density patterns from panels (al)-(a4). (c) The same as in (b), for for S = 1 anc different values of N. All vortex QDs shown in (c) are stable.

FIG. 3. (a) A zoom (in domain |z,y| < 8) of the density pattern, |q (a, y)|?, for a QD with (S,N) = (3, 1000), which was initially perturbed as per Eq. (25), with s = +1 and « = 0.0013+0.00237. The pattern is produced by the simulation of Eq. (29) up to t = 5000. (b) The corresponding phase pattern clearly identifies three unitary vortices, whose pivots (phase singularities) are located at zeros of the local amplitude. (c) and (d): The same as in (a) and (b), but with the initial perturbation carrying vorticity s = —1, as per Eq. (27). In this case, the amplitude and phase patterns demonstrate splitting in a set of five pivots: one with winding number —1 located in the middle, surrounded by four satellites with winding numbers +1. Note the extremely small scale of the local amplitude, |7) (a, y)| ~ 107” in (a) and (c). On the normal scale, these splitting patterns remain invisible. The results are borrowed from Ref. [161].

FIG. 4. The evolution in time t of three-dimensional QDs (quantum droplets) with vorticity S = 2, as produced by simulation of Eq. (21) reported in Ref. [160], and shown by means of surfaces |) (x,y, z)| — const. The first three rows display example of splitting of the same unstable 3D QD (quantum droplet) into sets of three, four, or five fragments, initiated by differen eigenmodes of small perturbations initially added to the stationary QD. A random small perturbation causes the splitting int four fragments, as the respective instability eigenmode has the largest growth rate. The bottom row shows the evolution of stable QD with S = 2 and with a larger norm.

FIG. 5. Spontaneous splitting of an HV (hidden-vorticity) QD with vorticities of its components S; = —S2 = 1S = 2, as produced by simulations of the full two-component version of Eq. (21) in Ref. [160], and shown by means of constant-density surfaces of one component (for the second one, the picture seems identical).

FIG. 6. (a) The chemical potential and (b) energy, defined as per Eq. (33), versus norm N for the 3D vortex mode with S = 1, trapped in (anisotropic) HO potential (32) with Q = 10 and 1, as found from numerical solution of Eq. (31). Solid and dotted portions of the curves denote, respectively, stable and unstable parts of the solution families, as identified by a numerical solution of the eigenvalue problem based on Bogoliubov - de Gennes equations (45) in Ref. [78].

FIG. 7. Self-cleaning of a stable 3D vortex torus with S = 1, produced in [79] by simulations of the model based on Eq. (29) with isotropic (QQ = 1) potential (32), after the application of a random perturbation at the amplitude level of 10%. Panels (a) and (b) display, severally, the initially perturbed shape of the vortex state at t = 0, and its shape at t = 120. The unperturbed vortex has chemical potential 44 = 2 and norm N = 12.55.

FIG. 8. Periodic evolution of an initially perturbed 2D vortex with S = 1, w= 1.2 and N = 8.48, which belongs to interval (50), as per Ref. [78]. In this dynamical regime, the trapped mode periodically splits into two fragments and recombines, while its vorticity is conserved: (a) t = 0, (b) t = 100, (c) t = 140, and (d) t = 180. It is relevant to compare this dynamical state with its counterpart in the dissipative model based on Eq. (96), in which the unstable vortex with S = 1 splits into a stably rotating dipole, see Fig. 25 below. b. Two-component systems A noteworthy generalization of the above analysis was performed for a system of two nonlinearly coupled fields, which may be realized as a binary BEC, or as co-propagation of two optical beams in a bulk waveguide [187]. The respective coupled two-dimensional GPE/NLSE system is

FIG. 9. Density profiles of two components of a stable asymmetric vortex state with S = 1 and total norm N = Ni + No = 8.8 produced by Eqs. (53a) with « = 0.4, as per Ref. [163].

FIG. 10. The symmetry-breaking bifurcation diagram for two-component vortex solitons produced by Eqs. (53a), as per Ref. [163]. The asymmetry parameter (54) is shown as a function of the total norm, N. Dashed lines represent unstable vortex solitons.

FIG. 11. Panel (b): an example of a stable on-site-centered vortex soliton, with winding number S = 1, supported by th« lattice potential in the 2D variant of Eq. (37), with k = 2, « = 5, and g = 1/2, as per Ref. [177]. The plot displays the density) distribution, | (x,y)|? (the inset additionally shows its 1D cross section along the x axis). The 2D norm of the soliton i N = 2x. The vorticity is represented by the phase pattern (not shown here), with phase shifts 7/2 between four main peaks which corresponds to the global phase circulation 27. For comparison, panel (a) displays an example of a stable fundamenta soliton (S = 0, with norm N = 10), supported by the same model.

