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Simulations of Magnetic Monopole Collisions

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read BPS monopole collisions scatter at 90, 60, and 45 degrees for two, three, and four monopoles, matching moduli-space predictions, even at relativistic speeds and away from the BPS limit.

desk verdict Useful numerical confirmation of moduli-space monopole scattering with a new initial-condition ansatz; central angles are credible, but radiation estimate and numerical fidelity need more support. read the letter →

arxiv 2502.01756 v1 pith:65U7YWBF submitted 2025-02-03 hep-th hep-phmath-phmath.MP

Maximilian Bachmaier , Gia Dvali , Josef Seitz , Juan Sebastián Valbuena-Bermúdez This is my paper · ORCID
classification hep-thhep-phmath-phmath.MP MSC 81T1335Q5165M06 PACS 14.80.Hv11.27.+d
keywords magneticmonopolesBPSlimitmodulispaceapproximationAtiyah-Hitchinmanifoldmonopolescatteringnumericalsimulationnon-BPStopologicalsolitons
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper uses numerical simulations of 't Hooft–Polyakov magnetic monopoles to test whether low-energy scattering is described by geodesic motion on the moduli space of static solutions. It reports that head-on collisions of two, three, and four monopoles deflect by 90, 60, and 45 degrees respectively, matching the moduli-space approximation, and that non-planar collisions pass through the predicted tetrahedral and cubic intermediate energy configurations. The paper further claims these patterns persist at relativistic speeds up to $u=0.8$ and for non-BPS parameters with $m_h/m_v$ between 0 and 1.0, with radiation losses of roughly five percent even at $u=0.62$. A sympathetic reader would care because the results suggest the moduli-space picture is robust well beyond the slow-motion, BPS regime where it is formally derived.

What carries the argument

The load-bearing object is the moduli space approximation: slow monopole motion is treated as geodesic motion on the hyper-Kähler moduli space of static BPS solutions, with the two-monopole relative dynamics governed by the Atiyah-Hitchin manifold. The paper's numerical study relies on an analytic ansatz for multi-monopole initial data in which the scalar-field direction is built from sums of azimuthal angles around each monopole position, and on tracking the zeros of $\phi^a=0$ to measure monopole trajectories. The ansatz supplies initial conditions whose time evolution produces the toroidal, tetrahedral, and cubic intermediate states used to identify each scattering channel.

What would settle it

Repeat the two-monopole head-on collision at $u=0.62$ with lattice spacings $\delta = 0.125 m_v^{-1}$ and $0.0625 m_v^{-1}$, or with a box of side $120 m_v^{-1}$, and measure the final scattering angle and radiated energy; if the radiation fraction or the final velocity change differs from the reported $\Delta u = 0.04$ by a comparable amount, the quantitative radiation claim is a lattice artifact.

Watch

Extended reading notes

Core claim

The central claim is that full field-theory simulations reproduce the moduli-space approximation's scattering laws for multi-monopole systems, including in regimes beyond the approximation's formal validity. For two monopoles, a head-on collision forms a toroidal state and the monopoles emerge at 90 degrees, as predicted by the smoothed cone submanifold of the Atiyah-Hitchin metric. Three and four monopoles in cyclically symmetric planar configurations scatter at 60 and 45 degrees, and toroidal charge-$N$ monopoles colliding with single monopoles form tetrahedral or cubic energy configurations before splitting into individual monopoles. The authors take this agreement as evidence that the moduli-space description captures the essential dynamics of monopole collisions.

Load-bearing premise

The numerical lattice with spacing $0.25 m_v^{-1}$ inside a box of side $60 m_v^{-1}$ resolves the collisions well enough that the reported scattering angles, velocity changes, and the roughly five-percent radiation estimate are physical rather than numerical artifacts; the paper reports crosschecks with absorbing boundaries and larger lattices but shows no quantitative convergence study.

