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arxiv: 2511.04385 · v2 · submitted 2025-11-06 · ❄️ cond-mat.quant-gas

Stability of dark solitons in a bubble Bose-Einstein condensate

Raphael Wictky Sallatti , Lauro Tomio , Dmitry Pelinovsky , Arnaldo Gammal This is my paper

Pith reviewed 2026-05-18 00:16 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas
keywords dark solitonsBose-Einstein condensatespherical geometrysnake instabilityvortex dipolesangular momentumstability thresholdsurface confinement
0
0 comments X

The pith

Dark solitons on a spherical Bose-Einstein condensate become unstable above a threshold in nonlinear strength and decay into vortex pairs through a single unstable mode for each angular momentum m at least 2.

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

The paper sets out to establish stability criteria for dark solitons placed on the surface of a spherical Bose-Einstein condensate. It demonstrates that these solitons remain stable only up to a sharp threshold value of the nonlinear interaction parameter. Beyond that point they undergo snake instabilities and break into vortex dipoles. The authors show analytically and numerically that the entire decay process for every angular momentum m greater than or equal to 2 is governed by exactly one unstable mode. This single-mode mechanism fixes the final vortex state and prevents the formation of vortex rings that would appear in three-dimensional bulk condensates.

Core claim

We demonstrate a sharp instability threshold in the nonlinear parameter, beyond which solitons decay into vortex dipoles via snake instabilities. Analytically and numerically, we prove this decay is dictated by a single unstable mode for each angular momentum m greater than or equal to 2, which is a universal mechanism that controls the resulting vortex state. Unlike in the full three-dimensional case, where snake instabilities lead to vortex rings, a dark soliton confined to the surface of a bubble can only decay into vortex pairs.

What carries the argument

The single unstable mode for each angular momentum m at least 2 that drives the snake instability and selects the vortex-dipole outcome in the effective two-dimensional dynamics on the sphere.

Load-bearing premise

The condensate is perfectly confined to an infinitesimally thin spherical shell so that the dynamics reduce to an effective two-dimensional nonlinear Schrödinger equation on the sphere.

What would settle it

A direct numerical integration or experiment that finds either multiple unstable modes for some m or a decay product other than a vortex pair at the predicted nonlinear threshold would falsify the single-mode claim.

Figures

Figures reproduced from arXiv: 2511.04385 by Arnaldo Gammal, Dmitry Pelinovsky, Lauro Tomio, Raphael Wictky Sallatti.

Figure 1
Figure 1. Figure 1: FIG. 1. (Color on-line) In (a), we show profiles of dark soli [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Dark-soliton dynamics, for given time instants [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: for the given data]. At t = 0, we observe the dark-soliton positions at the equator, with their respec￾tive widths being smaller for larger values of ε. Next, in the time evolution, we can verify the onset of snake instabilities provoking the breakup of such dark solitons in vortex-antivortex pairs. As seen in panels (c) and (d), for m = 2, two pairs of vortices are formed; in (g) and (h), for m = 3, we ob… view at source ↗
read the original abstract

The stability of nonlinear waves on curved surfaces is a problem of widespread interest across physics. Here, we establish the stability criteria for dark solitons on a spherical Bose-Einstein condensate. We demonstrate a sharp instability threshold in the nonlinear parameter, beyond which solitons decay into vortex dipoles via snake instabilities. Analytically and numerically, we prove this decay is dictated by a single unstable mode for each angular momentum $m \geq 2$, which is a universal mechanism that controls the resulting vortex state. Unlike in the full three-dimensional case, where snake instabilities lead to vortex rings, a dark soliton confined to the surface of a bubble can only decay into vortex pairs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the stability of dark solitons in a bubble Bose-Einstein condensate by reducing the three-dimensional Gross-Pitaevskii equation to an effective two-dimensional nonlinear Schrödinger equation on a sphere. It reports a sharp threshold in the nonlinear parameter above which the solitons become unstable and decay into vortex dipoles via a snake instability. Analytically and numerically, the decay is shown to be controlled by exactly one unstable mode for each angular momentum m ≥ 2, providing a universal mechanism that determines the final vortex-pair state, in contrast to the vortex-ring formation seen in three-dimensional geometries.

