pith. sign in

arxiv: 2605.04485 · v1 · submitted 2026-05-06 · 🧮 math.NA · cs.NA

Analysis of gradient flow for computing defocusing action ground states of rotating nonlinear Schr\"odinger equations

Wei Liu , Tingfeng Wang , Yongjun Yuan , Xiaofei Zhao This is my paper

Pith reviewed 2026-05-08 16:40 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords direct gradient flowaction ground statesrotating nonlinear Schrödingerunconditional stabilityexponential convergencephase-shift distancevorticesdefocusing
0
0 comments X

The pith

Direct gradient flow on the action functional is unconditionally stable and converges to defocusing ground states of rotating nonlinear Schrödinger equations.

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

The paper shows that a direct gradient flow discretization applied to the action functional for rotating nonlinear Schrödinger equations decreases the functional at every step no matter how large the time step is chosen. It then proves that the discrete sequence converges globally to a phase-shift class of the ground state whenever sublevel sets remain bounded, and converges exponentially fast locally once a non-degeneracy condition on the target state holds. The analysis is built around a specially chosen H^1 distance that identifies states differing only by a constant phase factor, which is essential because the ground states carry quantized vortices and live in the complex plane. This combination removes the usual need to restrict the time step for stability and supplies a concrete rate of convergence once the flow enters a neighborhood of the ground state.

Core claim

The authors prove that the direct gradient flow scheme is unconditionally stable in the sense that the discrete action is monotonically non-increasing for arbitrary step sizes. They further establish global convergence of the iterates to a phase-shift equivalence class of the action ground state under the assumption that sublevel sets of the action are uniformly bounded, together with local exponential convergence under an additional non-degeneracy condition on the ground state. The proof introduces a tailored H^1 distance between phase-shift classes to control the flow for complex-valued solutions containing vortices and develops a new analytic framework to obtain the exponential rate.

What carries the argument

The direct gradient flow iteration that produces a monotonically decreasing sequence for the action functional, measured in a phase-shift-invariant H^1 distance on equivalence classes.

If this is right

  • The action functional decreases at every iteration regardless of step size, removing any CFL-type restriction on the discretization.
  • The iterates are guaranteed to enter a neighborhood of the ground state whenever the action sublevels are bounded.
  • Once inside that neighborhood and under non-degeneracy, the distance to the ground state decays exponentially in the tailored metric.
  • The same scheme therefore supplies both a stable integrator and a provably convergent method for vortex-carrying states without auxiliary projection steps.

Where Pith is reading between the lines

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

  • The phase-shift invariant distance may extend directly to other gauge-invariant or phase-invariant variational problems in nonlinear optics or quantum mechanics.
  • If the non-degeneracy condition can be verified analytically for a range of rotation speeds, the exponential rate becomes a practical a-priori error estimator.
  • The unconditional stability result offers a template for constructing structure-preserving flows on other energy or action functionals that lack maximum principles.

Load-bearing premise

Sublevel sets of the action functional remain uniformly bounded and the target ground state satisfies a non-degeneracy condition that makes the phase-shift H^1 distance well-defined and controlling.

What would settle it

An explicit numerical run with a fixed time step in which the computed action value increases at some iteration, or a sequence of iterates that remains bounded away from every ground state while the boundedness and non-degeneracy assumptions are satisfied.

Figures

Figures reproduced from arXiv: 2605.04485 by Tingfeng Wang, Wei Liu, Xiaofei Zhao, Yongjun Yuan.

Figure 1
Figure 1. Figure 1: The change of SΩ,ω(φ n )−SΩ,ω(φg) (left) and H1 -norm error kφ n −φgkH1 (right) of DGF (2.4) along iteration (on logarithmic scale) under different τ with ω = −10 fixed in Example 4.1. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 0 10 20 30 40 10-10 10-5 100 0 20 40 60 80 10-10 10-5 100 view at source ↗
Figure 2
Figure 2. Figure 2: The profiles of GS φg (left), the change of SΩ,ω(φ n ) − SΩ,ω(φg) (middle) and H1 -norm error kφ n − φgkH1 (right) of DGF (2.4) along iteration (on logarithmic scale) under different ω with τ = 0.1 fixed in Example 4.1. In view at source ↗
Figure 3
Figure 3. Figure 3: Contour plots of |φg| 2 for different Ω and ω with φ 0 of (4.1) in Exam￾ple 4.2. 0 100 200 300 400 500 -11 -10 -9 -8 -7 -6 -5 -4 0 20 40 60 80 100 -200 0 200 400 600 800 1000 1200 0 2000 4000 6000 8000 10000 -16 -14 -12 -10 -8 -6 -4 -2 view at source ↗
Figure 4
Figure 4. Figure 4: SΩ,ω(φ n ) for different Ω and ω with φ 0 of (4.1) in Example 4.2 view at source ↗
Figure 5
Figure 5. Figure 5: The change of SΩ,ω(φ n ) − SΩ,ω(φg) and kφ n − φgk 2 H1 of DGF (2.4) along iteration (on logarithmic scale) under different Ω and ω with φ 0 of (4.1) in Exam￾ple 4.2. monotonic decrease of SΩ,ω in all cases view at source ↗
read the original abstract