Editorial extensions

If this is right

  • Head-on two-monopole collisions in the $x$-$y$ plane scatter at 90 degrees, with the monopole speeds nearly unchanged ($\Delta u \approx 0.01$ at $u=0.2$).
  • Cyclically symmetric three- and four-monopole planar collisions scatter at 60 and 45 degrees respectively.
  • Right-angle scattering persists at velocities up to $u=0.8$ and for non-BPS parameters with $m_h/m_v$ up to 1.0, despite the magnetic repulsion present when the scalar force is screened.
  • Non-planar collisions of toroidal charge-$N$ monopoles with single monopoles pass through tetrahedral or pyramidal states and then split into individual monopoles, while two charge-two tori pass through a cubic state and split into four monopoles.
  • The observed zero and anti-zero dynamics in non-planar scattering agree with the predictions for monopole zeros.

Reading between the lines

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

  • An implication the authors leave implicit is that the moduli-space geodesic picture may remain quantitatively trustworthy in regimes where its formal assumptions are violated, which could justify using effective moduli-space descriptions for cosmological or collider-scale monopole interactions.
  • The analytic multi-monopole initial configurations introduced here could be adapted to test the predicted modifications of moduli-space trajectories caused by long-lived semi-bound excitations, analogous to the multi-bounce windows seen in vortex scattering.
  • The reported five-percent radiation estimate at $u=0.62$ rests on the lattice resolution; a quantitative convergence study would tell whether that number is physical or partly numerical.
  • Tracking zeros of the scalar field is a gauge-invariant diagnostic that could be applied to other soliton collisions, including skyrmion and vortex systems, to map scattering channels.
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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 / 5 minor

Summary. The paper reports numerical simulations of 't Hooft–Polyakov magnetic monopole scattering in an SU(2) gauge theory, using approximate multi-monopole ansätze as initial data. The authors study planar head-on collisions of two, three, and four monopoles, observing scattering angles of 90, 60, and 45 degrees respectively, and non-planar processes in which a charge-one monopole collides with a toroidal monopole or two toroidal monopoles collide, passing through tetrahedral and cubic intermediate states. They also extend the simulations to relativistic velocities up to u=0.8 and to non-BPS masses m_h/m_v in [0,1.0], and estimate that radiation emission is small. The central claim is that these observations validate the moduli-space approximation for monopole scattering.

Significance. If the numerical results are robust, this is a valuable direct test of the moduli-space approximation in a regime where exact analytic scattering is unavailable. A notable strength of the paper is that the scattering angles are not fitted or imposed; they emerge from independent time evolution of the field equations and match external predictions from the Atiyah–Hitchin metric and rational-map constructions. The paper also provides useful approximate initial configurations for multi-monopole collisions, and the non-planar simulations target nontrivial symmetric monopole states. However, the quantitative claims, particularly the radiation estimate, currently rest on a single lattice setup without a convergence study or error estimates, so the significance is conditional on additional numerical substantiation.