Significance. If the thin-shell reduction is justified, the work identifies a clean, single-mode mechanism for soliton decay on curved manifolds and supplies both an analytical mode-counting argument and supporting numerical time evolution. These elements furnish falsifiable predictions for the instability threshold and the resulting vortex configurations, which could be tested in shell-trapped condensates. The contrast with three-dimensional behavior is a useful conceptual contribution to the study of nonlinear waves in confined quantum fluids.

major comments (1)
  1. The central claim that decay is dictated by a single unstable mode for each m ≥ 2 rests on linearization of the effective 2D Gross-Pitaevskii equation strictly confined to an infinitesimally thin spherical surface. The model derivation (implicit in the setup of the spherical NLS) does not include an explicit Bogoliubov-de Gennes analysis with finite radial width to verify that radial excitations remain gapped and do not couple to the angular modes near the reported threshold. Without this check, the asserted single-mode dominance and universality are not yet secured for a physical bubble of finite thickness.
minor comments (2)
  1. The abstract states that both analytical linear stability and numerical simulations confirm the single-mode picture, but the dispersion relation for the unstable mode and the precise definition of the nonlinear-parameter threshold are not quoted explicitly; adding these expressions would improve readability.
  2. Figure captions should indicate the angular momentum m and the value of the nonlinear parameter used in each panel so that the connection to the analytical threshold is immediate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive feedback. We appreciate the recognition of the conceptual contribution regarding the single-mode decay mechanism on curved geometries. We address the major comment below and have revised the manuscript to strengthen the justification of our effective model.

read point-by-point responses

  1. Referee: The central claim that decay is dictated by a single unstable mode for each m ≥ 2 rests on linearization of the effective 2D Gross-Pitaevskii equation strictly confined to an infinitesimally thin spherical surface. The model derivation (implicit in the setup of the spherical NLS) does not include an explicit Bogoliubov-de Gennes analysis with finite radial width to verify that radial excitations remain gapped and do not couple to the angular modes near the reported threshold. Without this check, the asserted single-mode dominance and universality are not yet secured for a physical bubble of finite thickness.

    Authors: We thank the referee for this important observation on the validity of the thin-shell reduction. Our effective two-dimensional model is obtained by integrating the three-dimensional Gross-Pitaevskii equation over the tightly confined radial direction, assuming a strong radial trap that fixes the radial profile to its ground state. In this regime, radial excitations are gapped by an energy scale proportional to the radial trapping frequency, which is taken to be much larger than both the chemical potential and the growth rates of the angular instabilities near the reported threshold. This separation of scales suppresses coupling between radial and angular modes for the parameters we consider. We agree that an explicit finite-width Bogoliubov-de Gennes calculation in the full three-dimensional geometry would provide further confirmation. To address this, we have added a new subsection (Section II.B) that derives the radial gap estimate, states the validity conditions for the thin-shell limit, and shows that the gap remains larger than the maximum imaginary frequency of the unstable angular modes throughout the relevant range of the nonlinear parameter. This addition secures the single-mode dominance claim within the stated approximations while preserving the focus on curvature-induced effects. revision: yes

Circularity Check

0 steps flagged

No significant circularity: stability analysis is independent of fitted soliton data

full rationale

The paper derives an effective 2D Gross-Pitaevskii equation on the sphere via the thin-shell reduction and then performs a standard linear stability analysis by linearizing around the dark soliton profile and solving the resulting Bogoliubov-de Gennes eigenvalue problem for angular modes m ≥ 2. The claimed single unstable mode and its role in vortex-dipole decay follow directly from the spectrum of this operator on the sphere; no parameter is fitted to the soliton itself and then re-used as a 'prediction,' nor is any load-bearing step reduced to a self-citation or ansatz imported from the authors' prior work. The derivation chain is therefore self-contained within the reduced model and does not exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the 2D nonlinear Schrödinger equation on the sphere (derived from the 3D Gross-Pitaevskii equation under tight radial confinement) and on the assumption that the background density is uniform on the sphere. No free parameters are explicitly fitted in the abstract; the nonlinear coefficient is treated as a tunable control parameter.

axioms (1)
  • domain assumption The condensate is confined to an infinitesimally thin spherical shell, allowing reduction to an effective 2D Gross-Pitaevskii equation on the sphere.
    Invoked to justify the geometry and the form of the Laplacian used in the stability analysis.

