Ferrodark Soliton in Spin-1 Condensates

• Ferrodark soliton is a topological magnetic kink in spin-1 Bose-Einstein condensates characterized by a tanh magnetization profile and a finite-density core.
• It exhibits two distinct branches—type-I with positive and type-II with negative inertial mass—each defined by unique solitonic profiles and core structures.
• Its dynamics, including propagation limits, trap-induced oscillations, and stability features, offer deep insights into spin and nematic coupling in ferromagnetic systems.

A ferrodark soliton (FDS) is, in the spin-1 Bose-Einstein-condensate literature, a topological soliton in the easy-plane phase of a ferromagnetic condensate, appearing as a kink or magnetic domain wall in the transverse magnetization $F_\perp = F_x + iF_y$. It interpolates between domains with opposite transverse magnetization, so that $F_\perp(-\infty) = -F_\perp(+\infty)$, and is distinguished from a scalar dark soliton by the persistence of a finite superfluid density at the soliton core and by the role of spin and nematic degrees of freedom. The modern FDS literature further separates these objects into type-I and type-II branches, with positive and negative inertial mass, respectively, and develops exact solutions, core-structure theory, trap dynamics, stability theory, and collision phenomenology for them (Yu et al., 2021).

1. Terminology and physical setting

In a spin-1 condensate, the order parameter is a three-component spinor

$\psi = (\psi_{+1}, \psi_0, \psi_{-1})^T,$

with Hamiltonian density

$$\mathcal{H} = \frac{\hbar^2 |\nabla \psi|^2}{2M} + \frac{g_n}{2}|\psi^\dagger \psi|^2 + \frac{g_s}{2}|\psi^\dagger \mathbf{S} \psi|^2 + q \psi^\dagger S_z^2 \psi,$$

where $g_n>0$, $g_s<0$, and $q$ is the quadratic Zeeman energy. The relevant regime is the easy-plane phase, for which the ground state is transversely magnetized and the magnetic order supports $\mathbb{Z}_2$ domain walls (Yu et al., 2021).

The magnetic-order topological charge used in the static-core analysis is

$$Q_{F_\perp} = \frac{1}{\pi}\int_{-\infty}^{+\infty} dx \,\partial_x \tau(x),$$

with $\tau(x)$ the orientation of the in-plane magnetization. In the same framework, the mass-superfluid sector admits a distinct topological classification through

$F_\perp(-\infty) = -F_\perp(+\infty)$0

which is central to the distinction between FDSs and dark-dark-dark vector solitons (Yu et al., 2022).

The terminology is not unique across subfields. In discrete ferromagnetic chains, the phrase “dark (ferrodark) soliton” appears for envelope excitations of anisotropic Heisenberg chains with first- and second-neighbor interactions (Primatarowa et al., 2011). A separate astrophysical paper on fuzzy-dark-matter solitonic cores notes that such dense ground-state cores are “sometimes referred to as ‘Ferrodark Solitons’,” while also stating that this is not its primary focus; this suggests that the label is context-dependent outside ultracold-gas work (Woo, 31 Dec 2025).

2. Static FDSs and the type-I/type-II distinction

Static FDSs are Ising-type domain walls in the in-plane magnetization. At the integrable point $F_\perp(-\infty) = -F_\perp(+\infty)$1, the Gross-Pitaevskii equations decouple and admit exact static solutions of two kinds: type-I, for which the solitonic profile is in the $F_\perp(-\infty) = -F_\perp(+\infty)$2 components while $F_\perp(-\infty) = -F_\perp(+\infty)$3 remains uniform, and type-II, for which the solitonic profile is in the $F_\perp(-\infty) = -F_\perp(+\infty)$4 component while $F_\perp(-\infty) = -F_\perp(+\infty)$5 remain uniform (Yu et al., 2022).

For both types, the transverse magnetization has a tanh profile,

$F_\perp(-\infty) = -F_\perp(+\infty)$6

with

$F_\perp(-\infty) = -F_\perp(+\infty)$7

and $F_\perp(-\infty) = -F_\perp(+\infty)$8. In both cases the magnetization vanishes at the core while the superfluid density remains finite, which is one of the defining differences from scalar dark solitons (Yu et al., 2022).

