FIG. 14.35 Shapes of | ψ ( x , y ) | in stable vortex solitons with S = 1, produced as numerical solutions of Eq. ( 14.43 ) with parameters A = 2, Γ₁ = 0.025, Γ₂ = 0.1, Γ₃ = 0.05, and q₀ = 1 in (a) (a tightly bound vortex soliton) and q₀ = 1.6 in (b) (a loosely bound one) ( Sakaguchi and Malomed, 2009 ). More about this image found in Shapes of | ψ ( x , y ) | in stable vortex solitons with

FIG. 14.44 The energy of the vortex solitons with S = 1 vs the nonlinear-gain coefficient, ε . The solutions are produced by Eq. ( 14.60 ) with normal GVD, D = −0.2 in (a), and anomalous GVD, D = 1, in (b). Other parameters are δ = 0.4 , β = 0.1 , γ = 0.5 , μ = 1 , and ν = 0.1 . Stable (black) and unstable (red) portions of the solution branches are marked by symbols “s” and “u,” in addition to using the different colors in them. Arrows indicate stability boundaries ( Mihalache et al., 2007a ). More about this image found in The energy of the vortex solitons with S = 1 vs the nonlin...

FIG. 14.46 Merger of the vortex solitons with winding numbers S = 2, which are set in motion as per initial conditions ( 14.65 ) with χ = 0.5 . Configurations are displayed at propagation distances z = 25 (a), 28 (b), 30 (c), 35 (d), 40 (e), and 60 (f). The figure corresponds to panel (a) in Fig. 14.45 ( Mihalache et al., 2008a ). More about this image found in Merger of the vortex solitons with winding numbers S = 2, ...

FIG. 3.4 Spontaneous splitting of an unstable vortex-soliton solution of Eq. ( 3.1 ) with δ = 0.14 and S = 1, see Eq. ( 3.12 ), in two fragments, each being close to a 2D soliton with S = 0. The values of propagation distance z corresponding to each frame are indicated near it. Because δ = 0.14 is very close to the critical value, δ(S = 1) ≈ 0.1487, beyond which the vortex soliton with S = 1 is stable, according to Eq. (3.16), the instability develops very slowly, the vortex remaining practically stable at z < 570. Coordinates used in this figure are ( X , Y ) ≡ 2 ( x , y ) ( Malomed et al., 2002 ). More about this image found in Spontaneous splitting of an unstable vortex-soliton solution of Eq. ( 3.1 )...

FIG. 3.5 The same as in Fig. 3.4, but for an unstable vortex soliton with δ = 0.14 and S = 2, which spontaneously splits in three fragments ( Malomed et al., 2002 ). More about this image found in The same as in Fig. 3.4, but for an unstable vortex soliton with δ = 0...

FIG. 3.6 Radial profiles, ρ ( r ) [see Eq. ( 3.12 )], of vortex-soliton solutions to Eq. ( 3.1 ) (alias vortex annuli), with winding numbers S = 1, 2, 3, 4, 5 (from left to right), each corresponding to the propagation constant δ taken at the respective stability boundary in Eq. (3.16) ( Pego and Warchall, 2002 ). More about this image found in Radial profiles, ρ ( r ) [see Eq. ( 3.12 )], of vortex-soliton sol...

FIG. 3.8 Self-trapping of a stable higher-order vortex soliton, with winding number S = 3, from an arbitrarily shaped initial vortex ring with the same S, produced by simulations of Eqs. ( 3.1′ ) and (3.1^″) with mismatch q = 0.2. The total power of the input is P₀ = 885 [see Eq. (3.3b)]. Panels (a) and (b) display the distribution of the FF local intensity, |u(x, y)|², and phase in the input, at z = 0. Panels (c) and (d) present the result of the evolution at z = 4000 ( Mihalache et al., 2004c ). More about this image found in Self-trapping of a stable higher-order vortex soliton, with winding number ...

FIG. 5.5 Self-cleaning of a stable vortex soliton with S = 1 (“donut”) in the isotropic model [Ω = 1 in Eq. ( 5.1 )] after the application of a random initial perturbation at the amplitude level of 10%. The left and right panels display, respectively, the shape of the perturbed vortex at the initial moment, t = 0, and at t = 120. The unperturbed vortex soliton has chemical potential μ = 2 and norm N = 12.55 (Malomed et al., 2007). More about this image found in Self-cleaning of a stable vortex soliton with S = 1 (“donu...

FIG. 6.10 The gyroscopic precession of the 3D vortex soliton (torus) with S = 1 and μ = 16 , initiated by the application of torque ( 6.30 ) with β = 4 and x₀ = 5. The plots are produced by simulations of Eqs. ( 6.1 ) and ( 6.14 ) with α = 1 / 2 . Three images in the top row correspond to times t = 90, 90.8, and 91.5. Time Δt = 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, and the green ones corresponding to those displayed in the top row ( Driben et al., 2014a ). More about this image found in The gyroscopic precession of the 3D vortex soliton (torus) with S...

