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Gravity reshapes kink collisions: resonance windows climb in speed

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2026-08-04 04:16 UTC pith:LCJ6MGG3

load-bearing objection First systematic collision study of self-gravitating φ⁴ kinks in 2D MMSS gravity; the resonance-window shift is solid, the narrowing is not yet measured. the 3 major comments →

arxiv 2607.13620 v2 pith:LCJ6MGG3 submitted 2026-07-15 hep-th

Kink collisions in a two-dimensional gravity model

Zhen-Tao He , Yuan Zhong This is my paper
classification hep-th
keywords self-gravitating kinkskink-antikink collisionstwo-dimensional dilaton gravityphi^4 modelresonance windowsquasi-bound statessingularity formation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper asks what happens to classic kink-antikink scattering in the φ^4 scalar theory when the field is coupled to a two-dimensional dilaton gravity. Numerically evolving head-on collisions, it finds that gravity changes the scattering structure: the resonance windows—velocity intervals in which the pair bounces several times before escaping—move toward higher initial velocities and become narrower as the gravitational coupling κ grows, while the critical escape velocity rises slightly. The paper proposes that the flat-space kink's localized shape mode becomes a long-lived quasi-bound state that leaks energy, providing a dissipative storage channel that modifies the windows. It also observes a clear geometrical response: after a collision the conformal factor decreases, meaning the local proper spatial scale contracts, and the Ricci scalar develops transient but finite peaks, indicating no singularity formation in the simulated regime. This connects soliton scattering to two-dimensional gravity toy models and suggests gravitational backreaction leaves observable signatures in defect dynamics.

Core claim

In this self-gravitating φ^4 model, the kink-antikink collision dynamics is governed not by a normalizable shape mode, which does not exist (the stability operator's zero mode is non-normalizable), but by a long-lived open-channel resonance. For κ=0.1, linear perturbation analysis yields a complex frequency dω_res ≈ 1.70583 + 7.05×10⁻⁴i, with quality factor Q ≈ 1.2×10³, close to the flat-space shape-mode frequency √3. This quasi-bound state stores energy during collisions and slowly leaks it, which the paper argues shifts and narrows the bounce windows and mildly raises the critical velocity. The numerical scans across κ=0.1, 0.3, 0.5 confirm the qualitative trend, and the geometry responds

What carries the argument

The load-bearing object is the quasi-bound state, an open-channel resonance of the scalar perturbation equation. In flat space the φ^4 kink has a discrete shape mode below the continuum; here the stability potential V_eff(z)=f''/f with f=φ'/(2A') tends to zero on the AdS-side infinity, so the would-be bound state becomes a resonance with a complex frequency, matched to an outgoing wave on the open side and an evanescent tail on the flat side. This leaky mode replaces the shape-mode mechanism of flat-space kink physics, and the paper fits the bounce intervals through ω_fit τ_n = δ + 2πn, obtaining ω_fit ≈ 1.6261, close to Re(ω_res), supporting the identification.

Load-bearing premise

The entire evolution fixes the dilaton to equal twice the conformal factor (φ=2A), even though the dilaton equation allows other modes, and the paper offers no evidence that this sector is preserved for generic initial data.

What would settle it

Run the same collisions starting from generic initial data that satisfy the constraint equations exactly and do not impose φ=2A; if the resonance windows, their shift with κ, and the finiteness of the Ricci scalar survive, the central claims hold. Alternatively, compute the complex resonance frequency for κ=0.3 and 0.5 using a method that resolves strongly damped modes and check whether the bounce-interval frequency matches Re(ω_res); a mismatch would disprove the quasi-bound state as the mediator.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the quasi-bound-state mechanism holds, the flat-space resonant energy-exchange picture becomes dissipative: windows persist because of a leaky mode, not a normalizable one, so they should systematically shrink as the leakage rate grows.
  • The measured bounce intervals at κ=0.1 reproduce the resonance frequency, meaning the same quantity that governs linear stability also sets the collision timescales.
  • Increasing κ makes the resonance shorter-lived and the windows harder to resolve; the trend predicts that beyond some κ the fractal window structure disappears entirely, leaving only the critical-velocity threshold.
  • The observed conformal-factor decrease implies a persistent, cumulative contraction of local spatial scale whenever kinks collide, which should affect any observable built from proper distances.
  • The finiteness of the Ricci scalar across the scanned (κ,v) range gives the first evidence that two-dimensional self-gravitating kink collisions avoid the singularities seen in AdS5 thick-brane collisions.