FIG. 12. An example of the density pattern in an off-site-centered vortex soliton with S = 1, supported by the 2D variant of Eq. (37), as per Ref. [178].

FIG. 14. A stable higher-order vortex soliton, with winding number S = 4 and total norm N = 33.4, produced by the 2D variant of Eq. (37) with k = 2, « = 3, and g = 1, as per Ref. [196]. The left and right panels display, respectively, the profile of | (x, y)| and the pattern of the phase vector with components {Re (~ (a, y)) , Im (# (a, y))} /| (a, y)|, at some moment of time. The total phase circulation in this pattern is 87, which corresponds to S = 4.

FIG. 15. A stable supervortex complex, with local and global vorticities s = S = 1, and total norm N = 55.5, produced by the 2D variant of Eq. (37) with k = 2, « = 10, and g = 1, as per Ref. [196]. Panels (a) and (b) display, respectively, the 2D profile of |q (a, y)| and the same in the 1D cross section, |~ (#,0)| (the latter one directly shows that each local constituent o: the complex is a compact vortex with the inner hole).

Stationary solutions of Eq. (58) for 2D solitons with real chemical potential jz are looked for as W. = exp (—ipt) us (2, y), where complex stationary wave functions are determined by equations where o,,y,, are the Pauli matrices, is a real coefficient of the SOC of the Rashba type (a combination with the SOC of the Dresselhaus type, which is modeled by combination 0,0/0x — o,0/Oy, instead of —o,0/0x + o,0/Oy in Eq. (58), is not considered here, as it tends to destroy 2D solitons [81]), 7 is the relative strength of the cross-attraction between the components (cf. Eq. (51a)), while the strength of the self-attraction is normalized to be 1. Coefficient 1/A has the dimension of length, defining a fixed scale which breaks the scale invariance of the NLSE/GPE in the free 2D space and thus makes it possible, as mentioned above, to create stable solitons with norms falling below the collapse threshold.

FIG. 16. A stable 3D vortex soliton with S = 1 and total 3D norm N = 22.3, produced by Eq. (37) in its 3D form, but with the quasi-2D lattice potential, i.c., 7 = 0, and k = 2, « = 1.25, g = 1, as per Ref. [167]. The top and bototm panels display, respectively, the 3D amplitude profile, defined by |v (x,y, z)| = 1.45, and the phase pattern in the midplane, z = 0, superimposed on the contour plot of |~ (a, y, 0)|.

Dynamical invariants of Eqs. (58) are the same total norm N as defined above, Hamiltonian, and linear momentum: The consistent derivation of the effective 2D SOC model from the full 3D system of GPEs may give rise to the 2D equations with nonpolynomial nonlinearity, as a generalization of the cubic terms in Eq. (58) [229]. Such a generalized system also creates stable solitons in the 2D free space. Another relevant generalization addresses a model of a dual-core nonlinear coupler in optics, where SOC is emulated by temporal dispersion of the linear inter-core coupling [240]. It was demonstrated that this model gives rise to stable 2D spatiotemporal solitons. The 21) model jg taken here with the FOC of the Wev] tyne [89]-

FIG. 17. (a) Cross sections of the fundamental, |+ (x, y)|, and vortical, |q_ (x, y)|, components of a 2D semi-vortex (SV) with N = 5, along axis y = 0, are shown by continuous and dashed lines, respectively, as per Refs. [80] and [81]. (b) Chemical potential 4 vs. norm N for the SV family. (c) The share of the norm of the zero-vorticity component, N+, in the total SV’s norm N, as a function of N. In this figure, X = 1 and 7 = 0 are fixed in Eq. (58). Lastly, it is relevant to mention that 2D solitons of the MM type, additionally stabilized by the LHY correction t the model of the binary condensate with SOC, were considered too [242].

FIG. 18. Panels (a) and (b) have the same meaning as in Fig. 17, but for a 2D mixed-mode (MM) soliton with 7 = 2 and A = 1 in Eq. (58). The norm of the soliton shown in panel (a) is N = 5. (c) Separation DX between peak positions of components |4| and |w_| vs. N. The results are borrowed from Refs. [80] and [81].