FIG. 7.26 An example of a stable χ ( 2 ) vortex soliton of the IS-centered type, with winding numbers 1 and 2 in its FF and SH components, stabilized by the lattice potential ( 7.56 ) with the modulation amplitude ε = 8 and period L = π / 2 . The stable four-peak distribution of the absolute values of FF and SH fields, plotted in panels (c) and (d), respectively, is produced by simulations of Eqs. ( 7.54 ) and ( 7.55 ) with zero mismatch ( β = 0 ). At z = 0, the input is launched in the FF component only, with the amplitude (absolute-value) and phase distributions shown in panels (a) and (b) ( Xu et al., 2005 ). More about this image found in An example of a stable χ ( 2 ) vortex soliton of the IS-cente...

FIG. 14.40 Relaxation of an initially perturbed 3D vortex soliton with S = 3, produced by simulation of Eq. ( 14.60 ). The left and middle panels display the intensity distribution, |U(x, y, t)|², in the 3D form, and in the temporal mid plane, t = 0. The right panels: the phase distribution, arg ( U ( x , y ) ) , in the same plane. The top and bottom rows show, respectively, the input and the result of the evolution at z = 800. Parameters are D = 1, δ = 0.4 , β = 0.5 , ε = 2.3 , μ = 1 , ν = 0.1 , and γ = 0 ( Mihalache et al., 2007b ). More about this image found in Relaxation of an initially perturbed 3D vortex soliton with S...

FIG. 8.7 The left column: the top and bottom panels display, respectively, the intensity and phase profiles of the stable numerical solution for the six-peak vortex soliton with S = 2, produced by the model with the self-focusing sign of the nonlinearity. This vortex soliton corresponds to the experimentally observed one, which is shown in Fig. 8.6 . The right panel: the same, but for the stable vortex solitons with S = 1, in the case of the self-defocusing nonlinearity ( Terhalle et al., 2009 ). More about this image found in The left column: the top and bottom panels display, respectively, the inten...

FIG. 8.9 The same as in Figs. 8.6 and 8.8 , in the case when the input is launched with S = 2 into the self-defocusing PhR medium. In this case, the vortex soliton with S = 2 is unstable. The experiment demonstrates spontaneous transformation of the input into a stable vortex soliton with S = 1 ( Terhalle et al., 2009) . More about this image found in The same as in Figs. 8.6 and 8.8 , in the case when the input is launche...

FIG. 3.11 (a) Numerically found dependences K(E) for families of 3D solitons with vorticities S = 0, 1, and 2, as produced by the numerical solution of Eq. ( 3.20 ). Solid and dashed lines represent, respectively, stable and unstable states. (b) The instability growth rate, Re ( λ ) , corresponding to values L = 1, 2, and 3 of the azimuthal index of the perturbation eigenmode, vs K, as obtained from the numerical solution of Eq. ( 3.23 ) for the vortex solitons with S = 1. In interval ( 3.38 ), the instability growth rate is zero, the vortex solitons being stable in this region ( Mihalache et al., 2002a ). More about this image found in (a) Numerically found dependences K(E) fo...

FIG. 8.3 The left panel: the intensity distribution in the experimentally created IS-centered vortex soliton, with winding number S = 1, in the PhR medium, supported by the square-shaped virtual photonic lattice. The middle and right panels: the intensity and phase structure of the same vortex soliton, as predicted by the numerical solution of the respective model. In the right panel, white circles denote the location of the intensity peaks from the middle panel ( Neshev et al., 2004 ). More about this image found in The left panel: the intensity distribution in the experimentally created IS...

FIG. 3.3 Dependence N ( δ ) for families of 2D fundamental and vortex solitons with winding number S, produced by the numerical solution of Eq. ( 3.13 ). Solid and dashed curves represent, respectively, stable solutions and those vortex solitons that are unstable against spontaneous splitting. Label “wavenumber” stands for a rescaled propagation constant, 9 δ , and “energy flow” is tantamount to (2/3)N in comparison with Eq. ( 3.6 ) ( Malomed et al., 2002 ). More about this image found in Dependence N ( δ ) for families of 2D fundamental and vortex solit...

FIG. 3.10 The left panel: an initially strongly perturbed stable donut-shaped 3D soliton (vortex torus) with embedded vorticity S = 1. The right panel: the vortex soliton self-cleaned in the course of the evolution governed by Eq. ( 3.18 ) ( Malomed et al., 2005 ). This example is similar to the one shown in Fig. 5.5 of Chap. 5; however, the latter one was produced by simulations of Eq. (5.1), which includes the trapping potential. More about this image found in The left panel: an initially strongly perturbed stable donut-shaped 3D soli...

FIG. 7.4 A typical example of a stable multi-cell pattern representing an IS (intersite)-centered vortex soliton with winding number S = 1, produced by a solution of Eq. ( 7.5 ) with ε = − 1 / 2 and μ = 0 . Panels (a) and (b) show the distribution of the density and phase in the soliton ( Yang and Musslimani, 2003 ). More about this image found in A typical example of a stable multi-cell pattern representing an IS (inters...