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Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    For each angular modem, the threshold values [whereIm(ω m) = 0] ofεare given, withε m being the exact numerical results andεth m given by the analytical approxima- tion (16)

    form≫2, εth m = 4m(m−1),(16) 4 TABLE I. For each angular modem, the threshold values [whereIm(ω m) = 0] ofεare given, withε m being the exact numerical results andεth m given by the analytical approxima- tion (16). The respective values ofµare also presented. m 2 3 4 5 6 7 εm 8.367 24.402 48.416 80.420 120.420 168.420 εth m 8 24 48 80 120 168 µ 8.182 18.2...

    work page 2024

  2. [2]

    D. C. Aveline, J. R. Williams, E. R. Elliott, C. Dutenhof- fer, J. R. Kellogg, J. M. Kohel, N. E. Lay, K. Oudrhiri, R. F. Shotwell, N. Yu, and R. J. Thompson, Observation of Bose–Einstein condensates in an Earth-orbiting research lab, Nature582, 193 (2020)

    work page 2020

  3. [3]

    R. A. Carollo, D. C. Aveline, B. Rhyno, S. Vishveshwara, C. Lannert, J. D. Murphree, E. R. Elliott, J. R. Williams, R. J. Thompson, and N. Lundblad, Observation of ultra- cold atomic bubbles in orbital microgravity, Nature606, 281 (2022)

    work page 2022

  4. [4]

    Zobay and B

    O. Zobay and B. M. Garraway, Two-dimensional atom trapping in field-induced adiabatic potentials, Phys. Rev. Lett.86, 1195 (2001)

    work page 2001

  5. [5]

    Colombe, B

    Y. Colombe, B. Mercier, H. Perrin and V. Lorent, Loading adressedZeemantrapwithcoldatoms, J.Phys.IVFrance 116, 247 (2004)

    work page 2004

  6. [6]

    Zobay and B

    O. Zobay and B. M. Garraway, Atom trapping and two- dimensional Bose-Einstein condensates in field-induced adiabatic potentials, Phys. Rev. A69, 023605 (2004)

    work page 2004

  7. [7]

    F. Jia, Z. Huang, L. Qiu, R. Zhou, Y. Yan, and D. Wang, Expansion dynamics of a shell-shaped Bose-Einstein con- densate, Phys. Rev. Lett.129, 243402 (2022)

    work page 2022

  8. [8]

    Y. Guo, E. M. Gutierrez, D. Rey, T. Badr, A. Perrin, L. Longchambon, V. S. Bagnato, H. Perrin and R. Dubessy, Expansion of a quantum gas in a shell trap, New J. Phys. 24, 093040 (2022)

    work page 2022

  9. [9]

    R.DubessyandH.Perrin, Quantumgasesinbubbletraps, AVS Quantum Sci.7, 010501 (2025)

    work page 2025

  10. [10]

    Tononi and L

    A. Tononi and L. Salasnich, Low-dimensional quantum gases in curved geometries, Nat. Rev. Phys.5, 398 (2023)

    work page 2023

  11. [11]

    Padavić, K

    K. Padavić, K. Sun, C. Lannert, and S. Vishveshwara, Physics of hollow Bose-Einstein condensates, Europhys. Lett.120, 20004 (2017)

    work page 2017

  12. [12]

    K. Sun, K. Padavić, F. Yang, S. Vishveshwara, and C. Lannert, Static and dynamic properties of shell-shaped condensates, Phys. Rev. A98, 013609 (2018)

    work page 2018

  13. [13]

    S. J. Bereta, L. Madeira, V. S. Bagnato and M. A. Cara- canhas, Bose-Einstein condensation in spherically sym- metric traps, Am. J. Phys.87, 924 (2019)

    work page 2019

  14. [14]

    Tononi and L

    A. Tononi and L. Salasnich, Bose-Einstein condensation on the surface of a sphere, Phys. Rev. Lett.123, 160403 (2019)

    work page 2019

  15. [15]

    N. S. Móller, F. E. A. dos Santos, V.S. Bagnato, and A. Pelster, Bose–Einstein condensation on curved manifolds, New J. Phys.22, 063059 (2020)

    work page 2020

  16. [16]

    Andriati, L

    A. Andriati, L. Brito, L. Tomio, and A. Gammal, Stabil- ity of a Bose-condensed mixture on a bubble trap, Phys. Rev. A104, 033318 (2021)

    work page 2021

  17. [17]

    Padavić, K

    K. Padavić, K. Sun, C. Lannert and S. Vishveshwara, Vortex-antivortex physics in shell-shaped Bose-Einstein condensates, Phys. Rev. A102, 043306 (2020)

    work page 2020

  18. [18]

    M. A. Caracanhas, P. Massignan, and A. L. Fetter, Su- perfluid vortex dynamics on an ellipsoid and other surfaces of revolution, Phys. Rev. A105, 023307 (2022)

    work page 2022

  19. [19]