major comments (4)
  1. [Appendix (Numerical simulation), Section VIII] The quantitative claims in Section VIII depend entirely on the fidelity of one lattice setup: cubic box x,y,z in [-30 m_v^{-1}, 30 m_v^{-1}], spacing delta = 0.25 m_v^{-1}, time step dt = 0.1 m_v^{-1}, and Dirichlet boundary conditions. No convergence study, error bars, or resolution dependence is shown for the scattering angles, final velocities, or radiation estimate. The only crosscheck reported is the sentence 'absorbing boundaries and larger lattices produced no significant improvements,' which is not quantitative. In the BPS limit the Higgs and photon are massless, so long-range fields reach the Dirichlet boundaries; a systematic study varying delta, box size, and dt is needed to establish that the observed 90/60/45-degree scattering and the velocity changes are not numerical artifacts.
  2. [Section VIII] The radiation estimate is inferred solely from changes in the velocity of the scalar-field zeros: the paper reports Delta u = 0.01 at u = 0.2 and Delta u = 0.04 at u = 0.62, and then states that this corresponds to 'roughly 5 percent of the monopole mass being emitted as radiation.' No independent energy-loss measurement is presented, such as the flux of energy through a distant surface or a deficit in total field energy. Since the zeros themselves carry no energy, changes in zero velocity are not a direct measure of radiated energy. In addition, the approximate initial ansätze constructed in Sections VI and VII may themselves radiate if they are not sufficiently close to exact solutions, contaminating the post-collision velocity change. The manuscript should quantify the radiated energy directly and also quantify the residual field-equation error of the initial data.
  3. [Sections VI and VII, Eq. (34)] The multi-monopole configurations used as initial data are explicitly approximate, but the paper does not quantify how well they satisfy the field equations. The relaxation method in the Appendix defines an error functional E[f] for static solutions, but no analogous residual check is reported for the boosted, time-dependent initial ansätze used in the collision simulations. If these initial data contain significant unphysical radiation, the outgoing trajectories and velocity changes could be affected. The authors note that the z-axis two-monopole ansatz is 'less precise' than the x-axis one, but no precision metric is given. A quantitative measure of the residual, or a relaxation step applied to the initial data before boosting, would substantially strengthen the validation.
  4. [Section IX, Section X] The non-planar tetrahedral and cubic scattering processes are simulated by 'starting with their outcomes': a toroidal charge-N monopole is initialized colliding with a single monopole or another toroidal monopole, and the observed final states (three or four separate charge-one monopoles) are the natural continuation of these initial data. This is a legitimate inverse-scattering test, and the paper does acknowledge in Section X that a generic analytic initial configuration for multiple single-charged monopoles undergoing non-planar scattering remains open. However, the wording in Section X that these simulations 'validated several predictions of the moduli approximation' goes somewhat beyond what this setup demonstrates. The paper should state more carefully that the non-planar runs test the consistency of the known outcome states with the symmetric intermediate configurations, rather than showing that generic initial conditions evolve into those outcomes.
minor comments (5)
  1. [Section III header] The section title contains a typo: 'MODULI SP ACE' should be 'MODULI SPACE.'
  2. [Figure 1 caption] The caption says 'The trumped describes the motion in the other two planes'; 'trumped' should be 'trumpet.'
  3. [Section IX, around Eq. (30)] The sentence 'we chose z1 = -20 m_v^{-1}, z1 = 20 m_v^{-1}' appears to have a typo: the second value should be z2 = 20 m_v^{-1}.
  4. [Section VIII, around Eq. (29)] The text refers to the 'Lorenz factor'; the standard spelling in this context is 'Lorentz factor.'
  5. [Section VIII] The statement that Delta u = 0.04 corresponds to 'roughly 5 percent of the monopole mass being emitted as radiation' is misleading: the velocity change is a measure of kinetic-energy loss, not of rest-mass emission. The sentence should be rewritten in terms of the radiated energy fraction relative to the initial kinetic energy.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the planar scattering angles emerge from full-field evolution and agree with external moduli-space predictions; non-planar runs are openly initialized from their endpoints but still produce non-trivial intermediate dynamics, and self-citations are contextual only.

full rationale

The central claims are numerical tests of external moduli-space predictions. The planar two-, three-, and four-monopole results are obtained by evolving the full field equations from approximate multi-monopole ansätze (Eqs. 22-28) and tracking zeros of the scalar field; no scattering angle is inserted, and the observed 90/60/45-degree angles match the geodesic predictions of the Atiyah-Hitchin metric and the moduli approximation [13,14,16]. The ansätze are approximate but are not fitted to the scattering outcome. The non-planar simulations are explicitly time-reversed: Section IX states "We investigated these types of scatterings by starting with their outcomes," so the endpoint-to-endpoint map is not an independent prediction; nevertheless, the intermediate tetrahedral/cubic states and the zero/anti-zero dynamics are non-trivial outputs of the evolution and are compared with external results [28-30]. Self-citations [4,5,7,8,9] occur only in the cosmology preamble and are not load-bearing; no uniqueness theorem from the authors' prior work is invoked. The missing quantitative convergence study and the velocity-change-based radiation estimate (Section VIII, Appendix) are numerical-fidelity concerns, not circularity. No parameter is fitted and renamed as a prediction, and no equation is defined in terms of the quantity it is supposed to produce.

