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Equivariant stability of vortices in Manton's Chern-Simons-Schr\"odinger system on the hyperbolic plane

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Pith's one-line read This paper proves that every sufficiently small equivariant perturbation of the degree-m vortex in the self-dual Chern-Simons-Schrödinger system on the hyperbolic plane evolves globally with a uniform Strichartz bound.

desk verdict Genuinely new stability theorem for vortices in Manton's model on H², but the spectral lemma underpinning the Strichartz estimates has a divergent boundary term and the abstract overclaims asymptotic stability. read the letter →

arxiv 2509.06090 v1 pith:WGK4WKRY submitted 2025-09-07 math.AP

Oussama Landoulsi , Sohrab Shahshahani This is my paper · ORCID
classification math.AP MSC 35Q5535B3535R01
keywords equivariantvorticeshyperbolicplaneChern-Simons-SchrödingersystemasymptoticstabilityDarbouxtransformStrichartzestimatesself-dualvortexequationsmagnetic
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

The paper's goal is to show that the degree-m equivariant magnetic vortices of the self-dual Chern-Simons-Schrödinger system on the hyperbolic plane are dynamically stable: any sufficiently small equivariant perturbation, measured in H^1_m, extends to a global-in-time solution and obeys a uniform Strichartz bound. The obstruction is that the linearized flow is governed by nonlocal, real-linear operators not amenable to direct spectral analysis. The authors bypass this with a nonlinear Darboux transform: differentiating the field equation once shifts the linearized problem to a new variable whose evolution is driven by a local, complex-linear, self-adjoint Schrödinger operator R_Q. They prove R_Q has purely absolutely continuous spectrum [5/4, ∞) with no eigenvalues or threshold resonance, which grants Strichartz estimates; a coupled elliptic equation with Green's function estimates then recovers the original perturbation. If correct, the theorem settles asymptotic stability of all equivariant vortices (m ≥ 1) in this self-dual model.

What carries the argument

The load-bearing object is the nonlinear Darboux transform. Writing the m-equivariant solution as Φ = e^{imθ}(Q+ε), the linearized evolution for ε involves the nonlocal, real-linear, non-complex-linear operator L_Q^* L_Q, where L_Q and L_Q^* are only formal adjoints with respect to the real inner product. Applying D_+ (the covariant Cauchy–Riemann operator) turns D_+Φ = e^{i(m+1)θ} ε_1 into a new unknown satisfying i∂_t ε_1 − ½ R_Q ε_1 = N(ε_1), where R_Q = −Δ_{H²} + V + 1 = A_Q^* A_Q − 1 is a local, complex-linear, self-adjoint Schrödinger operator with exponentially decaying potential. The spectral lemma for R_Q (purely absolutely continuous spectrum on [5/4, ∞), no eigenvalues, no thresho

What would settle it

Numerically integrate R_Q u = (5/4)u with the near-origin behavior u ~ r^{1/2+m}; if the far-field coefficient of the growing mode e^{+r/2} is found to be zero, the threshold is non-resonant, while a nonzero coefficient would exhibit a threshold resonance, contradicting Lemma 2.2 and invalidating the Strichartz input. Equivalently, constructing a normalized L² eigendata for R_Q with eigenvalue in [0, 5/4] would falsify the theorem's linear premise.

Watch

Extended reading notes

Core claim

Main result (Theorem 1.1): there is a δ > 0 such that any radial H^1_m datum ε_0 with ‖ε_0‖_{H^1_m} ≤ δ initiates a global solution ε(t) of the perturbed system, with the uniform control ‖ε‖_{S^1_m(ℝ)} ≲ δ; in particular the degree-m equivariant vortex is asymptotically stable under equivariant perturbations. The argument applies the covariant derivative D_+ to the full flow to produce the transformed variable ε_1 = D_+Φ, which satisfies i∂_t ε_1 − ½ R_Q ε_1 = N(ε_1) with R_Q local and self-adjoint. Lemma 2.2 identifies σ(R_Q) = [5/4, ∞) as purely absolutely continuous, with no eigenvalue in [0, 5/4] and no threshold resonance; this feeds Strichartz estimates via Corollary 2.3. The original

Load-bearing premise

The proof stands on the spectral lemma that the Darboux-transformed operator R_Q has purely absolutely continuous spectrum [5/4, ∞) with no eigenvalue below 5/4 and no threshold resonance at 5/4; in particular, the boundary-term computation at infinity in Lemma 2.2 must give a genuine cancellation, otherwise the Strichartz estimates and the whole bootstrap collapse.