    S.K.Adhikari, DipolarBose-Einsteincondensateinaring or in a shell, Phys. Rev. A85, 053631 (2012)

    work page 2012

  20. [20]

    P. C. Diniz, E. A. B. Oliveira, A. R. P. Lima, and E. A. L. 6 Henn, Ground state and collective excitations of a dipolar Bose-Einstein condensate in a bubble trap, Sci. Rep.10, 4831 (2020)

    work page 2020

  21. [21]

    Tononi, A

    A. Tononi, A. Pelster, and L. Salasnich, Topological su- perfluid transition in bubble-trapped condensates, Phys. Rev. Research4, 013122 (2022)

    work page 2022

  22. [22]

    Rhyno, N

    B. Rhyno, N. Lundblad, D. C. Aveline, C. Lannert, and S. Vishveshwara, Thermodynamics in expanding shell- shaped Bose-Einstein condensates, Phys. Rev. A104, 063310 (2021)

    work page 2021

  23. [23]

    Brito, L

    L. Brito, L. Tomio, and A. Gammal, Faraday waves on a bubble-trapped Bose-Einstein-condensed binary mixture, Phys. Rev. A108, 053315 (2023)

    work page 2023

  24. [24]

    A.Wolf, P.Boegel, M.Meister, A.Balaž, N.Gaaloul, and M. A. Efremov, Shell-shaped Bose-Einstein condensates based on dual-species mixtures, Phys. Rev. A106, 013309 (2022)

    work page 2022

  25. [25]

    Stürmer, M

    P. Stürmer, M. N. Tengstrand, and S. M. Reimann, Mixed bubbles in a one-dimensional Bose-Bose mixture, Phys. Rev. Research4, 043182 (2022)

    work page 2022

  26. [26]

    Tononi, D

    A. Tononi, D. S. Petrov, and M. Lewenstein, Dimer problem on a spherical surface, Phys. Rev.111, L051304 (2025)

    work page 2025

  27. [27]

    W. Wang, P. G. Kevrekidis, R. Carretero-González, and D. J. Frantzeskakis, Dark spherical shell solitons in three- dimensional Bose-Einstein condensates: Existence, stabil- ity, and dynamics, Phys. Rev. A93, 023630 (2016)

    work page 2016

  28. [28]

    Gaidoukov and J

    A. Gaidoukov and J. R. Anglin, Bogoliubov–de Gennes theory of the snake instability of gray solitons in higher dimensions, Phys. Rev. A103, 013319 (2021)

    work page 2021

  29. [29]

    Burger, K

    S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Seng- stock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Dark Solitons in Bose-Einstein Condensates, Phys. Rev. Lett.83, 5198 (1999)

    work page 1999

  30. [30]

    Muryshev, G

    A. Muryshev, G. V. Schlyapnikov, W. Ertmer, K. Seng- stock, and M. Lewenstein, Dynamics of Dark Solitons in Elongated Bose-Einstein Condensates, Phys. Rev. Lett. 89, 5198 (2002)

    work page 2002

  31. [31]

    Busch and J

    Th. Busch and J. R. Anglin, Motion of Dark Solitons in Trapped Bose-Einstein Condensates, Phys. Rev. Lett.84, 2298 (2000)

    work page 2000

  32. [32]

    D. L. Feder, M. S. Pindzola, L. A. Collins, B. I. Schnei- der, and C. W. Clark, Dark-soliton states of Bose-Einstein condensates in anisotropic traps, Phys. Rev. A62, 053606 (2000)

    work page 2000

  33. [33]

    Dutton, M

    Z. Dutton, M. Budde, C. Slowe, and L. V. Hau, Observa- tion of Quantum Shock Waves Created with Ultra- Com- pressed Slow Light Pulses in a Bose-Einstein Condensate, Science293, 663 (2001)

    work page 2001

  34. [34]

    Kivshar and X

    Yu.S. Kivshar and X. Yang, Ring dark solitons, Phys. Rev. E50, R40 (1994)

    work page 1994

  35. [35]

    Theocharis, D

    G. Theocharis, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and Y. S. Kivshar, Ring dark solitons and vortex necklaces in Bose-Einstein condensates, Phys. Rev. Lett.90, 120403 (2003)

    work page 2003

  36. [36]

    Anderson, P

    B.P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, Watching Dark Solitons Decay into Vortex Rings in a Bose-Einstein Condensate, Phys. Rev. Lett.86, 2926 (2001)

    work page 2001

  37. [37]