This work focuses on the numerical computation of defocusing action ground states for rotating nonlinear Schr\"odinger equations (RNLS) using a direct gradient flow (DGF) method. We address theoretical gaps in the existing literature concerning the stability and convergence of this DGF scheme. Firstly, we prove the unconditional stability of the DGF scheme, demonstrating that the action functional is monotonically non-increasing along the discrete flow for arbitrary time step sizes. Secondly, we establish a rigorous convergence analysis, proving global convergence under minor assumptions and local exponential convergence to the action ground state under a reasonable non-degeneracy condition. The analysis relies on the uniform boundedness of sublevel sets of the action functional and introduces a tailored $H^1$-distance between phase-shift equivalence classes to handle complex-valued ground states with quantized vortices. A novel analytical framework is also developed to establish the exponential convergence rate. Numerical experiments are presented to validate the theoretical findings, demonstrating both the global migration towards a neighborhood of the ground state and subsequent exponential convergence.

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 / 1 minor

Summary. The manuscript analyzes the direct gradient flow (DGF) scheme for computing defocusing action ground states of rotating nonlinear Schrödinger equations. It establishes unconditional stability by proving that the action functional decreases monotonically along the discrete trajectory for any positive time step. Convergence results include global convergence to the ground state under minor assumptions and local exponential convergence under a non-degeneracy condition. The analysis depends on the uniform boundedness of sublevel sets of the action functional and employs a custom H¹-distance defined on phase-shift equivalence classes to accommodate vortex-carrying states. Numerical simulations are included to illustrate the migration to the ground state and the exponential rate.

Significance. If the proofs hold, this provides a rigorous foundation for gradient-flow-based computation of action ground states in rotating NLS, with direct relevance to modeling rotating Bose-Einstein condensates and quantum fluids. The unconditional stability result (monotonicity for arbitrary step sizes) is a clear strength that improves practical robustness over many competing schemes. The tailored distance on phase-shift classes offers a technically appropriate way to quotient out the U(1) invariance while preserving vortex structure. The combination of global and local exponential rates under stated conditions, together with numerical validation, strengthens the contribution to numerical analysis of nonlinear dispersive PDEs.

major comments (2)
  1. [Abstract and convergence analysis] Abstract and convergence analysis: The uniform boundedness of sublevel sets of the action functional is invoked as a standing assumption for both the global convergence result and the local exponential rate. No proof or verification is supplied that this boundedness holds automatically for the rotating case; for large rotation frequency Ω, vortex-lattice configurations can produce sequences with bounded action yet unbounded H¹ norm, allowing the discrete flow to escape any compact set despite monotonicity. This assumption is load-bearing for the central claims and must either be proved under explicit conditions on Ω or replaced by a verifiable hypothesis.
  2. [Section introducing the tailored H¹-distance] Section introducing the tailored H¹-distance: The custom distance on phase-shift equivalence classes is used to control the flow and to obtain the exponential convergence rate. It is not shown that this distance is equivalent to the standard H¹ metric on the quotient space or that the gradient flow remains well-defined with respect to it. Any failure of equivalence would invalidate the local exponential framework and the non-degeneracy argument.
minor comments (1)
  1. [Numerical experiments] Numerical experiments: The reported tests should specify the spatial discretization (mesh size, finite-element or spectral basis) and the concrete values of Ω, nonlinearity strength, and initial data used, so that the observed global migration and subsequent exponential decay can be reproduced independently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and positive assessment of the manuscript's significance. We address each major comment below and will revise the manuscript to strengthen the presentation of assumptions and the properties of the distance metric.

read point-by-point responses

  1. Referee: [Abstract and convergence analysis] Abstract and convergence analysis: The uniform boundedness of sublevel sets of the action functional is invoked as a standing assumption for both the global convergence result and the local exponential rate. No proof or verification is supplied that this boundedness holds automatically for the rotating case; for large rotation frequency Ω, vortex-lattice configurations can produce sequences with bounded action yet unbounded H¹ norm, allowing the discrete flow to escape any compact set despite monotonicity. This assumption is load-bearing for the central claims and must either be proved under explicit conditions on Ω or replaced by a verifiable hypothesis.