Where Pith is reading between the lines

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

  • The evolution is restricted to the exact sector φ=2A; a natural follow-up is to evolve generic dilaton data. If the extra dilaton mode becomes excited, the resonance-window trend and the no-singularity conclusion could change, so the observed κ-dependence might be partly a truncation artifact.
  • Extrapolating the quasi-bound-state lifetime trend predicts a κ above which Q drops to O(1) and the fractal windows vanish; this is testable with a resonance-extraction method that resolves heavily damped modes.
  • The post-collision conformal-factor contraction resembles a two-dimensional gravitational memory effect; computing the asymptotic energy flux, which the paper leaves to future work, would determine whether the leaked energy truly escapes to the AdS2 boundaries.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies head-on kink-antikink collisions in the ϕ^4 scalar theory coupled to two-dimensional Mann-Morris-Sikkema-Steele (MMSS) dilaton gravity. The authors reduce the system by setting φ(t,z)=2A(t,z), construct approximate boosted-superposition initial data, and evolve the coupled scalar-warp equations in the conformal gauge (12) with fourth-order finite differencing and adaptive time integration. They scan initial velocities at κ=0.1, 0.3, 0.5 and report that gravity shifts the resonance windows to higher velocities and narrows them with increasing κ, while the critical escape velocity increases mildly. A separate linear-stability computation gives a long-lived open-channel resonance at κ=0.1, with Re(ω_res)d≈1.70583 and Q≈1.2×10^3, which the authors connect to the bounce-time intervals via a two-parameter fit. The paper also reports a post-collision decrease of the conformal factor e^A and finite Ricci scalar in all simulations, concluding there is no evidence of singularity formation in this model.

Significance. If the central claims hold, this is the first numerical study of self-gravitating kink collisions in MMSS gravity, and the qualitative picture—stronger gravity shifting and narrowing resonance windows, raising the escape velocity, and producing quasi-bound energy storage—would be a genuinely new result connecting thick-brane phenomenology, 2D dilaton gravity, and kink-scattering theory. The paper has several concrete strengths: the equations of motion are written explicitly; the constraint equations are monitored; the momentum constraint shows fourth-order convergence; the resonance frequency is checked against five different cutoffs; and the stability-operator factorization gives a rigorous non-negative spectrum. The φ=2A truncation, which at first sight looks restrictive, is actually harmless for the evolved sector because Ψ=φ−2A satisfies the free wave equation (13) and does not back-react on the A,ϕ system. However, the quantitative support for the central scattering claims is not yet at the level required: the velocity scan is too coarse to resolve the window narrowing it claims, and the non-convergent Hamiltonian constraint violation in the initial data is not tested against

major comments (3)
  1. [Sec. 3.1, Fig. 3]
  2. [Sec. 2.3]
  3. [Sec. 3.1]
minor comments (5)
  1. [Sec. 3.2, Eq. (31) and Fig. 5]
  2. [Sec. 3.1, Fig. 3]
  3. [Sec. 2.4, Eq. (22)]
  4. [Sec. 2.2, Eq. (14)]
  5. [Sec. 3.3]

Circularity Check

0 steps flagged

No significant circularity: the central scattering results are independent numerical evolutions from stated equations, and the only fitted quantity is explicitly a post-hoc consistency check.

