Pith. sign in

REVIEW 2 major objections 7 minor 78 references

Stable three-dimensional solitons in spin-orbit-coupled atomic-molecular condensates

T0 review · 2 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A flat spin-orbit coupling stabilizes 3D vortex solitons

desk verdict Plausible new mechanism for stable 3D vortex solitons, but the stability claim is not yet backed by the numerics as presented. read the letter →

arxiv 2608.07939 v1 pith:DPUTESOJ submitted 2026-08-08 quant-ph

Chao Kong , Jinqing Li , Guihua Chen , Guilong Li , Zhaopin Chen , Haiming Deng , Boris A. Malomed , Yongyao Li This is my paper · ORCID
classification quant-ph
keywords atomic-molecularcondensatesparametricinterconversionspin-orbitcouplingsemi-vorticesmixed-modesolitonscascadingapproximationsolitonstabilityquadraticnonlinearity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a planar (two-dimensional) spin-orbit coupling applied only to the atomic components of an atomic-molecular Bose-Einstein condensate is enough to stabilize fully three-dimensional solitons in free space, despite the quadratic (three-wave) nonlinearity that would otherwise make vortex states split apart. The solitons come in two types: semi-vortices, in which one atomic component is vortex-free and the other carries unit circulation, with the molecular field inheriting the vortex; and mixed modes, superpositions of zero- and unit-vorticity components with zero total angular momentum. Both families exist only up to a critical detuning $\alpha_c$, and in the semi-vortex case the vortex-carrying components hold more than half of the total norm. If correct, this addresses an open problem: stable vortex solitons in uniform quadratic media, previously thought to require competing nonlinearities or external structure.

What carries the argument

The load-bearing ingredient is the planar Rashba coupling $\lambda(\partial_x-i\partial_y)\varphi_2$ and $\lambda(\partial_x+i\partial_y)\varphi_1$ acting on the atomic components, combined with the phase-matched quadratic coupling $-\psi\varphi_1\varphi_2$. This pairing forces the molecular field to inherit the phase winding of the vortex atomic component, so the vortex topology is shared among all three fields. The conserved $z$-component of angular momentum $M_z$ ties the spin imbalance $|\varphi_1|^2-|\varphi_2|^2$ to the orbital angular momentum, providing the topological bookkeeping that makes the vortex-holding molecular component a natural partner. Stationary solutions are produced by imaginary-time propagation from SV and MM inputs, and the cascading approximation, which adiabatically eliminates $\psi$ at large negative detuning, is used to map the system onto effective two-component cubic equations, explaining both the formation mechanism and the nonexistence above $\alpha_c$.

What would settle it

Take the stationary SV and MM solutions shown in Figs. 1 and 2, add a random phase perturbation with amplitude $\epsilon$ between $10^{-4}$ and $10^{-2}$ of the local density, and evolve Eqs. (1)-(3) for at least an order of magnitude longer than the reported run; if the vortex ring fragments into several filaments or the angular momentum $M_z$ changes by more than numerical error, the stability claim is disproved. A cheaper quantitative check is the Bogoliubov spectrum of the linearized system around these stationary states: a positive imaginary part of any eigenfrequency would falsify asymptotic stability.

Watch

Extended reading notes

Core claim

The central discovery is that the mean-field system of two atomic components $\varphi_1,\varphi_2$ and a molecular field $\psi$, coupled by the quadratic three-wave interaction, supports stable free-space 3D solitons when a planar Rashba spin-orbit coupling acts on the atomic components alone. The solitons come in two symmetry classes: semi-vortices (SV), where $\varphi_1$ has zero vorticity and $\varphi_2$ and $\psi$ carry unit circulation, and mixed modes (MM), where each component superposes vorticities $0$ and $\pm1$ so that the total angular momentum vanishes. The paper reports existence up to a critical detuning $\alpha_c>0$, explains that boundary with the cascading approximation, and verifies stability by imaginary-time preparation followed by real-time propagation, supported by the monotonic decrease of chemical potential with norm. The vortex-carrying components in the SV states account for more than half of the total norm, a distinguishing feature relative to earlier SOC-supported solitons.