Editorial extensions

If this is right

  • For every winding number m ≥ 1, small equivariant perturbations of the vortex produce global solutions with a uniform Strichartz bound; in particular, no finite-time blow-up can arise from small equivariant data.
  • The global bound ‖ε‖_{S^1_m(ℝ)} ≲ δ implies time-integrated decay of the perturbation in the admissible L^p norms, which is exactly the asymptotic stability asserted in the abstract.
  • The nonlinearity is controlled by quadratic and cubic terms in the Strichartz norm, using the fact that the admissible range on H² is strictly larger than on ℝ²; the same control also yields local well-posedness for arbitrary H^1_m data (Appendix A).
  • Lemma 2.2's absolute-continuity result — no eigenvalues and no resonance at the threshold 5/4 — is what makes the linearized Strichartz estimates available, so the stability result is inherited from a purely spectral statement about a one-dimensional Schrödinger operator with exponentially decaying potential.

Reading between the lines

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

  • If the spectral lemma holds, the same Darboux-transform-plus-elliptic-inversion scheme may transfer to the classical self-dual Chern-Simons-Schrödinger equation on H² or to equivariant data on other noncompact rank-one symmetric spaces, where analogous dispersive estimates are available.
  • The boundary-term computation identifying 5/4 as a non-resonant threshold is delicate and involves the far-field behavior φ ~ c e^{-r/2}; a direct numerical check of the Jost-function coefficient at 5/4 for m = 1 would independently verify the paper's main spectral input.
  • The expected gap eigenvalues of H = −Δ_{H²} + Q² (Remark 2.5) do not enter the stability argument; this suggests that vortex stability on H² is governed by the absence of discrete spectrum of R_Q, not of H — a structural prediction one could test by computing both spectra numerically.
  • Because the perturbation class is equivariant and radial, the stability statement does not address non-equivariant or translational perturbations; extending the result would require additional modulation and radiation-damping analysis beyond the equivariant sector.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper studies Manton's Chern-Simons-Schrödinger system on the hyperbolic plane in the self-dual (Bogomolny) regime, restricted to equivariant perturbations of the degree-m vortex. In the Coulomb gauge the perturbation ε satisfies a nonlinear equation i∂tε - (1/2)L_Q^*L_Q ε = G(ε), where L_Q and L_Q^* are nonlocal and not complex-linear. The authors introduce a nonlinear Darboux transformation to a new variable ε1 satisfying i∂tε1 - (1/2)R_Q ε1 = N(ε1), with R_Q a local self-adjoint Schrödinger operator. They then derive elliptic equations relating Re ε and Im ε to ε1 and to the linearized gauge potential a_θ, solve these using Green's function for H=-Δ_{H^2}+Q^2, and use Strichartz estimates imported from [28] and [1] to close a bootstrap. Theorem 1.1 asserts that sufficiently small H^1_m data produce a global solution with uniform Strichartz bound. An appendix proves local well-posedness in H^1_m.

Significance. If the proof is completed, the result would be a meaningful first step in the stability theory of vortices in Manton's model on a curved background. The Darboux-transform strategy is attractive and mirrors recent work on Chern-Simons-Schrödinger equations, and the reduction of the nonlocal linearized operators to a local self-adjoint operator is a genuine structural insight. The paper also makes explicit use of the hyperbolic-space spectral and Strichartz theory from [28] and [1], which is appropriate and not circular. The main theorem, however, is currently not fully supported because the spectral analysis of R_Q and H contains gaps, and one nonlinear estimate in the bootstrap is not justified.