full rationale

The paper's central claims — shifted/narrowed resonance windows, mild increase in critical velocity, post-collision conformal-factor decrease, finite Ricci scalar — are obtained by time-evolving the stated PDE system (15)-(16) from superposed boosted kink initial data (19)-(21). No fitted parameter enters this evolution; κ and v are inputs, and the outcomes are read off the dynamics. The quasi-bound state frequency in Eq. (29) is computed by an eigenvalue/Wronskian shooting problem for the stated linear operator (23)-(25), not by fitting collision data. The subsequent fit ω_fit=1.6261, δ=15.1225 in Eq. (31) is explicitly labeled 'Fit' and used only as a consistency comparison with the independently computed resonance, so it is not a prediction forced by a fit. Self-citations to Refs. [93]-[99] supply the static background (6)-(9), potential (5), and stability factorization (26), but these ingredients are stated explicitly in this paper and are algebraically verifiable; no load-bearing assertion about the collision outcome is imported from a self-citation. The φ=2A sector restriction (14) is an explicit modeling assumption rather than a circular reduction; in fact the difference Ψ=φ-2A satisfies a free wave equation and does not back-react, so the truncation does not secretly encode the scattering conclusion. The manuscript's own caveats — under-resolved windows at κ=0.3,0.5 and non-convergent Hamiltonian constraint violation in the initial data — are numerical reliability concerns, not circularity. Accordingly, there are no circular steps to report.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The central new physics rests on two structural choices: the φ=2A truncation of the dilaton sector and the approximate superposition initial data. The only explicit fit to data is the bounce-time regression (ω_fit, δ). No new particles, forces, or entities are introduced.

free parameters (2)
  • ω_fit (effective oscillation frequency) = 1.6261 (with d=1)
    Obtained by linear fit of five bounce proper-time intervals to ω_fit τ_n = δ + 2πn in Fig. 5; used as support for the quasi-bound-state energy-storage mechanism.
  • δ (phase offset) = 15.1225
    Offset in the same linear fit; a nuisance parameter.
axioms (3)
  • ad hoc to paper The dilaton field is restricted to φ(t,z)=2A(t,z) for all times (Eq. 14).
    The paper states the dilaton equation (13) has more general solutions but chooses this special case for simplicity. All dynamical results, including the no-singularity conclusion, depend on this sector.
  • ad hoc to paper Initial data are constructed by linear superposition of boosted static solutions (Eqs. 20–21), which does not satisfy the constraints exactly.
    Sec. 2.3 notes that the Hamiltonian constraint violation remains essentially unchanged as N increases, indicating a limitation inherent in the superposition approximation. The physical validity of collision results relies on these violations being negligible.
  • domain assumption Artificial timelike boundaries at z=±L with Dirichlet/Neumann boundary conditions follow the asymptotic forms (Eqs. 10–11, 22).
    Boundary conditions may generate spurious reflections or constraint violations; the paper monitors them but they remain a modeling assumption, mitigated by restricting results to |z|<L−t.

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0 comments
read the original abstract

We numerically study kink-antikink collisions in the self-gravitating $\phi^4$ model coupled to the two-dimensional dilaton gravity theory proposed by Mann et al. The static kink solutions interpolate between an anti-de Sitter (AdS$_2$) region and a Minkowski region, and can be regarded as two-dimensional analogues of certain thick branes. By scanning the initial velocity for several gravitational couplings, we find that gravity modifies the scattering structure: the resonance windows shift toward higher initial velocities and become progressively narrower as the coupling $\kappa$ increases, while the critical escape velocity increases mildly. A linear perturbation analysis further indicates that the shape modes turn into long-lived quasi-bound states in the weak-gravity regime, which could leak energy during collisions and may therefore contribute to the shift and narrowing of the windows and the increase of the critical velocity. The collisions further produce a clear geometrical response: the conformal factor decreases after the collision, corresponding to a contraction of the local proper spatial scale in the conformal gauge, and this effect becomes stronger for larger $\kappa$. Meanwhile, the Ricci scalar develops transient peaks during kink encounters but remains finite in all simulations considered. Thus, in contrast to higher-dimensional thick-brane collisions, we find no evidence for spacetime singularity formation in this two-dimensional model.

Figures

Figures reproduced from arXiv: 2607.13620 by Yuan Zhong, Zhen-Tao He.