Load-bearing premise

The paper's stability verdict rests on watching imaginary-time-prepared solitons survive "long-term real-time propagation" without reporting the integration duration, perturbation size, or grid resolution, and without computing a linear-stability spectrum; if the numerical horizon is shorter than the growth time of an azimuthal instability, the blue stability regions would not represent true asymptotic stability.

Editorial extensions

If this is right

  • Stable free-space 3D vortex solitons become available in quadratic nonlinear media without a competing cubic nonlinearity, removing the azimuthal-instability obstruction that previously forced vortex fragmentation.
  • The same planar SOC mechanism should be realizable in a $^{39}$K condensate with Raman-laser-induced SOC and photoassociation, with the quoted scales giving about 6500 atoms for $N=100$ and a physical evolution time near one second for $t=1000$.
  • The negative slope $d\mu/dN$ persists across the studied parameter domains, so the predicted families should be identifiable as attractors in quench experiments that suddenly turn on the SOC and molecular coupling.
  • Both soliton families disappear above a critical mismatch: for the SV case the reported cutoff is $\alpha>1.06$ at the studied parameters, while for the MM case the critical detuning grows with atom number.
  • The large vortex-norm share, exceeding 50%, means the angular-momentum-carrying part dominates the wavefunction, which is favorable for applications that require a strong vortex response.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A Bogoliubov–de Gennes linear-stability calculation around the stationary SV and MM solutions would sharpen the regime boundaries: if unstable modes with small growth rates exist, the blue stability regions in Fig. 3 would shrink to metastability on the numerical horizon.
  • The cascading reduction suggests that for large negative detuning the atomic subsystem alone obeys a two-component cubic SOC model with attraction proportional to $1/|\alpha|$; comparing full soliton profiles with this reduced model's predictions would provide a quantitative test of the stabilization mechanism.
  • The same geometry may transfer to photonic systems with synthetic spin-orbit coupling and three-wave mixing, so that the optical-bullet counterpart would inherit the stabilization; the paper names this as a future direction but does not demonstrate it.
  • The conserved angular momentum expression contains a spin-imbalance term $\frac12(|\varphi_1|^2-|\varphi_2|^2)$, so a targeted perturbation that changes the relative populations of the two atomic components should have a measurable effect on the vortex stability, which could be tested in simulations today.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 7 minor

Summary. The paper proposes a mean-field model of an atomic-molecular Bose-Einstein condensate with planar Rashba spin-orbit coupling applied to the atomic components and quadratic three-wave coupling to a molecular component. Using imaginary-time propagation to construct stationary states, it reports semi-vortex (SV) and mixed-mode (MM) solitons in free 3D space, studies their dependence on the atomic-molecular mismatch alpha, explains the existence cutoff via a cascading approximation, and claims that the SV solitons carry a dominant vortex fraction of the norm. The central physical claim is that planar SOC alone stabilizes 3D vortex-carrying solitons in a quadratic medium, a regime previously thought to be dominated by azimuthal splitting instability. The existence and qualitative structure of the soliton families are plausible, but the stability claim is supported only by finite-time real-time propagation with no numerical parameters, no linear stability analysis, and no documented symmetry-breaking perturbation protocol.

Significance. If the stability claim is correct, the result is significant: it would identify a mechanism for stable vortex solitons in free space with quadratic nonlinearity, going beyond the usual reliance on competing cubic nonlinearities or external potentials. The model and cascading reduction are standard, and the existence families in Figs. 3-5 would be useful additions to the soliton literature. However, the manuscript currently does not supply the evidence needed to distinguish a genuinely stable 3D vortex soliton from a metastable state whose azimuthal instability has not yet developed over the numerical horizon. The central claim therefore needs to be either proven or carefully downgraded before publication.

