Multi-Vortex States and Interactions

• Multi-vortex refers to composite states containing multiple interacting vortex degrees of freedom, manifesting in optics, quantum, fluid, and superconducting systems.
• In nonlinear optics, multi-vortex solitons balance diffraction, Kerr self-focusing, gain, and absorption, enabling stable formations like vortex twins, anti-twins, and necklace vortices.
• In quantum and fluid dynamics, multi-vortex behavior emerges through symmetry breaking, bifurcations, and complex pairwise as well as collective interactions that govern stability.

Searching arXiv for recent and foundational papers relevant to the different technical meanings of “multi-vortex.” “Multi-vortex” denotes a class of states, solutions, or configurations containing more than one vortex degree of freedom, but its precise meaning depends strongly on context. In nonlinear optics it refers to dissipative soliton complexes composed of several phase singularities sustained by structured gain landscapes (Borovkova et al., 2011); in quantum mechanics it can mean multi-valued wavefunctions built from knotted vortex filaments (Davidson, 2019); in fluid dynamics it denotes interacting assemblies of vortices, including deformable viscous vortex particles and polygonal co-rotating configurations (Uminsky et al., 2010, Swaminathan et al., 2015); in Bose–Einstein condensates it encompasses bifurcating polygonal vortex states, vortex crystals, multiply charged vortices that split into many singly charged lines, and bound vortex clusters in multicomponent gases (García-Azpeitia et al., 2017, Xie et al., 2017, Telles et al., 2015, Dantas et al., 2015, Teles et al., 2015, Li et al., 2016); in superconductivity it includes controllable multi-flux configurations and vortex matter with multi-scale interactions (Polshyn et al., 2019, Meng et al., 2016, Meng et al., 2014); and in photonics and electromagnetics it also describes multiple vector-vortex beams and multiplexed orbital-angular-momentum channels (Schulz et al., 2013, Zhao et al., 18 Feb 2025). The common structure is the coexistence and interaction of several vortical singularities, circulations, or topological windings within a single composite system, but the governing equations, stabilization mechanisms, and observable consequences vary substantially across subfields.

1. Optical and photonic multi-vortex states

In nonlinear optics, one of the most explicit uses of the term is the multi-vortex soliton complex supported by structured gain in a focusing Kerr medium with strong two-photon absorption (Borovkova et al., 2011). The governing equation is a dissipative nonlinear Schrödinger equation,

$$i\,\frac{\partial q}{\partial \xi} = -\frac{1}{2}\left(\frac{\partial^2 q}{\partial \eta^2} + \frac{\partial^2 q}{\partial \zeta^2}\right) - |q|^2\,q + i\,p\,R(\eta,\zeta)\,q - i\,\alpha\,|q|^2\,q,$$

with annular gain channels

$$R(\eta,\zeta) = \sum_{k=1}^{N} \exp\!\left[-\frac{\big(r_k(\eta,\zeta) - r_c\big)^2}{w^2}\right].$$

Here the multi-vortex state is a dissipative attractor produced by a double balance: diffraction against Kerr self-focusing, and localized gain against two-photon absorption (Borovkova et al., 2011). The topological charge is

$$m = \frac{1}{2\pi} \oint \nabla \phi \cdot d\boldsymbol{\ell},$$

and the work distinguishes “vortex twins” with identical charges and “vortex anti-twins” with opposite charges. Stable complexes with two, three, or four vortices exist above a minimal gain and minimal ring separation. In two-ring geometries, vortex twins are always asymmetric in intensity, whereas anti-twins can be symmetric near threshold and become asymmetric as gain increases or ring separation decreases (Borovkova et al., 2011).

The same broad optical theme reappears in integrated photonics, where “multi-vortex” refers not to interacting singularities in one nonlinear beam, but to the simultaneous generation of several vector-vortex beams by concentric ring resonators (Schulz et al., 2013). In that setting the emitted orbital angular momentum order is fixed by the angular phase-matching rule

$\ell = m - q,$

with $m$ the ring azimuthal mode number and $q$ the number of grating elements. Several concentric rings, each with its own vertically displaced bus waveguide, generate multiple OAM channels and arbitrary qudit superpositions $|\psi\rangle = \sum_\ell c_\ell |\ell\rangle$ (Schulz et al., 2013). The same device can also operate as an OAM sorter with an average cross-talk of $-10\,\mathrm{dB}$ between different OAM channels (Schulz et al., 2013).