major comments (4)
  1. [§2.2, Lemma 2.2 (resonance argument)] The boundary-term computation after (2.21) is not valid. With φ~c e^{-r/2}, β=tanh(r/2)→1, ∂_r^*β→-1, sh(R)~e^R/2 and ∂_rφ(R)~-(c/2)e^{-R/2}, the displayed term sh(R)∂_rφ(R)(∂_r^*β)(R) behaves like (c/4)e^{R/2}, which diverges. If an omitted factor φ(R) was intended, the claimed cancellation -c^2/8 - c^2/8 + c^2/4 = 0 still does not correspond to the actual asymptotics, since the third boundary term is O(e^{-2R}), not -c^2/8. Thus the exclusion of a threshold resonance is unproved. Because Corollary 2.3 and Proposition 3.1 rely on Lemma 2.2, the main theorem is not supported as written.
  2. [§2.2, Lemma 2.4 (zero not an eigenvalue of H)] The proof that 0 is not an eigenvalue of H is not rigorous. The sentence 'Since v∈L^2(0,∞), it must vanish at some point' is false: a nonzero L^2 function need not have any zero. Moreover, with Hv=0 the displayed inequality '0<∫_0^{r0} Hu v dx' is unjustified (the integral is 0 if Hv=0). The Green representation (2.26) requires a reliable exclusion of the L^2 kernel of H. Please replace this argument with a standard Sturm-Liouville/positivity comparison or provide a correct reference.
  3. [§3.1, proof of Proposition 3.1] The displayed nonlinear estimate ∥(a_θ/sh)ε1∥_{L^{4/3}L^{4/3}} ≤ ∥a_θ/sh∥^2_{L∞L4} ∥ε∥_{L^{4/3}L4} ≲ X^3+X^5 does not follow from Lemma 3.6, which has ∥ε1∥_{L^{p2}} on the right-hand side, nor from any previous estimate. The power of a_θ/sh is also inconsistent with the left-hand side. This line is needed for the bootstrap (3.13). Please correct the Hölder/Lemma combination or the displayed inequality.
  4. [Abstract and Introduction (claim of asymptotic stability)] The paper states in the abstract and introduction that the vortex is 'asymptotically stable,' but Theorem 1.1 only proves global existence and the uniform Strichartz bound ∥ε∥_{S^1_m(R)} ≲ δ. No decay as t→∞ or convergence to the vortex is established. If the intended statement is Lyapunov/global stability, the terminology should be softened; if asymptotic stability is intended, additional dispersive decay must be proved.
minor comments (6)
  1. [Abstract] Typo: 'Abalian-Higgs' should be 'Abelian-Higgs'; later 'Boglomony' should be 'Bogomolny'.
  2. [Corollary 2.3] The statement says 'for any two pairs of exponents (p1,q1) and (p1,q1)' but only one pair is named. It should presumably be (p1,q1) and (p2,q2).
  3. [Lemma 2.4] The phrase 'Let 0<r0≤∞ be the first value such that v(r0)=0' needs care when r0=∞; the integration by parts requires decay conditions at infinity that are not stated.
  4. [Appendix A, Theorem A.1] The uniqueness statement uses C((0,T), H^1_m)∩S^1_m((0,T)); for a well-posedness statement the interval should be [0,T] or the endpoint behavior should be specified.
  5. [Notation in §3.1] Norms such as ∥a_θ/sh∥^2_{L∞L4} are ambiguous; it should be stated explicitly whether the L∞ is in time and L^4 in the radial variable.
  6. [References] There are spacing artifacts in several reference entries ('La wrie', 'Hor v athy') that should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and relies on external published Strichartz benchmarks.

full rationale

The paper's central claim (Theorem 1.1) is proved by a nonlinear Darboux transform derived in Section 2, followed by bootstrap estimates in Section 3. The key spectral input, Lemma 2.2, is proved in the paper (spectrum of R_Q is purely absolutely continuous [5/4,∞), no eigenvalues, no threshold resonance); it is not imported from a self-citation. The Strichartz estimates for the transformed variable are imported from [28, Cor 1.19] and free-streaming estimates from [1]; [28] is a published AMS Memoir and is invoked with the spectral assumptions verified in Lemma 2.2, so it is an external benchmark rather than a restatement of the target result. Although [28] shares an author, its results are parameter-free, published, and independent of the present vortex-specific construction. The estimates relating ε and ε1 (Lemmas 3.2–3.8) are proven from the derived elliptic equations (2.16), (2.17), and (2.18) via Green's function representations; no fitted parameter is renamed as a prediction. The potential issue raised in the skeptic's note—the threshold-resonance boundary-term cancellation in Lemma 2.2—is a mathematical correctness concern, not a circular reduction: it does not make the conclusion equivalent to an input by construction. Under the hard rule that circularity requires a quotable reduction of the derivation to its own inputs, no such step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

Pure mathematical proof with no fitted parameters, no invented physical entities, and no data. The imported Strichartz estimates and standard vortex asymptotics are the main exogenous inputs. The least-supported ledger entry is the asserted monotonicity ∂_r V < 0, which is used critically in the spectral analysis.