FIG. 19. Contour plots of |W+ (x,y) | (a) and |W_ (x,y) | (b) of the 2D stable MM (mixed-mode) soliton with norm N = 3.1, moving at velocity vy = 0.5 , for 7 = 2 and \ = 1, as per Refs. [80] and [81]. (c) The amplitude of the moving solitons as a function of vy.

FIG. 20. (Color online) Energy Etot of the 3D solitons in the SOC system, as a function of condensate’s size L, predicted by Eq. (75) (as per Ref. [82]), with cint = cfc) + fF), The red solid, blue dashed, and green dashed-dotted lines represent the variation of the energy for \ = 0, and for A > 0 which does or does not satisfy condition (76), respectively.

FIG. 21. (Color online) Density profiles of stable 3D solitons numerically generated in the SOC system for N = 8 and \ = as per Ref. [82]. (a) A semi-vortex for 7 = 0.3, whose fundamental and vortical components, |7)+| and |_|, are plotted in (z and (a2), respectively. (b) A mixed mode for 7 = 1.5, with (b1), (b2) displaying |+| and |y_|, respectively. In each subpl different colors represent constant-magnitude surfaces, |¢+ (x, y, z) | = (0.96, 0.4, 0.04) x |b+|max-

The 2D system of the coupled GPEs for two components, ¢+,, which form a pseudo-spinor wave function of the binary BEC, was derived in Ref. [85]. In the scaled form, the system takes the form Here, the integral terms represent the action of the microwave field on the matter waves, the field itself being replaced by the solution of the respective Poisson equation, produced by means of the 2D Green’s function, y is the strength of the field-matter interaction (which appears as the effective long-range interaction in Eqs. (77) and (78)), and Ho is a background magnetic field. Further, 6 is the coefficient of the additional contact (collisional) interaction between the components of the wave function, which may be present too, 8 > 0 implying the attractive interactions. Tis A eed Se) pees pe ef fc of ee I PN ee ORIN ee PRION wee cee ee ee eee dS eo Another possibility to create stable 2D vortex solitons with arbitrarily large values of the vorticity was recently predicted in the hybrid model based on a system of two GPEs coupled by a microwave field through a magnetic-dipole transition. Two components of the matter waves, which are governed by the coupled GPEs, represent a BEC mixture of two different hyperfine states of the same atomic species, the microwave field being generated by the transition between the levels, which gives rise to the feedback of the matter waves on the microwave radiation. This physical model was introduced, in the 1D form, in Ref. [243].

TABLE II. Table II. Bconapse and Blonaly *) are numerically obtained and analytically predicted (see Eq. (86)) values of the collapse contact-interaction strength, 8, up to which the fundamental and vortex solitons exist, in the framework of Eq. (79). stability is the numerically identified stability boundary of the vortex solitons. FIG. 22. Top and bottom panels display, severally, examples of the stable and unstable perturbed evolution of the vortex solitons with indicated values of S and , in the framework of Eq. (79), as per Ref. [85]. The necklace-shaped set, observed in the latter case, remains confined (keeping the same overall radius) in the course of subsequent evolution. In this figure, time t is denoted rT.

FIG. 23. The left and right panels display the amplitude and spiral-phase shapes of a stable dissipative soliton with vorticity S = 1, produced by Eq. (93) with y = 8 = 0.5, e = 2.5, uw = 1, and vy = 0.1, as per Ref. [287].

FIG. 24. The bottom row displays the result of the self-cleaning of an initially perturbed 3D dissipative vortex (shown in the top row) with embedded vorticity S = 2, as per simulations of the three-dimensional CGLE reported in Ref. [290]. The left. middle, and right panels display, respectively, the 3D amplitude profile, its 2D cross section, and the phase profile in the same 2D plane.

FIG. 25. Spontaneous transformation of an unstable dissipative vortex soliton with winding number S = 1 into a stably rotating dipole state, produced by simulations of Eq. (96), as per Ref. [288]. The parameters are y = 0.5, ¢ = 1.8, w= 1, y= 0.1, and 0? = 0.25.

‘IG. 26. An example of a stable dynamical mode with winding number S = 1, in the form of the ring-shaped vortex whicl eatures elliptic deformation and eccentricity. The solution is generated by the model based on Eqs. (93) (without the diffusio1 erm, 3 = 0) and (97), as per Ref. [293]. Four configurations of the intensity profile, |u (x,y, z)| = ||, displayed at differen alues of z, demonstrate that it performs orbital motion around the origin and synchronous rotation around itself, with perioc r. =

Figure 13 – uploaded by Boris Malomed

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