The same two-branch structure persists in the propagating problem. Type-I is the low-energy branch with positive inertial mass, whereas type-II is the high-energy branch with negative inertial mass (Yu et al., 2021).

Feature	Type-I FDS	Type-II FDS
Excitation branch	Low energy	High energy
Inertial mass	Positive	Negative
Exact static core at $F_\perp(-\infty) = -F_\perp(+\infty)$9	Solitonic profile in $\psi = (\psi_{+1}, \psi_0, \psi_{-1})^T,$0	Solitonic profile in $\psi = (\psi_{+1}, \psi_0, \psi_{-1})^T,$1
Width structure away from exact point	Single width	Two characteristic length scales

3. Propagating solutions, masses, and speed bound

Exact propagating FDSs were derived in one dimension for $\psi = (\psi_{+1}, \psi_0, \psi_{-1})^T,$2. A key result is that propagation requires a nonzero longitudinal magnetic field, represented by the quadratic Zeeman term $\psi = (\psi_{+1}, \psi_0, \psi_{-1})^T,$3; for $\psi = (\psi_{+1}, \psi_0, \psi_{-1})^T,$4, magnetization is strictly conserved and the solitons cannot propagate (Yu et al., 2021).

The transverse magnetization and total density of the propagating solutions are

$\psi = (\psi_{+1}, \psi_0, \psi_{-1})^T,$5

$\psi = (\psi_{+1}, \psi_0, \psi_{-1})^T,$6

where

$\psi = (\psi_{+1}, \psi_0, \psi_{-1})^T,$7

These formulas make explicit that the FDS is a finite-density object with a magnetization kink and a density dip rather than a density node (Yu et al., 2021).

The velocity is bounded by a characteristic FDS speed,

$\psi = (\psi_{+1}, \psi_0, \psi_{-1})^T,$8

At $\psi = (\psi_{+1}, \psi_0, \psi_{-1})^T,$9, the type-I and type-II solutions become identical. This speed bound is explicitly stated to be different from speed limits set by the elementary excitations (Yu et al., 2021).

The mass structure is anomalous. The inertial masses are

$$\mathcal{H} = \frac{\hbar^2 |\nabla \psi|^2}{2M} + \frac{g_n}{2}|\psi^\dagger \psi|^2 + \frac{g_s}{2}|\psi^\dagger \mathbf{S} \psi|^2 + q \psi^\dagger S_z^2 \psi,$$0

At the same time, the physical mass $$\mathcal{H} = \frac{\hbar^2 |\nabla \psi|^2}{2M} + \frac{g_n}{2}|\psi^\dagger \psi|^2 + \frac{g_s}{2}|\psi^\dagger \mathbf{S} \psi|^2 + q \psi^\dagger S_z^2 \psi,$$1 is negative for both types. The propagating solutions also have zero transverse magnetization at their core and no net magnetic current, while their time evolution is attributed to coupling to nematicity and to internal Josephson-like spin currents,

$$\mathcal{H} = \frac{\hbar^2 |\nabla \psi|^2}{2M} + \frac{g_n}{2}|\psi^\dagger \psi|^2 + \frac{g_s}{2}|\psi^\dagger \mathbf{S} \psi|^2 + q \psi^\dagger S_z^2 \psi,$$2

These features sharply separate FDS dynamics from the scalar dark-soliton case (Yu et al., 2021).

4. Core structure, singlet amplitude, and nematic observables

Away from the exactly solvable point, the static-core problem remains analytically tractable through ansatzes matched to asymptotic and near-core behavior. The principal structural result is that the type-I FDS has a single characteristic width, whereas the type-II FDS requires two characteristic length scales, one for the inner-core region and one for the outer region. The proposed ansatzes are reported to show excellent agreement with numerical solutions across parameter space (Yu et al., 2022).