major comments (2)
  1. [Figs. 1-3 and related captions] Stability is the load-bearing claim, but the evidence is not quantitative. The captions of Figs. 1 and 2 state that the solitons are "confirm[ed]" to be stable after long-term real-time propagation, and Fig. 3 draws stable and unstable regions from that criterion, yet the paper reports no integration time, grid resolution, boundary conditions, perturbation amplitude, or whether the evolution was carried out in full 3D without imposed rotational symmetry. The azimuthal splitting instability of vortex solitons in quadratic media (Refs. [12,13,24,31,32]) is precisely a non-axisymmetric process; if the solver or the initial state preserves continuous rotational symmetry, that instability cannot develop. Please add a Bogoliubov-de Gennes linear stability analysis with the azimuthal-mode spectrum, or at least full-3D real-time runs initialized with explicit non-axisymmetric perturbations, for representative points in each region of Fig. 3, and state the quantitative stability criterion and the numerical protocol.
  2. [Fig. 4 and discussion] The Vakhitov-Kolokolov condition dμ/dN<0 mentioned in the discussion of Fig. 4 is necessary but not sufficient for stability of vortex solitons, because it only constrains rotationally symmetric perturbations and cannot detect the m≠0 azimuthal modes that are dangerous in quadratic media. Since the text itself calls it an indicator of "plausible stability," it cannot justify the stable/unstable boundary in Fig. 3. The stability classification should be based on the linear spectrum or on explicitly documented symmetry-breaking dynamics.
minor comments (7)
  1. [Abstract] "This is scheme for realizing stable vortex solitons" should read "This provides a scheme for realizing stable vortex solitons."
  2. [Eq. (8)] The angular-momentum integral is written as dxdy, but the fields depend on three coordinates; it should be dxdydz, or the z-integration convention should be explained.
  3. [Eqs. (9)-(10) and Fig. 1] The vortex structure of the molecular component ψ in the SV ansatz is imposed by the phase-matching condition ψ(0)=φ1φ2 and the e^{iθ} factor in ansatz (10); the numerical observation in Fig. 1 is therefore a consistency check of the stationary solution, and the conclusion should not imply that this vortex structure is an emergent prediction.
  4. [Throughout] There are several typographical errors: "eliminaring" near Eq. (13), "repsectively" in the Fig. 1 caption, "solitons do not exists" in the Fig. 4 caption, and "region is includes" in the Fig. 3 caption.
  5. [Reference [5]] Reference [5] is cited as "Mod. Rev. Phys."; the journal is Reviews of Modern Physics.
  6. [Fig. 4] Please state the branch-continuation procedure used to produce the families in Fig. 4, including how N and α are swept and how the imaginary-time convergence criterion is defined.
  7. [Eq. (14) and Fig. 5] Equation (14) defines Fv using N2+N3, and N3 carries the factor 2 in Eq. (5); please state explicitly that this factor is included and discuss whether the conclusion that the vortex share exceeds 50% depends on that normalization convention.

Circularity Check

1 steps flagged · score 2.0 of 10

Molecular-component vortex in the SV ansatz is imposed by construction; the central stability claim is otherwise self-contained.

  1. self definitional [Eqs. (9) and (10), Section '3D solitons of the MM type...' immediately before Fig. 1]
    "φ1(t = 0) = A1 exp(−α1r2 −β1z2), φ2(t = 0) = A2r exp(−α2r2 −β2z2 +iθ), ψ(t = 0) = φ1φ2, (9) ... the molecular component of the ansatz is selected according to the phase-matching assumption, so that components φ2 and ψ carry the same vorticity 1, while φ1 has no vorticity."

    The unit topological charge of the molecular component in the SV soliton is not an emergent finding: it is put into the initial seed by ψ(0) = φ1φ2 with φ2 carrying e^{iθ}, and it is then preserved by the SV ansatz ψ = e^{−2iµt+iθ}Ψ(r,z). The later statement that the molecular component displays a ring density and 2π phase circulation (Fig. 1) is therefore a consequence of the assumed form, not an independent derivation. This does not make the stability results circular, because the stability of that assumed vortex-bearing state is still determined by solving the full time-dependent system, and the MM molecular structure is not fixed in the same way.

full rationale

The paper's central claim is a numerical existence-and-stability statement: for the model (1)-(3), stationary SV and MM solitons are found by imaginary-time propagation from explicit seeded inputs, then propagated in real time, with stability regions plotted in Fig. 3. No parameter is fitted to the outcome, and the stability classification is not imported from the cited background papers [63,64]. The only clear constructional circularity is the molecular SV vortex: Eqs. (9) and (10) prescribe that φ2 and ψ carry the same unit vorticity, so the paper's observation that the molecular component is a 3D vortex is a restatement of the ansatz. This is a minor, local self-definitional feature and does not touch the broader claim that such solitons are stable. The more serious weakness, namely the absence of reported integration times, resolution, and Bogoliubov-de Gennes spectra, is a correctness/rigor concern rather than circularity, since the finite-time real-time runs are genuine numerical evidence, however incompletely documented. Overall, the derivation chain is largely self-contained, with one by-construction vortex statement.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim depends on the standard mean-field model and on numerical stability inference. There are no fitted free parameters and no newly postulated entities. The main unproved input is the identification of finite-time propagation with asymptotic stability, and the cascading approximation used to explain the cutoff.

