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arxiv: 2605.13629 · v1 · pith:JVWUNYB7new · submitted 2026-05-13 · 🧮 math.AP

Orbital stability of black solitons for quasilinear Schr\"odinger equations with nonzero conditions at infinity

Erwan Le Quiniou This is my paper

Pith reviewed 2026-05-14 18:01 UTC · model grok-4.3

classification 🧮 math.AP MSC 35Q5535B35
keywords black solitonsorbital stabilityquasilinear Schrödinger equationsVakhitov-Kolokolov conditionkink solutionsenergy space
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The pith

The black soliton is orbitally stable in the energy space for quasilinear Schrödinger equations when the Vakhitov-Kolokolov slope condition holds.

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

The paper proves orbital stability of the black soliton, also called a kink, for a broad class of one-dimensional quasilinear Schrödinger equations that satisfy nonzero boundary conditions at infinity. The result applies to general defocusing semilinear nonlinearities paired with either focusing or defocusing quasilinear terms, provided the Vakhitov-Kolokolov condition is met: the derivative of momentum with respect to speed must be negative at zero speed. First the authors give conditions on the nonlinearities that produce a local branch of finite-energy solitons parameterized by speed, with the stationary black soliton at the zero-speed end of the branch. They then control the size of perturbations by introducing a variational problem whose value bounds the supremum norm in terms of the conserved energy and momentum.

Core claim

The black soliton is orbitally stable in the energy space whenever the Vakhitov-Kolokolov condition holds. This is established by analyzing minimizing sequences for a variational problem that bounds the sup-norm of the perturbation in terms of the conserved energy and momentum, even though the infimum is not attained.

What carries the argument

A variational problem that bounds the supremum norm of a perturbation of the kink by the conserved energy and momentum.

If this is right

  • Small perturbations of the black soliton remain close to it in the energy norm for all time when the slope condition holds.
  • An explicit formula for the momentum derivative allows immediate verification of the condition for concrete nonlinearities.
  • The stability statement covers both focusing and defocusing quasilinear terms under the stated existence conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same variational control on supremum norm could be tested on other traveling-wave profiles in related dispersive equations.
  • When the slope condition fails, one expects orbital instability, giving a potential sharp threshold between stable and unstable regimes.

Load-bearing premise

Sufficient conditions on the quasilinear nonlinearities must hold to guarantee the existence of the local branch of finite-energy solitons parameterized by speed.

What would settle it

A direct calculation showing that the derivative of momentum with respect to speed is positive at zero speed for a given nonlinearity would place that equation outside the stability result.

Figures

Figures reproduced from arXiv: 2605.13629 by Erwan Le Quiniou.

Figure 1
Figure 1. Figure 1: Plot of the kink profile u0,κ for several nonlinearities and values of κ. The top left panel displays profiles in Case (i). In the top right, profiles in Case (ii). The bottom left and right panels show black solitons in Case (iii) with r0 = 1 and r0 = 2, respectively. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plot of P ′ κ (0), the derivative of the momentum with respect to the speed at 0, as a function of κ near κ˜, in Cases (i)–(iii). The top left panel displays the slope condition in Case (i). In the top right, the condition in Case (ii). The bottom left and right pannels show P ′ κ (0) with respect to κ in Case (iii) with r0 = 1 and r0 = 2, respectively. The following result supports this idea that the quas… view at source ↗
read the original abstract

We investigate the orbital stability of black solitons for a broad class of quasilinear Schr\"odinger equations in one space dimension, with nonzero boundary conditions at infinity. Namely, our framework handles general defocusing semilinear nonlinearities and focusing or defocusing quasilinear nonlinearities. First, we establish sufficient conditions on the quasi-linear nonlinearities ensuring the existence of a local branch of finite-energy solitons parameterized by their speed. Within this branch, the black soliton, also called kink, corresponds to the stationary solution. Our main result is the orbital stability of the black soliton in the energy space, provided that the Vakhitov-Kolokolov (VK) slope condition holds; namely, that the derivative of the momentum with respect to the speed is negative at zero. Moreover, we derive an explicit formula for verifying this VK condition. The proof relies on the analysis of a carefully designed variational problem, which allows us to control the sup-norm of the evolution of a perturbation of the kink in terms of the energy and momentum, both of which are conserved by the flow. A delicate part of the argument is the analysis of minimizing sequences for this variational problem, since the infimum is not attained.