assumptions (5)
  • domain assumption The potential V of R_Q satisfies ∂_r V < 0 globally
    Asserted in Lemma 2.2 with no derivation from vortex properties; used to rule out eigenvalues below 5/4. The displayed formula has an ambiguous sign for the Q²(m-A_θ[Q])/sh r term in the rendered text.
  • domain assumption The vortex Q and connection A_θ[Q] satisfy |A_θ[Q]| ≤ m, 1-Q² ≥ 0, Q ~ r^m near 0, and Q → 1 at infinity
    Standard vortex asymptotics cited from [37,44,45]; needed for the Green's function estimates and for the asserted sign of ∂_r V.
  • standard math Strichartz estimates for Schrödinger operators with purely absolutely continuous spectrum and no threshold resonance ([28, Cor. 1.19]) apply to R_Q
    Invoked in Corollary 2.3; imported from a published AMS Memoir. The second author is an author of [28], but the result is independently published, so this is an external benchmark rather than circularity.
  • standard math The linear Schrödinger flow on H² satisfies Strichartz estimates ([1, Theorem 3.6])
    Used in Appendix A for local well-posedness of the ε-equation.
  • domain assumption The elliptic operator H = -∆_{H²} + Q² has no zero eigenvalue
    Proven in Lemma 2.4 via a test function and a maximum-principle argument; load-bearing for inverting equation (2.16) to represent Re(ε) in terms of ε_1.

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Pith. "Pith review of Equivariant stability of vortices in Manton's Chern-Simons-Schr\"odinger system on the hyperbolic plane." pith.science (2026). https://pith.science/paper/WGK4WKRY

@misc{pith2026250906090,
  author       = {Pith},
  title        = {Pith review of: Equivariant stability of vortices in Manton's Chern-Simons-Schr\"odinger system on the hyperbolic plane},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WGK4WKRY}},
  note         = {Machine review of arXiv:2509.06090}
}
abstract

In this work we study magnetic vortices on the hyperbolic plane for a Chern-Simons-Schr\"odinger system introduced by Manton. The model can be thought of as the Schr\"odinger analogue of the Abalian-Higgs model. It consists of a system of partial differential equations, where the complex Higgs field $\Phi$ evolves according to a nonlinear Schr\"odinger equation coupled to an electromagnetic field $A$. We restrict attention to the self-dual (Bogomolny) case under equivariance symmetry. For each $m\geq 1$ we prove the asymptotic stability of the equivariant vortex of degree $m$. The main novelties are unraveling the favorable structure of the equations after a nonlinear Darboux transform, and the analysis of the elliptic operator relating the original and the transformed variables.

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Cited by 1 Pith paper

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  1. Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model

    math.AP 2026-08 conditional novelty 8.0 of 10

    Small equivariant perturbations of the degree-one vortex decay with radiation rate, while the internal mode damps like epsilon squared over one plus Gamma epsilon squared t, proving asymptotic stability.

Reference graph

Works this paper leans on

51 extracted references · 50 canonical work pages · cited by 1 Pith paper

  1. [28]

    uhrmann, J., Oh, S.-J., and Shahshahani, S. Local smoothing estimates for S chr\

    Lawrie, A., L\"uhrmann, J., Oh, S.-J., and Shahshahani, S. Local smoothing estimates for S chr\"odinger equations on hyperbolic space. Mem. Amer. Math. Soc. 291 , 1447 (2023), v+165

    work page 2023

  2. [1]

    Nonlinear Schr\"odinger Equation on Real Hyperbolic Spaces

    Anker, J.-P., and Pierfelice, V. Nonlinear Schr\"odinger Equation on Real Hyperbolic Spaces . Annales de l'I.H.P. Analyse non lin\'eaire 26 , 5 (2009), 1853--1869

    work page 2009

  3. [2]

    The nonlinear S chr\"odinger equation on hyperbolic space

    Banica, V. The nonlinear S chr\"odinger equation on hyperbolic space. Comm. Partial Differential Equations 32 , 10-12 (2007), 1643--1677

    work page 2007

  4. [3]

    On scattering for NLS : from E uclidean to hyperbolic space

    Banica, V., Carles, R., and Duyckaerts, T. On scattering for NLS : from E uclidean to hyperbolic space. Discrete Contin. Dyn. Syst. 24 , 4 (2009), 1113--1127

    work page 2009

  5. [4]