assumptions (4)
  • domain assumption The dilute BEC is accurately described by the mean-field Gross-Pitaevskii system (1)-(3) with quadratic three-wave coupling.
    The paper models coherent atom-molecule conversion and neglects losses, finite temperature, and beyond-mean-field effects. This is standard for this literature but is not validated against experiment here.
  • domain assumption A planar Rashba-type SOC can be imposed on the atomic components via two Raman lasers, as in Ref. [70].
    The assumed SOC acts only on phi1 and phi2, not on the molecular field; experimental realization relies on the cited scheme.
  • domain assumption For large negative detuning, the molecular field can be adiabatically eliminated, yielding the effective cubic equations (13).
    Used to explain the cutoff alpha_c; the approximation neglects derivatives of psi and is invoked outside a rigorous validity regime.
  • domain assumption The Vakhitov-Kolokolov criterion dmu/dN<0 is a meaningful stability indicator for these SOC systems.
    The criterion is rigorous for some NLS-type equations, but its sufficiency for multicomponent SOC GPEs is not established; the paper uses it as a plausibility check.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Stable three-dimensional solitons in spin-orbit-coupled atomic-molecular condensates." pith.science (2026). https://pith.science/paper/DPUTESOJ

@misc{pith2026260807939,
  author       = {Pith},
  title        = {Pith review of: Stable three-dimensional solitons in spin-orbit-coupled atomic-molecular condensates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DPUTESOJ}},
  note         = {Machine review of arXiv:2608.07939}
}
abstract

We elaborate a mechanism for the creation of stable three-dimensional (3D) solitons in spin-orbit-coupled (SOC) atomic-molecular Bose-Einstein condensate, modeled by the mean-field equations with the quadratic three-wave interaction, characterized by mismatch $\alpha $. The planar (effectively two-dimensional) SOC is applied to the soliton's atomic component, structuring it as a mixed mode (MM) or semi-vortex (SV). The molecular component of the SV soliton is shaped as a 3D vortex, while the molecular component in the MM soliton is an MM too. The solitons exist up to a critical value of $\alpha $. The system demonstrates a relatively large norm share of the vortex components, exceeding $50\%$ of the total norm, which is an essential feature of SOC-supported solitons. This is scheme for realizing stable vortex solitons in free space with the quadratic nonlinearity.

Figures

Figures reproduced from arXiv: 2608.07939 by the authors.

Figure 1
Figure 1. FIG. 1: A stable semi-vortex (SV) soliton with parameters [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: A stable soliton of the MM type for ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Parameter regions of the SV and MM solitons in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The dependence of the soliton chemical potential [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The norm ratio of the vortex components in the SV [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

78 extracted references · 71 canonical work pages

  1. [1]

    Corresponding author: woshidenghaiming@126.com cubic-quintic nonlinearities [39–46]. Furthermore, stable quantum droplets in binary BEC can also carry vorticity, due to the interplay of the cubic mean-field (MF) self- attraction and beyond-MF quartic self-repulsion [47–56]. Without the resort to the repulsive cubic nonlinear- ity, the stabilization of vort...

    work page Pith review arXiv 2026

  2. [2]

    C. T. Chen, B. C. Wu, A. D. Jiang, G. M. You, A New- Type Ultraviolet SHG Crystal-β-BaB2O4, Sci. Sin. B 28, 235(1985)

    work page 1985

  3. [3]

    F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, KxRb1−xTiOPO4: a new nonlinear optical material, J Appl Phys 47, 4980 (1976)

    work page 1976

  4. [4]

    R. A. Duine, H. T. C. Stoof, Atom-molecule coherence in Bose gases, Phys. Rep. 396, 115 (2004)

    work page 2004

  5. [5]