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

2 major / 2 minor

Summary. The manuscript establishes sufficient conditions on quasilinear nonlinearities to guarantee a local branch of finite-energy solitons parameterized by speed for a broad class of 1D quasilinear Schrödinger equations with nonzero boundary conditions at infinity. Within this branch the stationary black soliton (kink) is shown to be orbitally stable in the energy space whenever the Vakhitov-Kolokolov slope condition holds, i.e., the derivative of momentum with respect to speed is negative at zero speed. The proof proceeds by analyzing a carefully chosen variational problem whose infimum is not attained; conserved energy and momentum are used to control the sup-norm of perturbations of the kink.

Significance. If the central claims are correct, the work provides a technically substantial extension of orbital-stability results to quasilinear dispersive equations with general focusing or defocusing quasilinear terms. The explicit formula supplied for verifying the VK condition is a concrete, usable contribution. The variational treatment of non-attaining minimizing sequences, if fully rigorous, supplies a reusable technique for similar problems with nonzero conditions at infinity.

major comments (2)
  1. [Existence theorem for the soliton branch] The sufficient conditions on the quasilinear nonlinearities that guarantee the local soliton branch (stated in the existence theorem) are load-bearing for the entire stability result; the manuscript should verify that these conditions are compatible with the energy-space setting and do not inadvertently exclude physically relevant examples.
  2. [Variational characterization and minimizing-sequence analysis] In the analysis of minimizing sequences for the variational problem (the part flagged as delicate in the abstract), the argument that energy-momentum control prevents both concentration and escape to infinity must be made fully explicit; without a quantitative compactness statement it is unclear whether the sup-norm bound on perturbations follows directly from the conserved quantities.
minor comments (2)
  1. [Notation and VK formula] Notation for the momentum functional and its derivative with respect to speed should be introduced once and used consistently; the explicit VK formula would benefit from a short appendix deriving it from the traveling-wave ODE.
  2. [Abstract and introduction] The abstract and introduction should clarify that the result is conditional on the VK slope condition rather than unconditional; this avoids any impression that stability holds for all admissible nonlinearities.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via variational methods

full rationale

The central claim establishes orbital stability of the black soliton under the VK slope condition (dP/dc < 0 at c=0) using conserved energy and momentum together with a variational problem whose infimum is not attained. The paper supplies an explicit formula for verifying the VK condition directly from the momentum-speed relation on the soliton branch, without reducing the stability conclusion to a fitted parameter or self-referential definition. Sufficient conditions on the quasilinear terms for the local soliton branch are stated independently of the stability result. No load-bearing self-citations, no ansatz smuggled via prior work, and no renaming of known results as new derivations appear in the provided chain. The argument remains self-contained against external benchmarks in the energy space.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract supplies no explicit free parameters or invented entities; the central claim rests on standard Sobolev-type energy spaces and the existence of a soliton branch under conditions on the nonlinearity.

axioms (1)
  • domain assumption Existence of a local branch of finite-energy solitons parameterized by speed under sufficient conditions on the quasilinear nonlinearities
    Stated as the first step before the stability result.

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Works this paper leans on

60 extracted references · 60 canonical work pages

  1. [1]

    M. A. Alejo and A. J. Corcho. Orbital stability of the black soliton for the quintic Gross- Pitaevskii equation.Rev. Mat. Iberoam., 40(5):1731–1780, 2024

    work page 2024

  2. [2]

    C. O. Alves, Y. Wang, and Y. Shen. Soliton solutions for a class of quasilinear Schrödinger equations with a parameter.J. Differential Equations, 259(1):318–343, 2015

    work page 2015

  3. [3]

    C. Audiard. Small energy traveling waves for the Euler-Korteweg system.Nonlinearity, 30(9):3362–3399, 2017

    work page 2017

  4. [4]

    Audiard and B

    C. Audiard and B. Haspot. Global well-posedness of the Euler-Korteweg system for small irrotational data.Comm. Math. Phys., 351(1):201–247, 2017

    work page 2017

  5. [5]

    Baldelli, B

    L. Baldelli, B. Bieganowski, and J. Mederski. Traveling waves for nonlinear Schrödinger equations, 2024. Preprint arXiv:2406.03910. 44

    work page arXiv 2024

  6. [6]