    Scattering theory for radial nonlinear schr \"o dinger equations on hyperbolic space

    Banica, V., Carles, R., and Staffilani, G. Scattering theory for radial nonlinear schr \"o dinger equations on hyperbolic space. Geometric and Functional Analysis 18 , 2 (2008), 367--399

    work page 2008

  6. [5]

    Weighted S trichartz estimates for radial S chr\"odinger equation on noncompact manifolds

    Banica, V., and Duyckaerts, T. Weighted S trichartz estimates for radial S chr\"odinger equation on noncompact manifolds. Dyn. Partial Differ. Equ. 4 , 4 (2007), 335--359

    work page 2007

  7. [6]

    Global existence, scattering and blow-up for the focusing NLS on the hyperbolic space

    Banica, V., and Duyckaerts, T. Global existence, scattering and blow-up for the focusing NLS on the hyperbolic space. Dyn. Partial Differ. Equ. 12 , 1 (2015), 53--96

    work page 2015

  8. [7]

    Blowing up time-dependent solutions of the planar, C hern- S imons gauged nonlinear S chr\"odinger equation

    Berg\'e, L., De Bouard, A., and Saut, J.-C. Blowing up time-dependent solutions of the planar, C hern- S imons gauged nonlinear S chr\"odinger equation. Nonlinearity 8 , 2 (1995), 235--253

    work page 1995

Show all 51 references
  1. [8]

    Collapse of C hern- S imons-gauged matter fields

    Berg\'e, L., De Bouard, A., and Saut, J.-C. Collapse of C hern- S imons-gauged matter fields. Phys. Rev. Lett. 74 , 20 (1995), 3907--3911

    1995

  2. [9]

    Borthwick, D., and Marzuola, J. L. Dispersive estimates for scalar and matrix S chr\"odinger operators on H ^ n+1 . Math. Phys. Anal. Geom. 18 , 1 (2015), Art. 22, 26

    2015

  3. [10]

    Bradlow, S. B. Vortices in holomorphic line bundles over closed k \"a hler manifolds. Communications in mathematical physics 135 , 1 (1990), 1--17

    1990

  4. [11]

    Global existence in the C auchy problem of the relativistic C hern- S imons- H iggs theory

    Chae, D., and Choe, K. Global existence in the C auchy problem of the relativistic C hern- S imons- H iggs theory. Nonlinearity 15 , 3 (2002), 747--758

    2002

  5. [12]

    Dichotomy results for the electromagnetic S chr\"odinger equation

    Czubak, M., Miller, I., and Roudenko, S. Dichotomy results for the electromagnetic S chr\"odinger equation. Preprint arXiv:2412.20239\/ (2024)

    2024 arXiv

  6. [13]

    Global existence for a nonlinear S chroedinger- C hern- S imons system on a surface

    Demoulini, S. Global existence for a nonlinear S chroedinger- C hern- S imons system on a surface. Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire 24 , 2 (2007), 207--225

    2007

  7. [14]

    Adiabatic limit and the slow motion of vortices in a C hern- S imons- S chr\"odinger system

    Demoulini, S., and Stuart, D. Adiabatic limit and the slow motion of vortices in a C hern- S imons- S chr\"odinger system. Comm. Math. Phys. 290 , 2 (2009), 597--632

    2009

  8. [15]

    A direct existence proof for the vortex equations over a compact R iemann surface

    Garc\'ia-Prada, O. A direct existence proof for the vortex equations over a compact R iemann surface. Bull. London Math. Soc. 26 , 1 (1994), 88--96

    1994

  9. [16]

    Geometric analysis on symmetric spaces , vol

    Helgason, S. Geometric analysis on symmetric spaces , vol. 39 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 1994

    1994

  10. [17]

    A., and Zhang, P

    Horvathy, P. A., and Zhang, P. Vortices in (abelian) C hern- S imons gauge theory. Phys. Rep. 481 , 5-6 (2009), 83--142

    2009

  11. [18]

    Energy solution to the C hern- S imons- S chr\"odinger equations

    Huh, H. Energy solution to the C hern- S imons- S chr\"odinger equations. Abstr. Appl. Anal.\/ (2013), Art. ID 590653, 7

    2013

  12. [19]