    M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, Quasi-phase-matched second harmonic generation: tun- ing and tolerances, IEEE J. Quantum Electron. 28, 2631- 2654 (1992)

    work page 1992

  6. [6]

    T. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Observation of Two-Dimensional Spatial Soli- tary Waves in a Quadratic Medium, Phys. Rev. Lett. 74, 5036 (1995)

    work page 1995

  7. [7]

    K¨ ohlerand, K

    T. K¨ ohlerand, K. G´ oral, P. S. Julienne, Production of cold molecules via magnetically tunable Feshbach reso- nances, Mod. Rev. Phys. 78, 1311 (2006)

    work page 2006

  8. [8]

    A. A. Kanashov and A. M. Rubenchik, On diffraction and dispersion effect on three wave interaction, Physica D 4, 122-134 (1981)

    work page 1981

Show all 78 references
  1. [9]

    W. E. Torruellas, Z. Wang, L. Torner, and G. I. Stege- man, Observation of mutual trapping and dragging of two-dimensional spatial solitary waves in a quadratic medium, Opt. Lett. 20, 1949 (1995)

    1995

  2. [10]

    B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak. Spatio-temporal solitons in op- tical media with a quadratic nonlinearity. Phys. Rev. E 56, 4725-4735 (1997)

    1997

  3. [11]

    G. I. Stegeman, D. J. Hagan, and L. Torner, Cascad- ing phenomena and their applications to all-optical sig- nal processing, mode-locking, pulse compression and soli- tons, Opt. Quant. Elect. 28, 1691-1740 (1996)

    1996

  4. [12]

    Etrich, F

    C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, Optical solitons in media with a quadratic nonlinearity, Prog. Opt. 41, 483 (2000)

    2000

  5. [13]

    X. Liu, K. Beckwitt, and F. W. Wise, Two-dimensional optical spatiotemporal solitons in quadratic media, Phys. Rev. E 62, 1328-1340 (2000)

    2000

  6. [14]

    Hayata and M

    K. Hayata and M. Koshiba, Multidimensional solitons in quadratic nonlinear media, Phys. Rev. Lett. 71, 3275 (1993)

    1993

  7. [15]

    A. V. Buryak, P. D. Trapani, D. V. Skryabin, and S. Trillo, Optical solitons due to quadratic nonlinearities: From basic physics to futuristic applications, Phys. Rep. 370, 63 (2002)

    2002

  8. [16]

    Lutsky and B

    V. Lutsky and B. A. Malomed, One- and two-dimensional solitons supported by singular modulation of quadratic nonlinearity, Phys. Rev. A 91, 023815 (2015)

    2015

  9. [17]

    J. F. Corney and O. Bang, Solitons in quadratic nonlinea r photonic crystals, Phys. Rev. E 64, 047601 (2001)

    2001

  10. [18]

    Ba˘ gci, Partially-symmetric lattice solitons in quadratic nonlinear media, Phys

    M. Ba˘ gci, Partially-symmetric lattice solitons in quadratic nonlinear media, Phys. Rev. A 103, 023530 (2021)

    2021

  11. [19]

    Di Menza, and L

    M Colin, L. Di Menza, and L. C. Saut, Solitons in quadratic media, Nonlinearity 29, 1000-1035 (2016)

    2016

  12. [20]

    P. D. Drummond, K. V. Kheruntsyan, and H. He, Co- herent Molecular Solitons in Bose-Einstein Condensates, 6 Phys. Rev. Lett. 81, 3055(1998)

    1998

  13. [21]

    Ole´ s, K

    B. Ole´ s, K. Sacha, Solitons in coupled atomic-molecul ar Bose-Einstein condensates, J. Phys. B 40, 1103 (2007)

    2007

  14. [22]

    B. A. Malomed, Multidimensional Solitons (AIP Pub- lishing, Melville, NY, 2022)

    2022

  15. [23]

    T. G. Vaughan, K. V. Kheruntsyan, and P. D. Drum- mond, Three-dimensional solitons in coupled atomic- molecular Bose-Einstein condensates, Phys. Rev. A 70, 063611 (2004)

    2004

  16. [24]

    W. J. Firth, and D. V. Skryabin, Optical solitons car- rying orbital angular momentum, Phys. Rev. Lett. 79, 2450(1997)

    1997

  17. [25]

    Allen, M

    L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 45, 8185 (1992)