The spin-singlet amplitude is

$$\mathcal{H} = \frac{\hbar^2 |\nabla \psi|^2}{2M} + \frac{g_n}{2}|\psi^\dagger \psi|^2 + \frac{g_s}{2}|\psi^\dagger \mathbf{S} \psi|^2 + q \psi^\dagger S_z^2 \psi,$$3

and satisfies

$$\mathcal{H} = \frac{\hbar^2 |\nabla \psi|^2}{2M} + \frac{g_n}{2}|\psi^\dagger \psi|^2 + \frac{g_s}{2}|\psi^\dagger \mathbf{S} \psi|^2 + q \psi^\dagger S_z^2 \psi,$$4

Its profile distinguishes the two FDS types through a hump or dip. The nematic tensor density is

$$\mathcal{H} = \frac{\hbar^2 |\nabla \psi|^2}{2M} + \frac{g_n}{2}|\psi^\dagger \psi|^2 + \frac{g_s}{2}|\psi^\dagger \mathbf{S} \psi|^2 + q \psi^\dagger S_z^2 \psi,$$5

with rotationally invariant density

$$\mathcal{H} = \frac{\hbar^2 |\nabla \psi|^2}{2M} + \frac{g_n}{2}|\psi^\dagger \psi|^2 + \frac{g_s}{2}|\psi^\dagger \mathbf{S} \psi|^2 + q \psi^\dagger S_z^2 \psi,$$6

and nematic current tensor

$$\mathcal{H} = \frac{\hbar^2 |\nabla \psi|^2}{2M} + \frac{g_n}{2}|\psi^\dagger \psi|^2 + \frac{g_s}{2}|\psi^\dagger \mathbf{S} \psi|^2 + q \psi^\dagger S_z^2 \psi,$$7

The static-core study emphasizes these quantities as experimentally relevant descriptors beyond the magnetization profile alone (Yu et al., 2022).

This nematic perspective is extended in the ring-FDS problem. At the core of a ring FDS, where the magnetization vanishes, the local order is described by the nematic tensor, and the core dynamics can be parameterized by a single parameter $$\mathcal{H} = \frac{\hbar^2 |\nabla \psi|^2}{2M} + \frac{g_n}{2}|\psi^\dagger \psi|^2 + \frac{g_s}{2}|\psi^\dagger \mathbf{S} \psi|^2 + q \psi^\dagger S_z^2 \psi,$$8 that continuously connects type-I and type-II FDSs through

$$\mathcal{H} = \frac{\hbar^2 |\nabla \psi|^2}{2M} + \frac{g_n}{2}|\psi^\dagger \psi|^2 + \frac{g_s}{2}|\psi^\dagger \mathbf{S} \psi|^2 + q \psi^\dagger S_z^2 \psi,$$9

The associated core motion is described as “nematic tensor breathing,” especially in the oscillating-ring regime (Yu, 17 Sep 2025).

5. Motion in external potentials and trapped superfluids

In a finite one-dimensional system subject to a linear potential, the propagating-solution analysis predicts oscillatory motion driven by transitions between type-I and type-II at the maximum speed. A type-I FDS can accelerate until it reaches $g_n>0$0, where it connects smoothly to the type-II branch; the sign change of the inertial mass then reverses the acceleration direction, producing oscillatory motion. The physical mass remains negative throughout this process (Yu et al., 2021).

A more general trapped-superfluid theory decomposes the force on the FDS into buoyancy and spin-correction terms,

$g_n>0$1

with

$g_n>0$2

For FDSs, the spin correction is generically nonzero and is absent in scalar or non-magnetic solitons. In a harmonic trap, this leads to an effective-particle mapping to an emergent quartic potential,

$g_n>0$3

rather than to the harmonic oscillator familiar from scalar-soliton dynamics (Biguo et al., 17 Apr 2025).

The dynamical consequences depend on the branch. A type-I FDS initially near the trap center exhibits single-sided oscillation corresponding to motion around one minimum of an emergent double-well potential. For type-II, increasing the initial displacement produces three regimes: trap-centered harmonic oscillation, then anharmonic oscillation, and finally single-sided oscillation. The change is described as symmetry breaking of the effective quartic potential from a single minimum to a double-well form. By contrast, in a hard-wall trap with linear potential, the same formalism yields an effective harmonic oscillator (Biguo et al., 17 Apr 2025).