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

The central scattering-angle claims are checks of existing predictions and do not rest on fitted constants. The quantitative radiation-loss estimate depends on numerical convergence and the zero-tracking interpretation, both unquantified. No new physical entities are introduced.

assumptions (5)
  • standard math The Prasad-Sommerfield solution (Eq. 11) describes isolated BPS monopoles and is used to construct initial configurations.
    Taken as exact input from [10, 11] and used throughout Sections VI and VII.
  • domain assumption The moduli-space approximation of Manton and Stuart is valid for low-velocity monopole scattering.
    Section III and IV; the central comparison targets are derived from this approximation.
  • domain assumption The rational-map predictions of Hitchin, Manton, and Murray for tetrahedral and cubic configurations are correct.
    Section IX uses these predictions to interpret non-planar scattering results.
  • domain assumption The finite-difference discretization converges to the continuum classical field equations.
    Appendix: no quantitative convergence study is provided.
  • domain assumption Zeros of the scalar field phi^a = 0 track monopole cores sufficiently well to infer velocities and radiation loss.
    Section VIII: used for Delta u and the 5 percent radiation estimate.

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Pith. "Pith review of Simulations of Magnetic Monopole Collisions." pith.science (2026). https://pith.science/paper/65U7YWBF

@misc{pith2026250201756,
  author       = {Pith},
  title        = {Pith review of: Simulations of Magnetic Monopole Collisions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/65U7YWBF}},
  note         = {Machine review of arXiv:2502.01756}
}
read the original abstract

In this paper, we investigate the scattering of BPS magnetic monopoles through numerical simulations. We present an ansatz for various multi-monopole configurations suitable for analyzing monopole scattering processes. Our study includes planar scattering scenarios involving two, three, and four monopoles, as well as non-planar processes where three and four monopoles form intermediate tetrahedral and cubic states, respectively. Our observations align with the theoretical predictions of the moduli space approximation. Furthermore, we extend our analysis to relativistic velocities and explore parameters beyond the BPS limit.

Figures

Figures reproduced from arXiv: 2502.01756 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
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Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
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Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
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Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

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Works this paper leans on

37 extracted references · 26 canonical work pages · cited by 2 Pith papers

  1. [1]

    Magnetic Monopoles in Unified Gauge Theories,

    Gerard ’t Hooft, “Magnetic Monopoles in Unified Gauge Theories,” Nucl. Phys. B 79, 276–284 (1974)

    work page 1974

  2. [2]

    Particle Spectrum in Quantum Field Theory,

    Alexander M. Polyakov, “Particle Spectrum in Quantum Field Theory,” JETP Lett. 20, 194–195 (1974)

    work page 1974

  3. [3]

    The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,

    Alan H. Guth, “The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,” Phys. Rev. D 23, 347–356 (1981)

    1981

  4. [4]

    Is There a monopole problem?

    G. R. Dvali, Alejandra Melfo, and Goran Senjanovic, “Is There a monopole problem?” Phys. Rev. Lett. 75, 4559–4562 (1995), arXiv:hep-ph/9507230

    arXiv 1995

  5. [5]

    Large scale structure and supersymmetric inflation without fine tuning,

    G. R. Dvali, Q. Shafi, and Robert K. Schaefer, “Large scale structure and supersymmetric inflation without fine tuning,” Phys. Rev. Lett. 73, 1886–1889 (1994), arXiv:hep-ph/9406319

    arXiv 1994

  6. [6]

    Magnetic Monopoles in Grand Unified Theories,

    Paul Langacker and So-Young Pi, “Magnetic Monopoles in Grand Unified Theories,” Phys. Rev. Lett. 45, 1 (1980)

    work page 1980

  7. [7]

    Sweep- ing away the monopole problem,

    G. R. Dvali, Hong Liu, and Tanmay Vachaspati, “Sweep- ing away the monopole problem,” Phys. Rev. Lett. 80, 2281–2284 (1998), arXiv:hep-ph/9710301

    arXiv 1998

  8. [8]