    1992

  18. [26]

    P. D. Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, Observation of Quadratic Optical Vortex Solitons, Phys. Rev. Lett. 84, 3843-3846 (2000)

    2000

  19. [27]

    Towers, A

    I. Towers, A. V. Buryak, R. A. Sammut, and B. A. Mal- omed, Stable localized vortex solitons, Phys. Rev. E, 63, 055601(2001)

    2001

  20. [28]

    S. K. Adhikari, Stabilization of bright solitons and vo r- tex solitons in a trapless three-dimensional Bose-Einstei n condensate by temporal modulation of the scattering length, Phys. Rev. A 69, 063613 (2004)

    2004

  21. [29]

    C. Hang, V. V. Konotop, and B. A. Malomed, Gap vortex solitons in periodic media with quadratic nonlinearity, Phys. Rev. A 80, 023824 (2009)

    2009

  22. [30]

    Leblond, B

    H. Leblond, B. A. Malomed, and D. Mihalache, Three- dimensional vortex solitons in quasi-two-dimensional lat - tices, Phys. Rev. E 76, 026604(2007)

    2007

  23. [31]

    B. B. Baizakov, B. A. Malomed, and M. Salerno, Mul- tidimensional solitons in a low-dimensional periodic po- tential, Phys. Rev. A 70, 053613(2004)

    2004

  24. [32]

    D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, Observation of azimuthal mod- ulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal, Opt. Lett. 23, 1444 (1998)

    1998

  25. [33]

    Torner and D

    L. Torner and D. V. Petrov, Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second- harmonic generation, Electron. Lett. 33, 608 (1997)

    1997

  26. [34]

    X. Xu, F. Zhao, Y. Zhou, B. Liu, X. D. Jiang, B. A. Malomed, and Y. Y. Li. Vortex gap solitons in spin- orbit-coupled Bose-Einstein condensates with competing nonlinearities, Commun. Nonlinear. Sci. Numer. Simulat. 117, 106930(2023)

    2023

  27. [35]

    X. Xu, G. Ou, Z. Chen, B. Liu, W. C. Chen, B. A. Mal- omed, Y. Y. Li, Semidiscrete vortex soliton, Adv. Pho- tonics Res., 2, 2000082(2021)

    2021

  28. [36]

    DeSalvo, D

    R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stege- man, and E. W. Van Stryland, Self-focusing and self- defocusing by cascaded second-order effects in KTP, Opt. Lett. 17, 28 (1992)

    1992

  29. [37]

    Mihalache, D

    D. Mihalache, D. Mazilu, L. C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, Stable three-dimensional spinning optical solitons sup- ported by competing quadratic and cubic nonlinearities, Phys. Rev. E 66, 016613(2002)

    2002

  30. [38]

    T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, and P. S Julienne, Vortices in atomic-molecular Bose-Einstein condensates, J. Phys. B 4, S33 (2002)

    2002

  31. [39]

    Mihalache, D

    D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, Stable vortex solitons supported by competing quadratic and cubic nonlinearities, Phys. Rev. E 66, 066614 (2004)

    2004

  32. [40]

    Quiroga-Teixeiro and H

    M. Quiroga-Teixeiro and H. Michinel, Stable azimuthal stationary state in quintic nonlinear optical media, J. Opt. Soc. Am. B 14, 2004 (1997)

    1997

  33. [41]

    E. M. Wright, B. L. Lawrence, W. Torruellas, and G. Stegeman, Stable self-trapping and ring formation in polydiacetylene para-toluene sulfonate, Opt. Lett. 20, 2481(1995)

    1995

  34. [42]

    Mihalache, D

    D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, Stable Spinning Optical Solitons in Three Dimensions, Phys. Rev. Lett. 88, 073902 (2002)

    2002

  35. [43]

    V. I. Berezhiani, V. Skarka, and N. B. Aleksi´ c, Dy- namics of localized and nonlocalized optical vortex soli- tons in cubic-quintic nonlinear media, Phys. Rev. E 64, 057601(2001)

    2001

  36. [44]

    Mihalache, D

    D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kar- tashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Stable Vortex Tori in the Three-Dimensional Cubic- Quintic Ginzburg-Landau Equation, Phys. Rev. Lett. 97, 073904(2006)