Figure 1
Figure 1. Figure 1: Constraint violations of the initial data constructed by superposing [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Snapshots of the Hamiltonian constraint H. The plot shows that the constraint violations induced by the approximate initial data and by the boundary conditions propagate approximately along null characteristics. We have also monitored the momentum constraint M and found that it exhibits the same qualitative behavior, with smaller violations in the parameter range considered here. The violations induced by … view at source ↗
Figure 3
Figure 3. Figure 3: ϕ(t,z = 0) for κ = 0.1, 0.3, 0.5 and v ∈ [0.002, 0.3] with step size ∆v = 0.002. As κ increases, the bounce windows become narrower and shift toward higher initial velocities, and they are difficult to resolve in the strong-coupling regime. The critical velocity also increases with κ, although only mildly. where Veff(z) = f ′′ f , f = ϕ ′ 2A′ . (24) With the mode ansatz δϕ(t,z) ∼ e iωtψ(z), this becomes th… view at source ↗
Figure 4
Figure 4. Figure 4: Linear stability potentials and the long-lived open-channel resonance. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of ϕ (left column), eA (middle column), and the Ricci scalar R (right column) for κ = 0.1 and representative initial velocities v = 0.1, 0.235, and 0.25 from top to bottom. The scalar-field profiles distinguish bion formation and multi-bounce escape, while eA and R show the corresponding geometrical response. The post-collision decrease of eA reflects a contraction of the local conformal spatial … view at source ↗
Figure 5
Figure 5. Figure 5: Proper time interval τn between two consecutive bounces as a func￾tion of the bounce window index n for κ = 0.1. The proper time is computed via τn = R t2 t1 e A(t,z=0) dt between the two collision times. Black dots show the extracted data for n = 1, · · · , 5, and the red dashed line is the linear fit to ωfitτn = δ + 2πn, yielding ωfit = 1.6261 (we set d = 1 here) and δ = 15.1225. To test whether the long… view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the scalar field ϕ (left column), the conformal factor eA (middle column), and the Ricci scalar R (right column) for representative collisions with intermediate and strong gravitational coupling. From top to bottom, the rows correspond to (κ, v) = (0.3, 0.235), (0.5, 0.235), (0.3, 0.5), and (0.5, 0.5). The scalar-field profiles distinguish multi-collision dynamics at v = 0.235 from single-boun… view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of the scalar field ϕ (left column), the conformal factor eA (middle column), and the Ricci scalar R (right column) for representative collisions with intermediate and strong gravitational coupling. From top to bottom, the rows correspond to (κ, v) = (0.3, 0.235), (0.5, 0.235), (0.3, 0.5), and (0.5, 0.5). The scalar-field profiles distinguish multi-collision dynamics at v = 0.235 from single-boun… view at source ↗

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Works this paper leans on

107 extracted references · 78 linked inside Pith

  1. [1]

    Vachaspati, Kinks and domain walls: an introduction to classical and quantum solitons, Cambridge University Press, 2006.doi:10.1017/9781009290456

    T. Vachaspati, Kinks and domain walls: an introduction to classical and quantum solitons, Cambridge University Press, 2006.doi:10.1017/9781009290456

    doi:10.1017/9781009290456 2006

  2. [2]

    del Campo, W

    A. del Campo, W. H. Zurek, Universality of phase transition dynamics: topological defects from sym- metry breaking, Int. J. Mod. Phys. A 29 (8) (2014) 1430018.arXiv:1310.1600,doi:10.1142/ S0217751X1430018X

    Pith/arXiv arXiv 2014

  3. [3]

    T. W. B. Kibble, Topology of cosmic domains and strings, J. Phys. A 9 (1976) 1387–1398.doi:10.1088/ 0305-4470/9/8/029

    1976

  4. [4]

    T. W. B. Kibble, Some implications of a cosmological phase transition, Phys. Rept. 67 (1980) 183.doi:10. 1016/0370-1573(80)90091-5

    1980

  5. [5]

    Vilenkin, Cosmic strings, Phys

    A. Vilenkin, Cosmic strings, Phys. Rev. D 24 (1981) 2082–2089.doi:10.1103/PhysRevD.24.2082

    doi:10.1103/physrevd.24.2082 1981

  6. [6]

    Randall, R

    L. Randall, R. Sundrum, A large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370–3373.arXiv:hep-ph/9905221,doi:10.1103/ PhysRevLett.83.3370

    Pith/arXiv arXiv 1999

  7. [7]

    Randall, R

    L. Randall, R. Sundrum, An alternative to compactifi- cation, Phys. Rev. Lett. 83 (1999) 4690–4693.arXiv: hep-th/9906064,doi:10.1103/PhysRevLett.83. 4690

    Pith/arXiv arXiv 1999

  8. [8]

    Skenderis, P

    K. Skenderis, P. K. Townsend, Gravitational stabil- ity and renormalization group flow, Phys. Lett. B 468 (1999) 46–51.arXiv:hep-th/9909070,doi:10. 1016/S0370-2693(99)01212-5