    I. V. Barashenkov. Stability criterion for dark solitons.Phys. Rev. Lett., 77:1193–1197, Aug 1996

    work page 1996

  7. [7]

    Benzoni-Gavage, R

    S. Benzoni-Gavage, R. Danchin, S. Descombes, and D. Jamet. Structure of Korteweg models and stability of diffuse interfaces.Interfaces Free Bound., 7(4):371–414, 2005

    work page 2005

  8. [8]

    Berestycki and P.-L

    H. Berestycki and P.-L. Lions. Nonlinear scalar field equations. I. Existence of a ground state.Arch. Rational Mech. Anal., 82(4):313–345, 1983

    work page 1983

  9. [9]

    Berthoumieu

    J. Berthoumieu. Minimizing travelling waves for the one-dimensional nonlinear Schrödinger equation with non-zero condition at infinity, 2023. Preprint arXiv:2305.17516

    work page arXiv 2023

  10. [10]

    Berthoumieu

    J. Berthoumieu. Asymptotic stability of travelling waves for general nonlinear schrödinger equations with non-zero condition at infinity, 2025. Preprint arXiv:2504.03547

    work page arXiv 2025

  11. [11]

    Béthuel, P

    F. Béthuel, P. Gravejat, and J.-C. Saut. Existence and properties of travelling waves for the Gross-Pitaevskii equation. InStationary and time dependent Gross-Pitaevskii equations, volume 473 ofContemp. Math., pages 55–103. Amer. Math. Soc., Providence, RI, 2008

    work page 2008

  12. [12]

    Béthuel, P

    F. Béthuel, P. Gravejat, and J.-C. Saut. Travelling waves for the Gross-Pitaevskii equation. II.Comm. Math. Phys., 285(2):567–651, 2009

    work page 2009

  13. [13]

    Béthuel, P

    F. Béthuel, P. Gravejat, J.-C. Saut, and D. Smets. Orbital stability of the black soliton for the Gross-Pitaevskii equation.Indiana Univ. Math. J., 57(6):2611–2642, 2008

    work page 2008

  14. [14]

    J. L. Bona and A. Soyeur. On the stability of solitary-waves solutions of model equations for long waves.J. Nonlinear Sci., 4(5):449–470, 1994

    work page 1994

  15. [15]

    Brüll and H

    L. Brüll and H. Lange. Stationary, oscillatory and solitary wave type solution of singular nonlinear Schrödinger equations.Math. Methods Appl. Sci., 8(4):559–575, 1986

    work page 1986

  16. [16]

    D. Chiron. Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one.Nonlinearity, 25(3):813–850, 2012

    work page 2012

  17. [17]

    D. Chiron. Stability and instability for subsonic traveling waves of the nonlinear Schrödinger equation in dimension one.Anal. PDE, 6(6):1327–1420, 2013

    work page 2013

  18. [18]

    Chiron and E

    D. Chiron and E. Pacherie. Smooth branch of travelling waves for the Gross-Pitaevskii equation inR 2 for small speed.Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 22(4):1937–2038, 2021

    work page 1937

  19. [19]

    M. Colin. Stability of stationary waves for a quasilinear Schrödinger equation in space dimension 2.Adv. Differential Equations, 8(1):1–28, 2003

    work page 2003

  20. [20]

    Colin, L

    M. Colin, L. Jeanjean, and M. Squassina. Stability and instability results for standing waves of quasi-linear Schrödinger equations.Nonlinearity, 23(6):1353–1385, 2010

    work page 2010

  21. [21]

    de Bouard, N

    A. de Bouard, N. Hayashi, P. I. Naumkin, and J.-C. Saut. Scattering problem and asymp- totics for a relativistic nonlinear Schrödinger equation.Nonlinearity, 12(5):1415–1425, 1999

    work page 1999

  22. [22]

    de Bouard, N

    A. de Bouard, N. Hayashi, and J.-C. Saut. Global existence of small solutions to a relativistic nonlinear Schrödinger equation.Comm. Math. Phys., 189(1):73–105, 1997

    work page 1997

  23. [23]

    de Laire

    A. de Laire. Minimal energy for the traveling waves of the Landau-Lifshitz equation.SIAM J. Math. Anal., 46(1):96–132, 2014. 45

    work page 2014

  24. [24]