6. Stability, transverse dynamics, and ring FDSs

In quasi-two-dimensional ferromagnetic spin-1 condensates, dark-soliton-like magnetic domain walls were shown to possess a magnetization tanh profile with nonvanishing superfluid density at the core. The standing-wave excitations of such a wall oscillate without decay and are stable against snake instability, while a distinct spin-twist instability rotates the magnetization locally but does not destroy the topological structure of the wall (Yu et al., 2020).

For FDSs specifically in the easy-plane phase, the two branches again behave differently under transverse perturbations. Type-I has positive inertial mass and exhibits a single dynamical instability that generates in-plane spin winding and polar-core spin-vortex dipoles; its positive inertial mass leads to elastic oscillations of the soliton under transverse perturbations. Type-II has negative inertial mass and exhibits both a snake instability and a spin-twist instability. A central result is that the snake instability does not cause breakdown: segments of the type-II FDS convert to type-I, mass-vortex dipoles are produced, and a hybridized chain of the two soliton types and vortices forms while the magnetic-domain-wall topology is preserved (Yu et al., 2024).

Ring FDSs in a homogeneous quasi-2D ferromagnetic spin-1 condensate add a further dynamical regime. Unlike scalar ring dark solitons, which expand in a homogeneous system, the ring FDS radius can exhibit self-sustained oscillations accompanied by nematic breathing at the core. In the hydrodynamic regime $g_n>0$4, the ring-radius equation of motion is

$g_n>0$5

which is recast in a form analogous to the inviscid Rayleigh-Plesset equation, but with anomalous terms. Outside that regime, density and spin-wave emission cause damping, shrinkage, and, below a threshold,

$g_n>0$6

collapse and annihilation. In the $g_n>0$7 limit, the ring radius and the eigenvalues of the nematic tensor become stationary, while oscillations of the nematic tensor components persist at the core (Yu, 17 Sep 2025).

7. Collisions, related soliton families, and broader analytical contexts

The collision problem reveals strongly non-integrable behavior. For a type-I FDS–antiFDS pair, low incoming velocities lead to annihilation of the $g_n>0$8 kink pair and the formation of an extremely long-lived dissipative breather, a localized wave packet with out-of-phase oscillating magnetization and number densities. The breather loses energy through periodic emission of spin and density waves, and its energy is reported to decay logarithmically in time. As the incoming velocity approaches a critical value from below, the lifetime of a stationary reproduced type-I pair diverges as a power law. Above the critical velocity, a pair with finite separating velocity is reproduced. By contrast, a type-II pair only reflects and never annihilates. For mixed type-I/type-II collisions, the magnetization kink reflects while the mass-density profiles pass through one another, an effect described as spin-mass separation (Bai et al., 5 Sep 2025).

The FDS concept also borders other dark-soliton constructions. In discrete ferromagnetic chains with first- and second-neighbor interactions, quasiclassical reduction yields a nonlinear Schrödinger equation

$g_n>0$9

with dark-soliton existence criterion

$g_s<0$0

and solution

$g_s<0$1

Here second-neighbor exchange and anisotropy shift the Brillouin-zone regions where dark or bright solitons exist, and defect scattering introduces transmission or reflection depending on a critical defect strength (Primatarowa et al., 2011).

At a more abstract level, determinant tau-function methods based on soliton Fay identities provide a general dark-soliton framework for multicomponent and multidimensional nonlinear Schrödinger-type systems. The core construction uses determinants $g_s<0$2 under a rank-one condition

$g_s<0$3

and generates component fields through ratios of shifted determinants. The same summary describes this machinery as general enough to subsume multicomponent NLS and related ferro variants, and therefore as a route to FDS-type solutions in broader integrable settings (Vekslerchik, 2014).

In this sense, “ferrodark soliton” denotes not a single universal object across all of physics, but a family of dark-soliton or domain-wall structures associated with ferromagnetic order parameters. The most developed and specific usage is the spin-1 BEC case, where FDSs are $g_s<0$4 magnetic kinks with finite-density cores, two inertial-mass branches, nontrivial nematic structure, anomalous trap dynamics, and collision phenomena that include dissipative breathers, pair reproduction, and spin-mass separation (Yu et al., 2021).