    Authors: We agree that the uniform boundedness of sublevel sets {u : J(u) ≤ J(u0)} is stated as a hypothesis (Assumption 2.3) rather than proved for arbitrary Ω. In the non-rotating case this follows from standard coercivity, but for rotating NLS the referee correctly notes that large-Ω vortex lattices may yield bounded action with unbounded H¹ norm. We do not claim automatic boundedness for all Ω. In the revision we will (i) restate the assumption explicitly as a verifiable hypothesis on the parameters (e.g., Ω small enough that the ground state remains vortex-free or satisfies a priori H¹ bounds), (ii) add a remark citing known existence results that guarantee the hypothesis for moderate rotation, and (iii) note that the unconditional stability theorem itself does not require the assumption—only the convergence statements do. This replaces the implicit standing assumption with an explicit, checkable condition. revision: yes

  2. Referee: [Section introducing the tailored H¹-distance] Section introducing the tailored H¹-distance: The custom distance on phase-shift equivalence classes is used to control the flow and to obtain the exponential convergence rate. It is not shown that this distance is equivalent to the standard H¹ metric on the quotient space or that the gradient flow remains well-defined with respect to it. Any failure of equivalence would invalidate the local exponential framework and the non-degeneracy argument.

    Authors: The tailored distance d(u,v) := inf_θ ||u − e^{iθ}v||_{H¹} is introduced precisely to work on the quotient space that removes the U(1) phase invariance while retaining vortex topology. We will add a short lemma in the revised manuscript proving metric equivalence: there exist constants c,C > 0 (depending only on the H¹-norm of the states) such that c · dist_{H¹/∼}([u],[v]) ≤ d(u,v) ≤ C · dist_{H¹/∼}([u],[v]). The proof uses the fact that the infimum is attained at a continuous phase θ(u,v) and that the phase circle is compact. We will also verify that the discrete gradient flow is invariant under simultaneous phase shifts of all iterates, so the vector field remains tangent to the quotient and the flow is well-defined in the metric d. These additions ensure the local exponential convergence argument (which relies on the non-degeneracy condition in the quotient) is rigorously justified. revision: yes

Circularity Check

0 steps flagged

No circularity: proofs rest on explicit assumptions without self-referential reductions

full rationale

The paper presents a functional-analytic derivation of unconditional stability (monotonicity of the action along the discrete flow for arbitrary steps) and convergence (global under minor assumptions, local exponential under non-degeneracy) for the direct gradient flow scheme on rotating NLS. All load-bearing steps are conditional on stated hypotheses such as uniform boundedness of action sublevel sets and a non-degeneracy condition on the ground state; these are not derived inside the paper but posited upfront. The tailored H^1-distance on phase-shift classes is introduced as an auxiliary metric to control vortex-carrying states and is not obtained by redefining the target quantity. No fitted parameters are relabeled as predictions, no self-citation chains carry the central claims, and no ansatz or uniqueness result is smuggled via prior author work. The derivation therefore remains self-contained as a set of conditional theorems whose conclusions do not collapse to their inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on two domain assumptions drawn from variational PDE theory; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Uniform boundedness of sublevel sets of the action functional
    Invoked to obtain global convergence of the discrete flow.
  • domain assumption Non-degeneracy condition on the action ground state
    Required to obtain the local exponential convergence rate.

pith-pipeline@v0.9.0 · 5486 in / 1342 out tokens · 82173 ms · 2026-05-08T16:40:51.182697+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    Aftalion and Q

    A. Aftalion and Q. Du , Vortices in a rotating Bose–Einstein condensate: Critical a ngular velocities and energy diagrams in the Thomas-Fermi regime , Phys. Rev. A, 64 (2001), p. 063603

    work page 2001

  2. [2]

    Altmann, P

    R. Altmann, P. Henning, and D. Peterseim , The J-method for the Gross–Pitaevskii eigen- value problem, Numer. Math., 148 (2021), pp. 575–610

    work page 2021

  3. [3]

    Antoine, A

    X. Antoine, A. Levitt, and Q. Tang , Efficient spectral computation of the stationary states of rotating Bose–Einstein condensates by preconditioned n onlinear conjugate gradient methods , J. Comput. Phys., 343 (2017), pp. 92–109

    work page 2017

  4. [4]

    A. H. Ardila and H. Hajaiej , Global well-posedness, blow-up and stability of standing w aves for supercritical NLS with rotation , J. Dyn. Differ. Equ., 35 (2023), pp. 1643–1665

    work page 2023

  5. [5]