    de Laire, G

    A. de Laire, G. Dujardin, and S. Tapia-Mandiola. Numerical simulations of the quasilin- ear Gross–Pitaevskii equation with vanishing and nonvanishing conditions at infinity. In preparation

    work page

  25. [25]

    de Laire, P

    A. de Laire, P. Gravejat, and D. Smets. Construction of minimizing traveling waves for the Gross-Pitaevskii equation onR×T.Tunis. J. Math., 6(1):157–188, 2024

    work page 2024

  26. [26]

    de Laire, P

    A. de Laire, P. Gravejat, and D. Smets. Minimizing travelling waves for the Gross-Pitaevskii equation onR×T.Ann. Fac. Sci. Toulouse Math. (6), 34(1):135–192, 2025

    work page 2025

  27. [27]

    de Laire and E

    A. de Laire and E. Le Quiniou. Exotic traveling waves for a quasilinear Schrödinger equation with nonzero background.Nonlinear Anal., 265:Paper No. 114027, 40, 2026

    work page 2026

  28. [28]

    de Laire and S

    A. de Laire and S. López-Martínez. Existence and decay of traveling waves for the nonlocal Gross-Pitaevskii equation.Comm. Partial Differential Equations, 47(9):1732–1794, 2022

    work page 2022

  29. [29]

    Di Menza and C

    L. Di Menza and C. Gallo. The black solitons of one-dimensional NLS equations.Nonlin- earity, 20(2):461–496, 2007

    work page 2007

  30. [30]

    Gallay and D

    T. Gallay and D. Pelinovsky. Orbital stability in the cubic defocusing NLS equation: II. The black soliton.J. Differential Equations, 258(10):3639–3660, 2015

    work page 2015

  31. [31]

    Genoud and S

    F. Genoud and S. Rota Nodari. Standing wave solutions of a quasilinear Schrödinger equa- tion. Part I: The low frequency limit.NoDEA Nonlinear Differential Equations Appl., 33(2):Paper No. 43, 2026

    work page 2026

  32. [32]

    Germain, B

    P. Germain, B. Harrop-Griffiths, and J. L. Marzuola. Compactons and their variational properties for degenerate KDV and NLS in dimension 1.Quart. Appl. Math., 78(1):1–32, 2020

    work page 2020

  33. [33]

    Gravejat and D

    P. Gravejat and D. Smets. Asymptotic stability of the black soliton for the Gross-Pitaevskii equation.Proc. Lond. Math. Soc. (3), 111(2):305–353, 2015

    work page 2015

  34. [34]

    Grillakis, J

    M. Grillakis, J. Shatah, and W. Strauss. Stability theory of solitary waves in the presence of symmetry. I.J. Funct. Anal., 74(1):160–197, 1987

    work page 1987

  35. [35]

    B. Guo, J. Chen, and F. Su. The “Blow up” problem for a quasilinear Schrödinger equation. Journal of Mathematical Physics, 46(7):073510, 06 2005

    work page 2005

  36. [36]

    Holmer, P

    J. Holmer, P. G. Kevrekidis, and D. E. Pelinovsky. Orbital stability of kinks in the NLS equation with competing nonlinearities, 2025. Preprint arXiv:2512.08840

    work page arXiv 2025

  37. [37]

    Ifrim and D

    M. Ifrim and D. Tataru. Global solutions for 1D cubic dispersive equations, Part III: the quasilinear Schrödinger flow, 2023. Preprint arXiv:2306.00570

    work page arXiv 2023

  38. [38]

    Ifrim and D

    M. Ifrim and D. Tataru. Global solutions for cubic quasilinear Schrödinger flows in two and higher dimensions, 2024. Preprint arXiv:2404.09970

    work page arXiv 2024

  39. [39]

    I. D. Iliev and K. P. Kirchev. Stability and instability of solitary waves for one-dimensional singular Schrödinger equations.Differential Integral Equations, 6(3):685–703, 1993

    work page 1993

  40. [40]

    C. E. Kenig, G. Ponce, and L. Vega. The Cauchy problem for quasi-linear Schrödinger equations.Invent. Math., 158(2):343–388, 2004

    work page 2004

  41. [41]

    P. G. Kevrekidis, D. E. Pelinovsky, and R. M. Ross. Stability of smooth solitary waves under intensity-dependent dispersion.IMA J. Appl. Math., 89(6):989–1005, 2024. 46

    work page 2024

  42. [42]