    Pith/arXiv arXiv 1999

  9. [9]

    DeWolfe, D

    O. DeWolfe, D. Z. Freedman, S. S. Gubser, A. Karch, Modeling the fifth-dimension with scalars and grav- ity, Phys. Rev. D 62 (2000) 046008.arXiv:hep-th/ 9909134,doi:10.1103/PhysRevD.62.046008

    doi:10.1103/physrevd.62.046008 2000

  10. [10]

    Gremm, Four-dimensional gravity on a thick domain wall, Phys

    M. Gremm, Four-dimensional gravity on a thick domain wall, Phys. Lett. B 478 (2000) 434–438.arXiv:hep-th/9912060,doi: 10.1016/S0370-2693(00)00303-8

    Pith/arXiv arXiv 2000

  11. [11]

    Dzhunushaliev, V

    V . Dzhunushaliev, V . Folomeev, M. Mina- mitsuji, Thick brane solutions, Rept. Prog. Phys. 73 (2010) 066901.arXiv:0904.1775, doi:10.1088/0034-4885/73/6/066901

    Pith/arXiv arXiv 2010

  12. [12]

    Liu, Introduction to extra dimensions and thick braneworlds, World Scientific, 2018, Ch

    Y .-X. Liu, Introduction to extra dimensions and thick braneworlds, World Scientific, 2018, Ch. 8, pp. 211–275. 7 Figure 7: Evolution of the scalar fieldϕ(left column), the conformal factor e A (middle column), and the Ricci scalarR(right column) for representative collisions with intermediate and strong gravitational coupling. From top to bottom, the row...

    Pith/arXiv arXiv 2018

  13. [13]

    Sugiyama, Kink-antikink collisions in the two- dimensionalφ 4 model, Prog

    T. Sugiyama, Kink-antikink collisions in the two- dimensionalφ 4 model, Prog. Theor. Phys. 61 (1979) 1550–1563.doi:10.1143/PTP.61.1550

    doi:10.1143/ptp.61.1550 1979

  14. [14]

    Moshir, Soliton-antisoliton scattering and capture in λφ4 theory, Nucl

    M. Moshir, Soliton-antisoliton scattering and capture in λφ4 theory, Nucl. Phys. B 185 (1981) 318–332.doi: 10.1016/0550-3213(81)90320-5

    doi:10.1016/0550-3213(81)90320-5 1981

  15. [15]

    D. K. Campbell, J. F. Schonfeld, C. A. Wingate, Reso- nance structure in kink-antikink interactions inφ4 theory, Physica D 9 (1983) 1.doi:10.1016/0167-2789(83) 90289-0

    doi:10.1016/0167-2789(83 1983

  16. [16]

    Anninos, S

    P. Anninos, S. Oliveira, R. A. Matzner, Fractal structure in the scalarλ(φ 2−1) 2 theory, Phys. Rev. D 44 (1991) 1147–1160.doi:10.1103/PhysRevD.44.1147

    doi:10.1103/physrevd.44.1147 1991

  17. [17]

    R. H. Goodman, R. Haberman, Chaotic scatter- ing and then-bounce resonance in solitary-wave interactions, Phys. Rev. Lett. 98 (2007) 104103. doi:10.1103/PhysRevLett.98.104103. URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.98.104103

    doi:10.1103/physrevlett.98.104103 2007

  18. [18]

    Dorey, A

    P. Dorey, A. Halavanau, J. Mercer, T. Roma´nczukiewicz, Y . Shnir, Boundary scattering in theϕ 4 model, JHEP 05 (2017) 107.arXiv:1508.02329,doi:10.1007/ JHEP05(2017)107

    Pith/arXiv arXiv 2017

  19. [19]

    D. K. Campbell, M. Peyrard, P. Sodano, Kink-antikink interactions in the double sine-Gordon equation, Phys- ica D 19 (2) (1986) 165–205.doi:10.1016/ 0167-2789(86)90019-9

    1986

  20. [20]

    V . A. Gani, A. E. Kudryavtsev, Kink-antikink inter- actions in the double sine-Gordon equation and the problem of resonance frequencies, Phys. Rev. E 60 (1999) 3305–3309.arXiv:cond-mat/9809015,doi: 10.1103/PhysRevE.60.3305