    Bao and Y

    W. Bao and Y. Cai , Mathematical theory and numerical methods for Bose-Einste in conden- sation, Kinet. Relat. Models, 6 (2013), pp. 1–135

    work page 2013

  6. [6]

    Bao and Q

    W. Bao and Q. Du , Computing the ground state solution of Bose–Einstein conden sates by a normalized gradient flow , SIAM J. Sci. Comput., 25 (2004), pp. 1674–1697

    work page 2004

  7. [7]

    Berestycki and P.-L

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

    work page 1983

  8. [8]

    existence of infinitely m any solutions , Arch

    , Nonlinear scalar field equations, II. existence of infinitely m any solutions , Arch. Ration. Mech. Anal., 82 (1983), pp. 347–375

    work page 1983

  9. [9]

    Canc` es, R

    E. Canc` es, R. Chakir, and Y. Maday , Numerical analysis of nonlinear eigenvalue problems , J. Sci. Comput., 45 (2010), pp. 90–117

    work page 2010

  10. [10]

    L. D. Carr, C. W. Clark, and W. P. Reinhardt , Stationary solutions of the one- dimensional nonlinear Schr¨ odinger equation. I. case of rep ulsive nonlinearity , Phys. Rev. A, 62 (2000), p. 063610

    work page 2000

  11. [11]

    Chen , Foundations for guided-wave optics , John Wiley & Sons, 2006

    C.-L. Chen , Foundations for guided-wave optics , John Wiley & Sons, 2006

    work page 2006

  12. [12]

    H. Chen, G. Dong, W. Liu, and Z. Xie , Second-order flows for computing the ground states of rotating Bose-Einstein condensates , J. Comput. Phys., 475 (2023), p. 111872

    work page 2023

  13. [13]

    Danaila and P

    I. Danaila and P. Kazemi , A new Sobolev gradient method for direct minimization of the Gross–Pitaevskii energy with rotation , SIAM J. Sci. Comput., 32 (2010), pp. 2447–2467

    work page 2010

  14. [14]

    Danaila and B

    I. Danaila and B. Protas , Computation of ground states of the Gross–Pitaevskii functi onal via Riemannian optimization , SIAM J. Sci. Comput., 39 (2017), pp. B1102–B1129

    work page 2017

  15. [15]

    C. M. Dion and E. Canc ` es, Ground state of the time-independent Gross–Pitaevskii equ ation, Comput. Phys. Commun., 177 (2007), pp. 787–798

    work page 2007

  16. [16]

    Dovetta, E

    S. Dovetta, E. Serra, and P. Tilli , Action versus energy ground states in nonlinear Schr¨ odinger equations, Math. Ann., 385 (2023), pp. 1545–1576

    work page 2023

  17. [17]

    Z. Feng, Q. Shu, and Q. Tang , Mass-preserving and energy-diminishing semi-implicit schemes for computing ground states of rotating Bose-Einst ein condensates , J. Sci. Comput., 106 (2026), p. 30

    work page 2026

  18. [18]

    Fetter, Rotating trapped bose-einstein condensates, Rev

    A. Fetter, Rotating trapped bose-einstein condensates, Rev. Mod. Phys., 81 (2009), p. 647–691

    work page 2009

  19. [19]

    Fukuizumi, Stability and instability of standing waves for the nonline ar Schr¨ odinger equation with harmonic potential , Discrete Contin

    R. Fukuizumi, Stability and instability of standing waves for the nonline ar Schr¨ odinger equation with harmonic potential , Discrete Contin. Dyn. Syst., 7 (2001), pp. 525–544

    work page 2001

  20. [20]

    Fukuizumi and M

    R. Fukuizumi and M. Ohta , Instability of standing waves for nonlinear Schr¨ odinger equations with potentials, Differ. Integral Equ., 16 (2003), pp. 691–706

    work page 2003

  21. [21]

    Gilbarg, N

    D. Gilbarg, N. S. Trudinger, D. Gilbarg, and N. Trudinger , Elliptic partial differential equations of second order , vol. 2, Springer, 1998. 20 W. LIU, T. W ANG, Y. YUAN, AND X. ZHAO

    work page 1998

  22. [22]

    Henning and D

    P. Henning and D. Peterseim , Sobolev gradient flow for the Gross–Pitaevskii eigenvalue problem: Global convergence and computational efficiency , SIAM J. Numer. Anal., 58 (2020), pp. 1744–1772

    work page 2020

  23. [23]

    Henning and M

    P. Henning and M. Yadav , On discrete ground states of rotating Bose–Einstein conden sates, Math. Comput., 94 (2025), pp. 1–32

    work page 2025

  24. [24]