A related, but distinct, optical development is the Bessel-lattice necklace-vortex soliton. In azimuthally modulated Bessel potentials, the field evolves according to

$$i\,\partial_{z}\psi + \tfrac{1}{2}\nabla_{\perp}^{2}\psi + \gamma\,|\psi|^{2}\psi + p\,V(r,\theta)\,\psi = 0,$$

with

$$V_{0}(r,\theta) = J_{1}^{2}\!\big(2\,b_{\mathrm{lin}}\,r\big)\,\cos(k\,\theta).$$

Here “multi-vortex” means a compound state with $$R(\eta,\zeta) = \sum_{k=1}^{N} \exp\!\left[-\frac{\big(r_k(\eta,\zeta) - r_c\big)^2}{w^2}\right].$$0 azimuthal beads per ring and total topological charge $$R(\eta,\zeta) = \sum_{k=1}^{N} \exp\!\left[-\frac{\big(r_k(\eta,\zeta) - r_c\big)^2}{w^2}\right].$$1, so the adjacent-bead phase step is $$R(\eta,\zeta) = \sum_{k=1}^{N} \exp\!\left[-\frac{\big(r_k(\eta,\zeta) - r_c\big)^2}{w^2}\right].$$2 (Zhao et al., 21 Feb 2026). The lattice discretizes the azimuthal degree of freedom and can stabilize high-winding-number necklace vortices that would be azimuthally unstable in uniform Kerr media. Stable sextupoles, breathing octupoles, and 12-pole necklaces with $$R(\eta,\zeta) = \sum_{k=1}^{N} \exp\!\left[-\frac{\big(r_k(\eta,\zeta) - r_c\big)^2}{w^2}\right].$$3 in a finite stability window are reported (Zhao et al., 21 Feb 2026). This suggests that, within nonlinear optics, “multi-vortex” spans both multi-core dissipative solitons and discretized many-bead vortex necklaces, unified by the role of structured media in suppressing azimuthal breakup.

2. Quantum and hydrodynamic formulations

In single-particle quantum mechanics, “multi-vortex” can mean multi-valued vortex solutions of the Schrödinger equation (Davidson, 2019). The wavefunction is written in Madelung form,

$$R(\eta,\zeta) = \sum_{k=1}^{N} \exp\!\left[-\frac{\big(r_k(\eta,\zeta) - r_c\big)^2}{w^2}\right].$$4

with Bohmian velocity $$R(\eta,\zeta) = \sum_{k=1}^{N} \exp\!\left[-\frac{\big(r_k(\eta,\zeta) - r_c\big)^2}{w^2}\right].$$5 and quantum potential

$$R(\eta,\zeta) = \sum_{k=1}^{N} \exp\!\left[-\frac{\big(r_k(\eta,\zeta) - r_c\big)^2}{w^2}\right].$$6

The paper constructs $$R(\eta,\zeta) = \sum_{k=1}^{N} \exp\!\left[-\frac{\big(r_k(\eta,\zeta) - r_c\big)^2}{w^2}\right].$$7 from vortex-filament potentials familiar from incompressible Euler flow, using a Seifert surface as the branch-cut structure, and shows that analytic continuation around a filament multiplies the wavefunction by $$R(\eta,\zeta) = \sum_{k=1}^{N} \exp\!\left[-\frac{\big(r_k(\eta,\zeta) - r_c\big)^2}{w^2}\right].$$8 (Davidson, 2019). The nodal set is enforced by

$$R(\eta,\zeta) = \sum_{k=1}^{N} \exp\!\left[-\frac{\big(r_k(\eta,\zeta) - r_c\big)^2}{w^2}\right].$$9