    2006

  37. [45]

    R. L. Pego and H. A. Warchall, Spectrally stable encap- sulated vortices for nonlinear Schr¨ odinger equations, J. Nonlinear Sci. 12, 347 (2002)

    2002

  38. [46]

    Y. Y. Wang, L. Chen, C. Q. Dai, J. Zheng, and Y. Fan, Exact vector multipole and vortex solitons in the me- dia with spatially modulated cubic-quintic nonlinearity, Nonlinear Dyn. 90, 1269 (2017)

    2017

  39. [47]

    Cartes, J

    C. Cartes, J. Cisternas, O. Descalzi, and H. R. Brand, Model of a Two-Dimensional Extended Chaotic System: Evidence of Diffusing Dissipative Solitons, Phys. Rev. Lett. 109, 178303 (2012)

    2012

  40. [48]

    C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas, P. Cheiney, and L. Tarruell, Quantum liquid droplets in a mixture of Bose-Einstein condensates, Science 359, 301 (2018)

    2018

  41. [49]

    Ferrier-Barbu, H

    I. Ferrier-Barbu, H. Kadau, M. Schmitt, M. Wenzel, and T. Pfau, Observation of Quantum Droplets in a Strongly Dipolar Bose Gas, Phys. Rev. Lett. 116, 215301 (2016)

    2016

  42. [50]

    Z. Luo, W. Pang, B. Liu, Y. Li, B. A. Malomed, A new form of liquid matter: Quantum droplets, Front. Phys. 16, 32201 (2021)

    2021

  43. [51]

    Z. Zhao, G. Chen, B. Liu, Y. Li, Discrete vortex quan- tum droplets, Chaos, Solitons, and Fractals, 162, 116481 (2022)

    2022

  44. [52]

    Nie, J-H

    Y. Nie, J-H. Zheng, and T. Yang, Spectra and dynamics of quantum droplets in an optical lattice, Phys. Rev. A 108, 053310 (2023)

    2023

  45. [53]

    T. A. Yoˇ gurt, U. Tanyeri, A. Kele¸ s, and M. ¨ o. Oktel, Vortex lattices in strongly confined quantum droplets, Phys. Rev. A 108, 033315 (2023)

    2023

  46. [54]

    Y. V. Kartashov, B. A. Malomed, L. Tarruell, and L. Torner, Three-dimensional droplets of swirling superflu- ids, Phys. Rev. A 98, 013612 (2018)

    2018

  47. [55]

    L-J. He, B. Liu, and Y-C. Zhang, Quantum droplets in dipolar condensate mixtures with arbitrary dipole orien- tations, Phys. Rev. A 110, 013308 (2024)

    2024

  48. [56]

    Zhang, X

    X. Zhang, X. Xu, Y. Zheng, Z. Chen, B. Liu, C. Huang, B. A. Malomed, Y. Li, Semidiscrete quantum droplets and vortices, Phys. Rev. Lett. 123, 133901 (2019)

    2019

  49. [57]

    Y. Li, Z. Chen, Z. Luo, C. Huang, H. Tan, W. Pang, and B. A. Malomed, Two-dimensional vortex quantum droplets, Phys. Rev. A 98,063602(2018)

    2018

  50. [58]

    F. Zhao, X. Xu, H. He, L. Zhang, Y. Zhou, Z. Chen, and B.A. Malomed, Y Li, Vortex Solitons in Quasi-Phase- Matched Photonic Crystals, Phys. Rev. Lett. 130, 157203 (2023)

    2023

  51. [59]

    V. E. Lobanov, A. A. Kalinovich, O. V. Borovkova, and B. A. Malomed, Fundamental and vortex dissipa- 7 tive quadratic solitons supported by spatially localized gain, Phys. Rev. A 105, 013519 (2022)

    2022

  52. [60]

    Hnizdo, Spin–orbit coupling and the conservation of angular momentum, Eur

    V. Hnizdo, Spin–orbit coupling and the conservation of angular momentum, Eur. J. Phys. 33, 407-416 (2012)

    2012

  53. [61]

    C. Kong, J. Li, X. Tang, X. Li, J. Jiao, J. Cao, and H. Deng, Composite solitary vortices of three-wave mixing in quasi-phase-matched photonic crystals, Chaos, Soli- tons and Fractals 187, 115358(2024)

    2024

  54. [62]

    W. J. Thompson, Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems, Vol. 1 (WileyVCH, Hoboken, 1994)

    1994

  55. [63]

    optical bullets

    suggest that MM solitons may be set in motion). It is relevant to construct stable “optical bullets” in the photonicl version of the system, based on the three-wave parametric interaction and the photonic counterpart of SOC (cf. Ref. [76]). A challenging possibility is to con-...