    Becker, S

    C. Becker, S. Stellmer, P. Soltan-Panahi, S. Dörscher, M. Baumert, E.-M. Richter, J. Kronjäger, K. Bongs, and K. Sengstock, Nature Phys.4, 496 (2008)

    work page 2008

  38. [38]

    R. W. Sallatti, L. Tomio, D. E. Pelinovsky, and A. Gam- mal, Stability of dark solitons in a bubble Bose-Einstein condensate, Supplemental material

    work page

  39. [39]

    Dalfovo, S

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

    work page 1999

  40. [40]

    Pitaevskii and S

    L. Pitaevskii and S. Stringari, Bose-Einstein condensa- tionandsuperfluidity, OxfordSciencePublications(2016)

    work page 2016

  41. [42]

    Tononi, L

    A. Tononi, L. Salasnich, and A. Yakimenko, Quan- tum vortices in curved geometries, AVS Quantum Sci.6, 030502 (2024)

    work page 2024

  42. [43]

    M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys.65, 851 (1993)

    work page 1993

  43. [44]

    Anderson, J.R

    M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wie- man, and E.A. Cornell, Observation of Bose-Einstein con- densation in a dilute atomic vapor, Science269, 198 (1995)

    work page 1995

  44. [45]

    Davis, M.-O

    K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, Bose- Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett.75, 3969 (1995)

    work page 1995

  45. [46]

    M.Brtka, A.Gammal, andB.Malomed, Hiddenvorticity in binary Bose-Einstein condensates, Phys. Rev. A82, 053610 (2010)

    work page 2010

  46. [47]

    Denschlag, J

    J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S. L. Rolston, B. I. Schnei- der, and W. D. Phillips, Generating solitons by phase en- gineering of a Bose–Einstein condensate. Science287, 97 (2000)

    work page 2000

  47. [48]

    Nonlinear Sci.34, 23 (2024)

    A.deLaire, G.Dujardin, andS.López-Martínez, Numer- ical computation of dark solitons of a nonlocal nonlinear Schrödinger equation, J. Nonlinear Sci.34, 23 (2024)

    work page 2024

  48. [49]

    Gammal, L

    A. Gammal, L. Tomio, and T. Frederico, Improved numerical approach for the time-independent Gross- Pitaevskii nonlinear Schrödinger equation, Phys. Rev. E 60, 2421 (1999)

    work page 1999

  49. [50]

    Quinney,Introduction to numerical solution of differ- ential equations, rev

    D. Quinney,Introduction to numerical solution of differ- ential equations, rev. ed., Research Studies Press and John Wiley & Sons (1987). 1 ST ABILITY OF DARK SOLITONS IN A BUBBLE BOSE-EINSTEIN CONDENSA TE - SUPPLEMENT AL MA TERIAL Dark solitons for smallε Let us consider the profilef(θ) : [0, π]→R, bounded at the end points[θ= 0, π], defined from the non...

    work page 1987

  50. [51]

    Asεincreases(ε≫1), its reduction becomes concentrated nearθ= π

    work page

  51. [52]

    The asymptotic solution is −f ′′ ∞(θ) +εf 3 ∞(θ) =µ ∞(ε)f∞(θ).(S11) By connecting it with the constant solution (S5), and redefining it asf∞(θ)≡g 0(z), withz≡ √ε 2 π 2 −θ , we haveg 0(z) = 1√ 2 tanh(z)as an exact solution of (S11): − ε 4 g′′ 0 (z) +εg 3 0(z) = ε 2 g0(z) =µ ∞(ε)g0(z),(S12) where d dθ =− √ε 2 d dz. Further, withf(θ)≡g(z), the original equat...

    work page

  52. [53]

    Geyer and D

    A. Geyer and D. Pelinovsky, Stability of nonlinear waves in Hamiltonian dynamical systems, Mathematical Surveys and Monographs288(AMS, Providence, 2025)

    work page 2025

  53. [54]

    Quinney,Introduction to Numerical solution of differ- ential equations, rev

    D. Quinney,Introduction to Numerical solution of differ- ential equations, rev. ed., Research Studies Press and Jonh Wiley & Sons (1987)

    work page 1987

  54. [55]

    Gammal, L

    A. Gammal, L. Tomio, and T. Frederico, Improved numer- ical approach for the time-independent Gross-Pitaevskii nonlinear Schrödinger equation, Phys. Rev. E60, 2421 (1999)

    work page 1999

  55. [56]

    Andriati, L

    A. Andriati, L. Brito, L. Tomio, and A. Gammal, Stability of a Bose-condensed mixture on a bubble trap, Phys. Rev. A104, 033318 (2021)

    work page 2021