    Jeanjean and S.-S

    L. Jeanjean and S.-S. Lu , On global minimizers for a mass constrained problem , Calc. Var. Partial Differ. Equ., 61 (2022), p. 214

    work page 2022

  25. [25]

    E. H. Lieb, R. Seiringer, and J. Yngvason , A rigorous derivation of the Gross–Pitaevskii energy functional for a two-dimensional bose gas , Comm. Math. Phys., 224 (2001), pp. 17–31

    work page 2001

  26. [26]

    Liu and Y

    W. Liu and Y. Cai , Normalized gradient flow with Lagrange multiplier for comput ing ground states of Bose–Einstein condensates , SIAM J. Sci. Comput., 43 (2021), pp. B219–B242

    work page 2021

  27. [27]

    W. Liu, C. W ang, and X. Zhao , On action ground states of defocusing nonlinear Schr¨ odinger equations, Math. Models Methods Appl. Sci., 35 (2025), pp. 39–74

    work page 2025

  28. [28]

    W. Liu, T. W ang, and X. Zhao , Convergence analysis of Lp+1-normalized gradient flow for action ground state of nonlinear Schr¨ odinger equation, arXiv preprint arXiv:2602.20820, (2026)

    work page arXiv 2026

  29. [29]

    W. Liu, Z. Wen, Y. Yuan, and X. Zhao , Computing defocusing action ground state of rotating nonlinear Schr¨ odinger equation: methods via var ious formulations and comparison , J. Comput. Phys., 538 (2025), p. 114193

    work page 2025

  30. [30]

    W. Liu, Y. Yuan, and X. Zhao , Computing the action ground state for the rotating nonlinear Schr¨ odinger equation, SIAM J. Sci. Comput., 45 (2023), pp. A397–A426

    work page 2023

  31. [31]

    Malomed , Soliton Management in Periodic Systems , Springer, New York, 2006

    B. Malomed , Soliton Management in Periodic Systems , Springer, New York, 2006

    work page 2006

  32. [32]

    S. I. Pohozaev , Eigenfunctions of the equation ∆ u +λf (u) = 0, Soviet Mathematics Doklady, 5 (1965), pp. 1408–1411

    work page 1965

  33. [33]

    Seiringer , Gross-pitaevskii theory of the rotating bose gas , Comm

    R. Seiringer , Gross-pitaevskii theory of the rotating bose gas , Comm. Math. Phys., 229 (2002), p. 491–509

    work page 2002

  34. [34]

    Shatah and W

    J. Shatah and W. Strauss , Instability of nonlinear bound states , Commun. Math. Phys., 100 (1985), pp. 173–190

    work page 1985

  35. [35]

    W. A. Strauss , Existence of solitary waves in higher dimensions , Commun. Math. Phys., 55 (1977), pp. 149–162

    work page 1977

  36. [36]

    Sulem and S

    C. Sulem and S. Pirre-Louis , The nonlinear Schr¨ odinger equation: self-focusing and wav e collapse, Springer New York, NY, 2006

    work page 2006

  37. [37]

    W ang, Computing the least action ground state of the nonlinear Schr ¨ odinger equation by a normalized gradient flow , J

    C. W ang, Computing the least action ground state of the nonlinear Schr ¨ odinger equation by a normalized gradient flow , J. Comput. Phys., 471 (2022), p. 111675

    work page 2022

  38. [38]

    Willem , Minimax theorems, vol

    M. Willem , Minimax theorems, vol. 24, Springer Science & Business Media, 2012

    work page 2012

  39. [39]

    Zhao , Numerical integrators for continuous disordered nonlinea r schr¨ odinger equation, J

    X. Zhao , Numerical integrators for continuous disordered nonlinea r schr¨ odinger equation, J. Sci. Comput., 89 (2021), p. 40

    work page 2021

  40. [40]

    Zhou , An analysis of finite-dimensional approximations for the gr ound state solution of Bose–Einstein condensates , Nonlinearity, 17 (2004), pp

    A. Zhou , An analysis of finite-dimensional approximations for the gr ound state solution of Bose–Einstein condensates , Nonlinearity, 17 (2004), pp. 541–550

    work page 2004

  41. [41]

    Zhuang and J

    Q. Zhuang and J. Shen , Efficient SAV approach for imaginary time gradient flows with applications to one-and multi-component Bose-Einstein co ndensates, J. Comput. Phys., 396 (2019), pp. 72–88. W. Liu: College of Science, National University of Defense T echnology, Changsha, 410073, China Email address : wl@nudt.edu.cn T. W ang: School of Mathematics and St...

    work page 2019