  56. [64]

    Sakaguchi and B

    H. Sakaguchi and B. A. Malomed, Rotating nonlinear states in trapped binary Bose-Einstein condensates under the action of the spin-orbit coupling, Physica Scripta 99, 125276 (2024)

    2024

  57. [65]

    Sakaguchi, B

    H. Sakaguchi, B. Li, and B. A. Malomed, Creation of two-dimensional composite solitons in spin-orbit-couple d self-attractive Bose-Einstein condensates in free space, Phys. Rev. E 89, 032920 (2014)

    2014

  58. [66]

    Zhang, Z.-W

    Y.-C. Zhang, Z.-W. Zhou, B. A. Malomed, and H. Pu, Stable solitons in three dimensional free space without the ground state: Self-trapped Bose-Einstein condensates with spin-orbit coupling, Phys. Rev. Lett. 115, 253902 (2015)

    2015

  59. [67]

    Huang, Y

    C. Huang, Y. Ye, S. Liu, H. He, W. Pang, B. A. Mal- omed, and Y. Li, Excited states of two-dimensional soli- tons supported by spin-orbit coupling and field-induced dipole-dipole repulsion, Phys. Rev. A 97, 013636 (2018)

    2018

  60. [68]

    Zhong, Z

    R. Zhong, Z. Chen, C. Huang, Z. Luo, H. Tan, B. A. Mal- omed, Y. Li, Self-trapping under two-dimensional spin- orbit coupling and spatially growing repulsive nonlinear- ity, Front. Phys. 13, 130311 (2018)

    2018

  61. [69]

    W. Z. Bao and Q. Du, Computing the ground state so- lution of Bose-Einstein condensates by a normalized gra- dient flow, SIAM J. Sci. Comp. 25, 1674 (2004)

    2004

  62. [70]

    Dalfovo, S

    F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. String ari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463 (1999)

    1999

  63. [71]

    M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor, Science 269, 5221(1995)

    1995

  64. [72]

    Y.-J. Lin, K. Jim´ enez-Garc ´ ıa, and I. B. Spielman, Spin-Orbit-Coupled Bose-Einstein Condensates, Nature 471,7336 (2011)

    2011

  65. [73]

    K. M. Jones, E. Tiesinga, P. D. Lett, and P. S. Juli- enne, Ultracold photoassociation spectroscopy: Long- range molecules and atomic scattering, Rev. Mod. Phys. 78, 483(2006)

    2006

  66. [74]

    C.-D. Mi, K. S. Nawaz, P.-J. Wang, L.-C. Chen, Z.-M. Meng, L. Huang and J. Zhang, Production of dual species Bose-Einstein condensates of 39K and 87Rb, Chin. Phys. B 30, 063401(2021)

    2021

  67. [75]

    N. G. Vakhitov and A. A. Kolokolov, Stationary solu- tions of the wave equation in a medium with nonlinearity saturation, Radiophys. Quantum Electron. 16, 783-789 (1973); https://doi.org/10.1007/BF01031343

    1973 doi

  68. [76]

    Berg´ e, Wave collapse in physics: principles and ap- plications to light and plasma waves, Phys

    L. Berg´ e, Wave collapse in physics: principles and ap- plications to light and plasma waves, Phys. Rep. 303, 259-370 (1998)

    1998

  69. [77]

    V. E. Zakharov and E. A. Kuznetsov, Solitons and col- lapses: two evolution scenarios of nonlinear wave sys- tems, Physics – Uspekhi 55, 535-556 (2012)

    2012

  70. [78]

    Y. V. Kartashov, B. A. Malomed, V. V. Konotop, V. E. Lobanov, and L. Torner, Stabilization of solitons in bulk Kerr media by dispersive coupling, Opt. Lett. 40, 1045-1048 (2015)

    2015

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.