with $$m = \frac{1}{2\pi} \oint \nabla \phi \cdot d\boldsymbol{\ell},$$0, so knotted or linked vortex filaments become nodal lines of $$m = \frac{1}{2\pi} \oint \nabla \phi \cdot d\boldsymbol{\ell},$$1 (Davidson, 2019). The analysis argues that multi-valued free-particle states radiate unless the single-valuedness condition $$m = \frac{1}{2\pi} \oint \nabla \phi \cdot d\boldsymbol{\ell},$$2 holds, and it further proposes radiation signatures in Aharonov–Bohm configurations when the induced phase is not an integer multiple of $$m = \frac{1}{2\pi} \oint \nabla \phi \cdot d\boldsymbol{\ell},$$3 (Davidson, 2019).

In classical viscous fluid dynamics, multi-vortex states appear as interacting localized vorticity concentrations. The multi-moment vortex method represents the vorticity by Hermite–Gaussian moments,

$$m = \frac{1}{2\pi} \oint \nabla \phi \cdot d\boldsymbol{\ell},$$4

with the base Gaussian

$$m = \frac{1}{2\pi} \oint \nabla \phi \cdot d\boldsymbol{\ell},$$5

For a multi-vortex flow,

$$m = \frac{1}{2\pi} \oint \nabla \phi \cdot d\boldsymbol{\ell},$$6

and each particle carries a hierarchy of Hermite moments encoding deformation and anisotropy (Uminsky et al., 2010). The central advance is a closed ODE system for self- and cross-interactions that avoids direct Biot–Savart evaluation for deformed particles. In merger problems this framework captures deformation, spiral-arm formation, and centroid drift far better than non-deforming Gaussian blobs (Uminsky et al., 2010).

A complementary hydrodynamic problem is the merger of $$m = \frac{1}{2\pi} \oint \nabla \phi \cdot d\boldsymbol{\ell},$$7 co-rotating vortices initially arranged on a regular polygon (Swaminathan et al., 2015). In that case the multi-vortex interaction is qualitatively distinct from the two-vortex problem because an annular vortical structure forms as an intermediate state. The axisymmetric annulus is described by the generalized Lamb–Oseen solution

$$m = \frac{1}{2\pi} \oint \nabla \phi \cdot d\boldsymbol{\ell},$$8

which diffuses inward until it becomes a standard Lamb–Oseen vortex (Swaminathan et al., 2015). At moderate Reynolds number, this annular stage slows merger significantly. At higher Reynolds number, odd and even vortex numbers behave differently: even-$$m = \frac{1}{2\pi} \oint \nabla \phi \cdot d\boldsymbol{\ell},$$9 configurations evolve via pairing, whereas odd-$\ell = m - q,$0 cases exhibit rapid and chaotic collapse (Swaminathan et al., 2015). A plausible implication is that multi-vortex interactions alter inverse-cascade timing in two-dimensional turbulence more strongly than pairwise-merger intuition would suggest.

3. Bose–Einstein condensates: bifurcations, crystals, and decay

In rotating trapped Bose–Einstein condensates, multi-vortex states arise by symmetry-breaking bifurcation from radially symmetric vortices. For the two-dimensional Gross–Pitaevskii equation

$\ell = m - q,$1

a charge-$\ell = m - q,$2 vortex $\ell = m - q,$3 has a Hessian with $\ell = m - q,$4 pairs of negative eigenvalues near $\ell = m - q,$5, and as $\ell = m - q,$6 increases there are $\ell = m - q,$7 global bifurcations (García-Azpeitia et al., 2017). These branches include an asymmetric single vortex for $\ell = m - q,$8, an asymmetric vortex pair for $\ell = m - q,$9, and polygonal vortex arrays for $m$0. The polygonal states are read off from zeros of Hermite–Gauss combinations and can include a central antivortex together with a ring of charge-one vortices (García-Azpeitia et al., 2017).