    Era- sure of strings and vortices,

    Gia Dvali and Juan Sebasti´ an Valbuena-Berm´ udez, “Era- sure of strings and vortices,” Phys. Rev. D 107, 035001 (2023), arXiv:2212.07535 [hep-th]

    arXiv 2023

Show all 37 references
  1. [9]

    Radiation emission during the era- sure of magnetic monopoles,

    Maximilian Bachmaier, Gia Dvali, and Juan Sebasti´ an Valbuena-Berm´ udez, “Radiation emission during the era- sure of magnetic monopoles,” Phys. Rev. D 108, 103501 (2023), arXiv:2306.12958 [hep-th]

    2023 arXiv

  2. [10]

    An Exact Classical Solution for the ’t Hooft Monopole and the Julia-Zee Dyon,

    M. K. Prasad and Charles M. Sommerfield, “An Exact Classical Solution for the ’t Hooft Monopole and the Julia-Zee Dyon,” Phys. Rev. Lett. 35, 760–762 (1975)

    1975

  3. [11]

    Stability of Classical Solutions,

    E. B. Bogomolny, “Stability of Classical Solutions,” Sov. J. Nucl. Phys. 24, 449 (1976)

    1976

  4. [12]

    The Force Between ’t Hooft-Polyakov Monopoles,

    N. S. Manton, “The Force Between ’t Hooft-Polyakov Monopoles,” Nucl. Phys. B 126, 525–541 (1977)

    1977

  5. [13]

    A Remark on the Scattering of BPS Monopoles,

    N. S. Manton, “A Remark on the Scattering of BPS Monopoles,” Phys. Lett. B 110, 54–56 (1982)

    1982

  6. [14]

    The Geodesic approximation for the Yang- Mills Higgs equations,

    D. Stuart, “The Geodesic approximation for the Yang- Mills Higgs equations,” Commun. Math. Phys. 166, 149– 190 (1994)

    1994

  7. [15]

    Low-Energy Scatter- ing of Nonabelian Monopoles,

    M. F. Atiyah and Nigel J. Hitchin, “Low-Energy Scatter- ing of Nonabelian Monopoles,” Phys. Lett. A 107, 21–25 (1985)

    1985

  8. [16]

    MICHAEL FRANCIS ATIYAH and NIGEL HITCHIN, The Geometry and Dynamics of Magnetic Monopoles (Princeton University Press, 1988)

    1988

  9. [17]

    TASI lectures on solitons: Instantons, monopoles, vortices and kinks,

    David Tong, “TASI lectures on solitons: Instantons, monopoles, vortices and kinks,” in Theoretical Ad- vanced Study Institute in Elementary Particle Physics: Many Dimensions of String Theory (2005) arXiv:hep- th/0509216

    2005

  10. [18]

    N. S. Manton and P. Sutcliffe, Topological solitons , Cambridge Monographs on Mathematical Physics (Cam- bridge University Press, 2004)

    2004

  11. [19]

    Vortex scattering,

    T. M. Samols, “Vortex scattering,” Commun. Math. Phys. 145, 149–180 (1992)

    1992

  12. [20]

    Classical and Quan- tum Dynamics of BPS Monopoles,

    G. W. Gibbons and N. S. Manton, “Classical and Quan- tum Dynamics of BPS Monopoles,” Nucl. Phys. B 274, 183–224 (1986)

    1986

  13. [21]

    Radiation From Monopole Scattering,

    N. S. Manton and T. M. Samols, “Radiation From Monopole Scattering,” Phys. Lett. B 215, 559–563 (1988)

    1988

  14. [22]

    Nonexistence of Spherically Symmetric Monopoles with Multiple Mag- netic Charge,

    Erick J. Weinberg and Alan H. Guth, “Nonexistence of Spherically Symmetric Monopoles with Multiple Mag- netic Charge,” Phys. Rev. D 14, 1660 (1976). 14

    1976

  15. [23]

    A Yang-Mills Higgs Monopole of Charge 2,

    R. S. Ward, “A Yang-Mills Higgs Monopole of Charge 2,” Commun. Math. Phys. 79, 317–325 (1981)

    1981

  16. [24]