    D., Pausader, B., and Staffilani, G

    Ionescu, A. D., Pausader, B., and Staffilani, G. On the global well-posedness of energy-critical S chr\"odinger equations in curved spaces. Anal. PDE 5 , 4 (2012), 705--746

    2012

  13. [20]

    D., and Staffilani, G

    Ionescu, A. D., and Staffilani, G. Semilinear S chr \"o dinger flows on hyperbolic spaces: scattering H^1 . Math. Ann. 345 , 1 (2009), 133--158

    2009

  14. [21]

    Vortices and monopoles

    Jaffee, A., and Taubes, C. Vortices and monopoles. structure of static gauge theories. Progress in Physics 2\/ (1980)

    1980

  15. [22]

    Construction of blow-up manifolds to the equivariant self-dual C hern- S imons- S chr\"odinger equation

    Kim, K., and Kwon, S. Construction of blow-up manifolds to the equivariant self-dual C hern- S imons- S chr\"odinger equation. Ann. PDE 9 , 1 (2023), Paper No. 6, 129

    2023

  16. [23]

    On pseudoconformal blow-up solutions to the self-dual C hern- S imons- S chr\"odinger equation: existence, uniqueness, and instability

    Kim, K., and Kwon, S. On pseudoconformal blow-up solutions to the self-dual C hern- S imons- S chr\"odinger equation: existence, uniqueness, and instability. Mem. Amer. Math. Soc. 284 , 1409 (2023), vi+128

    2023

  17. [24]

    Soliton resolution for equivariant self-dual C hern- S imons- S chr\"odinger equation equation in weighted sobolev class

    Kim, K., Kwon, S., and Oh, S.-J. Soliton resolution for equivariant self-dual C hern- S imons- S chr\"odinger equation equation in weighted sobolev class. Preprint arXiv:2202.07314\/ (2022)

    2022

  18. [25]

    Blow-up dynamics for radial self-dual C hern- S imons- S chr\"odinger equation with prescribed asymptotic profile

    Kim, K., Kwon, S., and Oh, S.-J. Blow-up dynamics for radial self-dual C hern- S imons- S chr\"odinger equation with prescribed asymptotic profile. Preprint arXiv:2409.13274\/ (2024)

    2024

  19. [26]

    Blow-up dynamics for smooth finite energy radial data solutions to the self-dual C hern- S imons- S chr\"odinger equation

    Kim, K., Kwon, S., and Oh, S.-J. Blow-up dynamics for smooth finite energy radial data solutions to the self-dual C hern- S imons- S chr\"odinger equation. Ann. Sci. \'Ec. Norm. Sup\'er. (4) 57 , 6 (2024), 1599--1691

    2024

  20. [27]

    uhrmann, J., Oh, S.-J., and Shahshahani, S. Asymptotic stability of harmonic maps on the hyperbolic plane under the S chr\

    Lawrie, A., L\"uhrmann, J., Oh, S.-J., and Shahshahani, S. Asymptotic stability of harmonic maps on the hyperbolic plane under the S chr\"odinger maps evolution. Comm. Pure Appl. Math. 76 , 3 (2023), 453--584

    2023

  21. [29]

    Stability of stationary equivariant wave maps from the hyperbolic plane

    Lawrie, A., Oh, S.-J., and Shahshahani, S. Stability of stationary equivariant wave maps from the hyperbolic plane. Amer. J. Math. 139 , 4 (2017), 1085--1147

    2017

  22. [30]

    Asymptotic stability of large energy harmonic maps under the wave map from 2 D hyperbolic spaces to 2 D hyperbolic spaces

    Li, Z. Asymptotic stability of large energy harmonic maps under the wave map from 2 D hyperbolic spaces to 2 D hyperbolic spaces. Adv. Math. 370\/ (2020), 107234, 87

    2020

  23. [31]

    On global dynamics of S chr\"odinger map flows on hyperbolic planes near harmonic maps

    Li, Z. On global dynamics of S chr\"odinger map flows on hyperbolic planes near harmonic maps. Comm. Math. Phys. 393 , 1 (2022), 279--345

    2022

  24. [32]

    On threshold solutions of the equivariant C hern- S imons- S chr\"odinger equation

    Li, Z., and Liu, B. On threshold solutions of the equivariant C hern- S imons- S chr\"odinger equation. Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire 39 , 2 (2022), 371--417

    2022

  25. [33]

    Lim, Z. M. Large data well-posedness in the energy space of the C hern- S imons- S chr\"odinger system. J. Differential Equations 264 , 4 (2018), 2553--2597