In the many-vortex regime, a separate reduction from the rotating GP equation produces an effective ODE for co-rotating vortex crystals in isotropic or anisotropic traps (Xie et al., 2017). In the isotropic case, the continuum limit yields an explicit areal density and a fold in the crystal branch, with maximum admissible vortex number

$m$1

where $m$2 (Xie et al., 2017). In strong anisotropy, vortices align on the long axis and the system reduces to an effective one-dimensional density problem. For two vortices, a pair on the long axis is linearly stable while a pair on the short axis is unstable, consistent with both the ODE reduction and full GP simulations (Xie et al., 2017).

A different BEC multi-vortex mechanism appears when a multiply charged vortex decays. In a cigar-shaped $m$3 condensate, a quadruply charged vortex imprinted by topological phase engineering decays into four singly charged lines that twist around one another as helical Kelvin waves (Telles et al., 2015). The governing model is the three-dimensional GP equation

$m$4

The decay is not a simple radial splitting: axial inhomogeneity causes different $m$5-slices to unwind at different times, so the emergent singly charged vortices acquire a helical structure (Telles et al., 2015). The resulting multi-vortex state is then used numerically to generate an almost isotropic turbulent tangle, whose velocity statistics display power-law tails rather than Gaussian ones (Telles et al., 2015).

The stability of multiply charged vortices can also be analyzed variationally in quasi-two-dimensional condensates (Teles et al., 2015). There the linearized collective-mode problem contains a vortex-coupled quadrupole mode $m$6 whose frequency can become imaginary, providing a decay channel into singly charged vortices. Static Gaussian pinning with width comparable to the vortex-core radius can stabilize the state, and a sinusoidally modulated Gaussian potential produces a Mathieu-type system whose parametric-resonance tongues delimit additional stability windows (Teles et al., 2015). This suggests that some multi-vortex configurations in BECs are not only interaction-selected but also dynamically engineered through mode control.

4. Multicomponent condensates and spin–orbit coupled vortex matter

In multicomponent homogeneous condensates, the decisive object is often the pairwise vortex interaction rather than the single-vortex spectrum (Dantas et al., 2015). For $m$7 components with coherent couplings,

$m$8

the energy contains density couplings $m$9 and Rabi couplings $q$0 (Dantas et al., 2015). In the miscible regime, intra-component vortices repel, but inter-component interactions can be attractive if $q$1, and the competition between density repulsion and coherent phase locking can produce non-monotonic pair potentials with a minimum at finite separation (Dantas et al., 2015). This generates vortex dimers and, in three-component systems, equilateral trimers with bond length obeying the empirical scaling

$q$2

for the parameter range studied (Dantas et al., 2015). The same pairwise mechanisms explain non-triangular many-vortex lattices and dimerized arrays under rotation, even though the rotating lattice problem itself is not solved directly in that work (Dantas et al., 2015).

Spin–orbit coupling plus an optical lattice yields another route to exotic multi-vortex order (Li et al., 2016). In a two-component BEC with Rashba SOC in a square lattice, the single-particle minima move with SOC strength $q$3 through four-fold, eight-fold, and Brillouin-zone-edge or corner degeneracies. When the minima touch the boundary of the first Brillouin zone, the interacting ground state can support simultaneously half-quantum vortex lattices, vortex–antivortex pair lattices, and fundamental vortex lattices (Li et al., 2016). The phase with this coexistence occurs when the corner minima $q$4 are occupied in a parity-symmetric superposition and $q$5, so that the two spin components populate complementary sublattices (Li et al., 2016). Here “multi-vortex” does not mean merely many vortices, but several topologically distinct vortex-lattice species embedded in the same ground state.

A related but distinct notion of layered vortex structure appears in generalized Maxwell–Higgs theories (Bazeia et al., 2019). There the visible sector obeys BPS equations such as

$q$6

or, in a two-$q$7 model,

$q$8

with a dielectric function $q$9 controlled by an auxiliary neutral or hidden field (Bazeia et al., 2019). The result is not a lattice of separate cores but a single radial vortex endowed with multiple magnetic rings. The number of rings is controlled by the dielectric modulation rather than by the winding $|\psi\rangle = \sum_\ell c_\ell |\ell\rangle$0, while the total flux remains quantized (Bazeia et al., 2019). This suggests a useful distinction: in field theory, “multi-vortex” can refer either to several separate topological cores or to a single topological object with multilayered internal vorticity structure.