    Pith/arXiv arXiv 1999

  21. [21]

    V . A. Gani, A. Gorina, I. Perapechka, Y . Shnir, Re- marks on sine-Gordon kink-fermion system: local- ized modes and scattering, Eur. Phys. J. C 82 (8) (2022) 757.arXiv:2205.13437,doi:10.1140/ epjc/s10052-022-10707-0

    Pith/arXiv arXiv 2022

  22. [22]

    Dorey, A

    P. Dorey, A. Gorina, I. Perapechka, T. Ro- ma´nczukiewicz, Y . Shnir, Resonance structures in kink-antikink collisions in a deformed sine-Gordon model, JHEP 09 (2021) 145.arXiv:2106.09560, doi:10.1007/JHEP09(2021)145. 8

    Pith/arXiv arXiv 2021

  23. [23]

    Carretero-Gonzalez, L

    R. Carretero-Gonzalez, L. A. Cisneros-Ake, R. Decker, G. N. Koutsokostas, D. J. Frantzeskakis, P. G. Kevrekidis, D. J. Ratliff, Kink–antikink stripe in- teractions in the two-dimensional sine–Gordon equa- tion, Commun. Nonlinear Sci. Numer. Simul. 109 (2022) 106123.arXiv:2108.03121,doi:10.1016/ j.cnsns.2021.106123

    Pith/arXiv arXiv 2022

  24. [24]

    da Hora, L

    E. da Hora, L. Pereira, C. dos Santos, F. C. Simas, Geometrically constrained sine-Gordon field: BPS soli- tons and their collisions, Commun. Nonlinear Sci. Nu- mer. Simul. 151 (2025) 109070.arXiv:2409.09767, doi:10.1016/j.cnsns.2025.109070

    Pith/arXiv arXiv 2025

  25. [25]

    Hoseinmardy, N

    S. Hoseinmardy, N. Riazi, Inelastic collision of kinks and antikinks in theϕ 6 system, Int. J. Mod. Phys. A 25 (2010) 3261–3270.doi:10.1142/S0217751X10049712

    doi:10.1142/s0217751x10049712 2010

  26. [26]

    Belendryasova, V

    E. Belendryasova, V . A. Gani, Resonance phenomena in theφ 8 kink scattering, J. Phys. Conf. Ser. 934 (1) (2017) 012059.arXiv:1712.02846,doi:10.1088/ 1742-6596/934/1/012059

    Pith/arXiv arXiv 2017

  27. [27]

    V . A. Gani, A. M. Marjaneh, K. Javidan, Exotic final states in theφ 8 multi-kink collisions, Eur. Phys. J. C 81 (12) (2021) 1124.arXiv:2106.06399,doi:10. 1140/epjc/s10052-021-09935-7

    Pith/arXiv arXiv 2021

  28. [28]

    Khare, A

    A. Khare, A. Saxena, Kink solutions with power law tails, Front. Phys. 10 (2022) 992915.arXiv:2207. 10876,doi:10.3389/fphy.2022.992915

    arXiv 2022

  29. [29]

    Moradi Marjaneh, F

    A. Moradi Marjaneh, F. C. Simas, D. Bazeia, Collisions of kinks in deformedφ 4 andφ 6 models, Chaos Solitons Fractals 164 (2022) 112723.arXiv:2207.00835,doi: 10.1016/j.chaos.2022.112723

    Pith/arXiv arXiv 2022

  30. [30]

    Bazeia, J

    D. Bazeia, J. G. F. Campos, A. Mohammadi, Kink- antikink collisions in theϕ 8 model: short-range to long- range journey, JHEP 05 (2023) 116.arXiv:2303. 12482,doi:10.1007/JHEP05(2023)116

    doi:10.1007/jhep05(2023)116 2023

  31. [31]

    Belendryasova, V

    E. Belendryasova, V . A. Gani, K. G. Zloshchastiev, Kink solutions in logarithmic scalar field theory: excitation spectra, scattering, and decay of bions, Phys. Lett. B 823 (2021) 136776.arXiv:2111.09096,doi:10.1016/ j.physletb.2021.136776

    Pith/arXiv arXiv 2021

  32. [32]