    Non- linear superposition of monopoles,

    Peter Forg´ acs, Zal´ an Horv´ ath, and L´ aszl´ o Palla, “Non- linear superposition of monopoles,” Nuclear Physics B 192, 141–158 (1981)

    1981

  17. [25]

    Rigorous Construction of Exact Multi - Monopole Solutions,

    M. K. Prasad and P. Rossi, “Rigorous Construction of Exact Multi - Monopole Solutions,” Phys. Rev. D 24, 2182 (1981)

    1981

  18. [26]

    Monopole- antimonopole Interaction Potential,

    Ayush Saurabh and Tanmay Vachaspati, “Monopole- antimonopole Interaction Potential,” Phys. Rev. D 96, 103536 (2017), arXiv:1705.03091 [hep-th]

    2017 arXiv

  19. [27]

    Vortex Scattering in Two-dimensions,

    E. P. S. Shellard and P. J. Ruback, “Vortex Scattering in Two-dimensions,” Phys. Lett. B 209, 262–270 (1988)

    1988

  20. [28]

    Symmetric monopoles,

    Nigel J. Hitchin, N. S. Manton, and M. K. Murray, “Symmetric monopoles,” Nonlinearity 8, 661–692 (1995), arXiv:dg-ga/9503016

    1995 arXiv

  21. [29]

    Tetrahedral and cubic monopoles,

    Conor J. Houghton and Paul M. Sutcliffe, “Tetrahedral and cubic monopoles,” Commun. Math. Phys. 180, 343– 362 (1996), arXiv:hep-th/9601146

    1996 arXiv

  22. [30]

    Monopole zeros,

    Paul M. Sutcliffe, “Monopole zeros,” Phys. Lett. B 376, 103–110 (1996), arXiv:hep-th/9603065

    1996 arXiv

  23. [31]

    Rational maps, monopoles and Skyrmions,

    Conor J. Houghton, Nicholas S. Manton, and Paul M. Sutcliffe, “Rational maps, monopoles and Skyrmions,” Nucl. Phys. B 510, 507–537 (1998), arXiv:hep-th/9705151

    1998 arXiv

  24. [32]

    Collective coordinate models for 2-vortex shape mode dynamics,

    A. Alonso Izquierdo, N. S. Manton, J. Mateos Guilarte, and A. Wereszczynski, “Collective coordinate models for 2-vortex shape mode dynamics,” Phys. Rev. D 110, 085006 (2024), arXiv:2405.20249 [hep-th]

    2024 arXiv

  25. [33]

    Scattering of vortices with excited normal modes,

    Steffen Krusch, Morgan Rees, and Thomas Winyard, “Scattering of vortices with excited normal modes,” Phys. Rev. D 110, 056050 (2024), arXiv:2406.04164 [math-ph]

    2024 arXiv

  26. [34]

    Feshbach resonances and dynamics of BPS solitons,

    Alberto Garc ´ ıa Mart ´ ın-Caro, Jose Queiruga, and An- drzej Wereszczynski, “Feshbach resonances and dynamics of BPS solitons,” (2025), arXiv:2501.02589 [hep-th]

    2025 arXiv

  27. [35]

    Structure of electroweak dumbbells,

    Teerthal Patel and Tanmay Vachaspati, “Structure of electroweak dumbbells,” Phys. Rev. D 107, 093010 (2023), arXiv:2302.04886 [hep-ph]

    2023 arXiv

  28. [36]

    On the stability of the iterated Crank-Nicholson method in numerical relativity,

    Saul A. Teukolsky, “On the stability of the iterated Crank-Nicholson method in numerical relativity,” Phys. Rev. D 61, 087501 (2000), arXiv:gr-qc/9909026

    2000 arXiv

  29. [37]

    Numba: A llvm-based python jit compiler,

    Siu Kwan Lam, Antoine Pitrou, and Stanley Seibert, “Numba: A llvm-based python jit compiler,” in Proceed- ings of the Second Workshop on the LL VM Compiler In- frastructure in HPC (2015) pp. 1–6

    2015

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