5. Superconductors, plasmas, and curved-surface vortex ensembles

In multiply connected superconducting structures, multi-vortex states can be encoded not as visible Abrikosov cores but as sets of loop winding numbers constrained by fluxoid quantization (Polshyn et al., 2019). For a partitioned ring, the gauge-invariant phase differences $|\psi\rangle = \sum_\ell c_\ell |\ell\rangle$1 satisfy

$|\psi\rangle = \sum_\ell c_\ell |\ell\rangle$2

and the free energy is a quadratic form in $|\psi\rangle = \sum_\ell c_\ell |\ell\rangle$3 (Polshyn et al., 2019). The resulting stability regions form hexagonal honeycombs in flux space, each cell corresponding to a distinct multi-vortex configuration $|\psi\rangle = \sum_\ell c_\ell |\ell\rangle$4. A micron-size magnetic particle on a cantilever locally tunes $|\psi\rangle = \sum_\ell c_\ell |\ell\rangle$5, allowing stabilization of specific multi-vortex states and even braiding of individual vortices within a larger ensemble (Polshyn et al., 2019). The braiding demonstrated is geometric rather than topologically protected, but it realizes deterministic exchange operations in a superconducting network (Polshyn et al., 2019).

A different superconducting use of “multi-vortex” concerns vortex matter with multi-scale inter-vortex interactions (Meng et al., 2016, Meng et al., 2014). A phenomenological pair potential of the form

$|\psi\rangle = \sum_\ell c_\ell |\ell\rangle$6

encodes short-range repulsion, intermediate-range attraction, and long-range Pearl-like repulsion (Meng et al., 2016). Phase diagrams in the $|\psi\rangle = \sum_\ell c_\ell |\ell\rangle$7 plane then contain not only triangular lattices, but dimer, trimer, tetramer, stripe, honeycomb, kagome, and cluster phases (Meng et al., 2016). Closely related layered-superconductor modeling shows that honeycomb, square, hexagonal, and kagomé vortex lattices can be stabilized by engineering multiple screening and core-overlap scales (Meng et al., 2014). A plausible implication is that “multi-vortex” in superconductors increasingly refers to designer vortex matter whose unit objects may themselves be molecular clusters rather than isolated flux lines.

In dusty plasmas, the term describes several co-rotating vortices self-organized in an extended dust cloud (Choudhary et al., 2017). The mechanism is a charge-gradient-driven vorticity source,

$|\psi\rangle = \sum_\ell c_\ell |\ell\rangle$8

generated by an inhomogeneous plasma background (Choudhary et al., 2017). The steady rotation frequency follows the Vaulina scaling

$|\psi\rangle = \sum_\ell c_\ell |\ell\rangle$9

and the characteristic vortex diameter is

$-10\,\mathrm{dB}$0

As RF power decreases, the dust column shortens and the characteristic vortex size increases, driving a transition from three vortices to two and then one (Choudhary et al., 2017). Here the multi-vortex state is a dissipative pattern of macroscopic circulation cells rather than a set of phase singularities.

On curved surfaces, point-vortex clusters on a torus introduce yet another meaning (R et al., 16 Jun 2025). The toroidal geometry generates a geometric self-force through the Robin function, and the vortex dynamics is written in terms of the Schottky–Klein prime function and its logarithmic derivative

$-10\,\mathrm{dB}$1

with $-10\,\mathrm{dB}$2 the $-10\,\mathrm{dB}$3-digamma function (R et al., 16 Jun 2025). Chiral same-sign clusters remain geometrically confined and undergo collective toroidal drift, whereas achiral zero-net-circulation clusters disperse over the surface (R et al., 16 Jun 2025). This suggests that on curved manifolds, confinement or dispersion of multi-vortex matter may be controlled as much by geometry and topology as by pairwise circulation signs.