    2018

  26. [34]

    Global wellposedness of the equivariant C hern- S imons- S chr\"odinger equation

    Liu, B., and Smith, P. Global wellposedness of the equivariant C hern- S imons- S chr\"odinger equation. Rev. Mat. Iberoam. 32 , 3 (2016), 751--794

    2016

  27. [35]

    Local wellposedness of C hern- S imons- S chr\"odinger

    Liu, B., Smith, P., and Tataru, D. Local wellposedness of C hern- S imons- S chr\"odinger. Int. Math. Res. Not. IMRN , 23 (2014), 6341--6398

    2014

  28. [36]

    On the decaying property of quintic NLS on 3 D hyperbolic space

    Ma, C., Wang, H., Yu, X., and Zhao, Z. On the decaying property of quintic NLS on 3 D hyperbolic space. Nonlinear Anal. 247\/ (2024), Paper No. 113599, 12

    2024

  29. [37]

    Topological solitons

    Manton, N., and Sutcliffe, P. Topological solitons . Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 2004

    2004

  30. [38]

    Manton, N. S. First order vortex dynamics. Ann. Physics 256 , 1 (1997), 114--131

    1997

  31. [39]

    Nonlinear waves on 3 D hyperbolic space

    Metcalfe, J., and Taylor, M. Nonlinear waves on 3 D hyperbolic space. Trans. Amer. Math. Soc. 363 , 7 (2011), 3489--3529

    2011

  32. [40]

    Dispersive wave estimates on 3 D hyperbolic space

    Metcalfe, J., and Taylor, M. Dispersive wave estimates on 3 D hyperbolic space. Proc. Amer. Math. Soc. 140 , 11 (2012), 3861--3866

    2012

  33. [41]

    Decay and scattering for the C hern- S imons- S chr\"odinger equations

    Oh, S.-J., and Pusateri, F. Decay and scattering for the C hern- S imons- S chr\"odinger equations. Int. Math. Res. Not. IMRN , 24 (2015), 13122--13147

    2015

  34. [42]

    Methods of modern mathematical physics

    Reed, M., and Simon, B. Methods of modern mathematical physics. IV . A nalysis of operators . Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978

    1978

  35. [43]

    Samols, T. M. Vortex scattering. Comm. Math. Phys. 145 , 1 (1992), 149--179

    1992

  36. [44]

    Hyperbolic vortices and some non-self-dual classical solutions of SU (3) gauge theory

    Schiff, J. Hyperbolic vortices and some non-self-dual classical solutions of SU (3) gauge theory. Phys. Rev. D 44\/ (Jul 1991), 528--535

    1991

  37. [45]

    Strachan, I. A. B. Low-velocity scattering of vortices in a modified abelian H iggs model. J. Math. Phys. 33 , 1 (1992), 102--110

    1992

  38. [46]

    Dynamics of abelian H iggs vortices in the near B ogomolny regime

    Stuart, D. Dynamics of abelian H iggs vortices in the near B ogomolny regime. Comm. Math. Phys. 159 , 1 (1994), 51--91

    1994

  39. [47]

    Stuart, D. M. A. Periodic solutions of the abelian H iggs model and rigid rotation of vortices. Geom. Funct. Anal. 9 , 3 (1999), 568--595

    1999

  40. [48]

    Global well-posedness and scattering for the defocusing mass-critical S chr\"odinger equation in the three-dimensional hyperbolic space

    Wilson, B., and Yu, X. Global well-posedness and scattering for the defocusing mass-critical S chr\"odinger equation in the three-dimensional hyperbolic space. Preprint arXiv:2310.12277\/ (2025)

    2025 arXiv

  41. [49]

    Some exact multipseudoparticle solutions of classical yang-mills theory

    Witten, E. Some exact multipseudoparticle solutions of classical yang-mills theory. Phys. Rev. Lett. 38\/ (Jan 1977), 121--124

    1977

  42. [50]

    Global aspects of current algebra

    Witten, E. Global aspects of current algebra. Nuclear Phys. B 223 , 2 (1983), 422--432

    1983

  43. [51]

    C., Hansson, T

    Zhang, S. C., Hansson, T. H., and Kivelson, S. Effective-field-theory model for the fractional quantum hall effect. Phys. Rev. Lett. 62\/ (Jan 1989), 82--85

    1989

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