    G. N. Koutsokostas, T. P. Horikis, P. G. Kevrekidis, and D. J. Frantzeskakis. Universal reductionsandsolitarywavesofweaklynonlocaldefocusingnonlinearSchrödingerequations. J. Phys. A, 54(8):Paper No. 085702, 17, 2021

    work page 2021

  43. [43]

    Solitonsinnonlocalnonlinearmedia: Exactsolutions.Physical Review E, 63(1):016610, 2000

    W.KrólikowskiandO.Bang. Solitonsinnonlocalnonlinearmedia: Exactsolutions.Physical Review E, 63(1):016610, 2000

    work page 2000

  44. [44]

    Krolikowski, O

    W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller. Modulational instability in nonlocal nonlinear Kerr media.Phys. Rev. E, 64:016612, Jun 2001

    work page 2001

  45. [45]

    Kurihara

    S. Kurihara. Large-amplitude quasi-solitons in superfluid films.Journal of the Physical Society of Japan, 50(10):3262–3267, 1981

    work page 1981

  46. [46]

    H. Lange. Nonexistence of solutions to singular nonlinear Schrödinger equations.Mathe- matische Modellierung, pages 114–124, 1986

    work page 1986

  47. [47]

    Le Quiniou

    E. Le Quiniou. Local well-posedness for quasilinear Schrödinger equations with nonzero conditions at infinity by energy methods. In preparation

    work page

  48. [48]

    Le Quiniou

    E. Le Quiniou. Stability and instability of the quasilinear Gross-Pitaevskii dark solitons. In ESAIM Proc. Surveys, volume 79, pages 42–57. EDP Sci., Les Ulis, 2025

    work page 2025

  49. [49]

    Z. Lin. Stability and instability of traveling solitonic bubbles.Adv. Differential Equations, 7(8):897–918, 2002

    work page 2002

  50. [50]

    Z. Lin, Z. Wang, and C. Zeng. Stability of traveling waves of nonlinear Schrödinger equation with nonzero condition at infinity.Arch. Ration. Mech. Anal., 222(1):143–212, 2016

    work page 2016

  51. [51]

    M. Mariş. Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity.Ann. of Math. (2), 178(1):107–182, 2013

    work page 2013

  52. [52]

    J. L. Marzuola, J. Metcalfe, and D. Tataru. Quasilinear Schrödinger equations III: Large data and short time.Arch. Ration. Mech. Anal., 242(2):1119–1175, 2021

    work page 2021

  53. [53]

    H. Mohamad. Hydrodynamical form for the one-dimensional Gross-Pitaevskii equation. Electron. J. Differential Equations, pages No. 141, 27, 2014

    work page 2014

  54. [54]

    D. E. Pelinovsky and M. Plum. Stability of black solitons in optical systems with intensity- dependent dispersion.SIAM J. Math. Anal., 56(2):2521–2568, 2024

    work page 2024

  55. [55]

    Porkolab and M

    M. Porkolab and M. V. Goldman. Upper-hybrid solitons and oscillating-two-stream insta- bilities.The Physics of Fluids, 19(6):872–881, 06 1976

    work page 1976

  56. [56]

    J. E. Rutledge, W. L. McMillan, J. M. Mochel, and T. E. Washburn. Third sound, two- dimensional hydrodynamics, and elementary excitations in very thin helium films.Phys. Rev. B, 18:2155–2168, Sep 1978

    work page 1978

  57. [57]

    Shen and S

    J. Shen and S. Jiang. Multiple solutions for quasilinear schrödinger equations with nonlin- earity sublinear at zero.Bulletin of the Australian Mathematical Society, page 1–13, 2026

    work page 2026

  58. [58]

    Shu and J

    J. Shu and J. Zhang. On a class of quasilinear Schrödinger equations.Appl. Math. Mech. (English Ed.), 28(7):981–986, 2007

    work page 2007

  59. [59]

    C. A. Stuart. Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation.Milan J. Math., 76:329–399, 2008

    work page 2008

  60. [60]

    Varholm, E

    K. Varholm, E. Wahlén, and S. Walsh. On the stability of solitary water waves with a point vortex.Communications on Pure and Applied Mathematics, 73(12):2634–2684, 2020. 47

    work page 2020