6. Broader meanings, applications, and recurring mechanisms

Across photonics, condensed matter, fluid mechanics, plasma physics, and quantum foundations, several recurring mechanisms organize multi-vortex behavior. One is balance or competition among multiple physical scales. In dissipative optical media, stable multi-vortex solitons require balance between diffraction, Kerr self-focusing, gain, and two-photon absorption (Borovkova et al., 2011). In superconductors and multicomponent condensates, unconventional multi-vortex matter emerges from competing inter-vortex forces acting on different length scales (Meng et al., 2016, Dantas et al., 2015, Meng et al., 2014). In layered Maxwell–Higgs models, auxiliary fields reshape a single vortex into a multilayer object through a tunable dielectric profile (Bazeia et al., 2019).

Another recurring mechanism is symmetry breaking from a simpler parent state. Rotating BEC multi-vortex polygons bifurcate from radially symmetric vortices as negative Hessian directions are removed by increasing rotation (García-Azpeitia et al., 2017). Multiply charged BEC vortices decay into multi-vortex sets when quadrupolar modes or axial inhomogeneity destabilize the original core (Telles et al., 2015, Teles et al., 2015). In SOC optical lattices, several vortex-lattice species coexist because corner degeneracies in the single-particle spectrum enforce particular coherent superpositions (Li et al., 2016). In superconducting networks, distinct multi-vortex states appear as neighboring cells of a discrete fluxoid honeycomb in control-parameter space (Polshyn et al., 2019).

A third theme is the tension between pairwise and collective descriptions. Some systems are well described by explicit pairwise interactions: multicomponent BEC vortex molecules (Dantas et al., 2015), superconducting multi-scale vortex matter (Meng et al., 2016), and co-rotating vortex polygons in viscous fluids (Swaminathan et al., 2015). Other systems require intrinsically collective formulations: Hermite-moment closures for deformable viscous vortices (Uminsky et al., 2010), continuum crystal densities in rotating BECs (Xie et al., 2017), or toroidal Hamiltonians with global geometric regularization (R et al., 16 Jun 2025). This suggests that “multi-vortex” is less a single object than a hierarchy of descriptions ranging from few-body topological molecules to many-body vortex matter.

Applications likewise reflect this breadth. Multi-ring gain landscapes permit deterministic generation of robust topological light structures (Borovkova et al., 2011). Concentric integrated resonators generate and sort multiple OAM channels on chip (Schulz et al., 2013), while reconfigurable metasurfaces convert several coaxial vortex-wave modes into spatially separated near-field focal spots for low-interference multi-user communication (Zhao et al., 18 Feb 2025). Superconducting multi-vortex control offers a route toward programmable braiding primitives (Polshyn et al., 2019). Layered-superconductor vortex lattices have been proposed as magnetic templates for ultracold-atom quantum emulators (Meng et al., 2014), and dusty-plasma multi-vortices provide a macroscopic testbed for driven dissipative vorticity organization (Choudhary et al., 2017).

A common misconception is that “multi-vortex” always means many identical, separate vortex cores. The literature does not support that restriction. It may denote multi-core dissipative solitons (Borovkova et al., 2011), multi-valued vortex-filament wavefunctions (Davidson, 2019), molecularly bound clusters (Dantas et al., 2015), multilayer internal ring structures (Bazeia et al., 2019), fluxoid configurations in multiply connected superconductors (Polshyn et al., 2019), or several concurrent OAM channels in photonics (Schulz et al., 2013, Zhao et al., 18 Feb 2025). A plausible implication is that the most robust definition is not geometric multiplicity alone, but the presence of multiple coupled topological or vortical degrees of freedom within one physical state or dynamical system.

In that broad sense, multi-vortex research traces how topology, interaction range, geometry, dissipation, and external structuring conspire to create composite vortex matter. The specific equations differ—from dissipative nonlinear Schrödinger models to Gross–Pitaevskii, Ginzburg–Landau, point-vortex Hamiltonians, and constrained multicomponent GP systems—but the central problem remains the same: how multiple vortices coexist, interact, stabilize, bifurcate, or reorganize into higher-order structures.