    J. G. F. Campos, A. Mohammadi, Wobbling double sine- Gordon kinks, JHEP 09 (2021) 067.arXiv:2103. 04908,doi:10.1007/JHEP09(2021)067

    doi:10.1007/jhep09(2021)067 2021

  33. [33]

    Alonso Izquierdo, L

    A. Alonso Izquierdo, L. M. Nieto, J. Queiroga-Nunes, Scattering between wobbling kinks, Phys. Rev. D 103 (4) (2021) 045003.arXiv:2007.15517,doi:10.1103/ PhysRevD.103.045003

    Pith/arXiv arXiv 2021

  34. [34]

    Alonso-Izquierdo, L

    A. Alonso-Izquierdo, L. M. Nieto, J. Queiroga- Nunes, Asymmetric scattering between kinks and wob- blers, Commun. Nonlinear Sci. Numer. Simul. 107 (2022) 106183.arXiv:2109.13904,doi:10.1016/ j.cnsns.2021.106183

    Pith/arXiv arXiv 2022

  35. [35]

    Y . S. Kivshar, Z. Fei, L. Vázquez, Resonant soliton- impurity interactions, Phys. Rev. Lett. 67 (1991) 1177– 1180.doi:10.1103/PhysRevLett.67.1177

    doi:10.1103/physrevlett.67.1177 1991

  36. [36]

    Z. Fei, Y . S. Kivshar, L. Vázquez, Resonant kink-impurity interactions in the sine-gordon model, Phys. Rev. A 45 (1992) 6019–6030. doi:10.1103/PhysRevA.45.6019. URLhttps://link.aps.org/doi/10.1103/ PhysRevA.45.6019

    doi:10.1103/physreva.45.6019 1992

  37. [37]

    Y . Zhou, B. G.-g. Chen, N. Upadhyaya, V . Vitelli, Kink-antikink asymmetry and impurity interactions in topological mechanical chains, Phys. Rev. E 95 (2) (2017) 022202.arXiv:1608.02127,doi:10.1103/ PhysRevE.95.022202

    Pith/arXiv arXiv 2017

  38. [38]

    Lizunova, J

    M. Lizunova, J. Kager, S. de Lange, J. van Wezel, Emer- gence of oscillons in kink-impurity interactions, J. Phys. A 54 (2021) 315701.arXiv:2012.07281,doi:10. 1088/1751-8121/ac0d36

    Pith/arXiv arXiv 2021

  39. [39]

    Zhong, X.-L

    Y . Zhong, X.-L. Du, Z.-C. Jiang, Y .-X. Liu, Y .-Q. Wang, Collision of two kinks with inner structure, JHEP 02 (2020) 153.arXiv:1906.02920,doi:10.1007/ JHEP02(2020)153

    Pith/arXiv arXiv 2020

  40. [40]

    H. Yan, Y . Zhong, Y .-X. Liu, K.-i. Maeda, Kink-antikink collision in a Lorentz-violatingϕ 4 model, Phys. Lett. B 807 (2020) 135542.arXiv:2004.13329,doi:10. 1016/j.physletb.2020.135542

    Pith/arXiv arXiv 2020

  41. [41]

    C. E. S. Santos, J. G. F. Campos, A. Mohammadi, On the localized and delocalized modes in kink-antikink in- teractions: a toy model, JHEP 01 (2025) 035.arXiv: 2408.00945,doi:10.1007/JHEP01(2025)035

    Pith/arXiv arXiv 2025

  42. [42]

    Dorey, K

    P. Dorey, K. Mersh, T. Roma ´nczukiewicz, Y . Shnir, Kink-antikink collisions in theϕ 6 model, Phys. Rev. Lett. 107 (2011) 091602.arXiv:1101.5951,doi: 10.1103/PhysRevLett.107.091602

    Pith/arXiv arXiv 2011

  43. [43]

    V . A. Gani, A. E. Kudryavtsev, M. A. Lizunova, Kink interactions in the (1+1)-dimensionalϕ 6 model, Phys. Rev. D 89 (12) (2014) 125009.arXiv:1402.5903, doi:10.1103/PhysRevD.89.125009

    Pith/arXiv arXiv 2014

  44. [44]

    Yan, Kink scattering in a Lorentz-violatingϕ 6 model, EPL 138 (2022) 14001.arXiv:2110.13381,doi:10

    H. Yan, Kink scattering in a Lorentz-violatingϕ 6 model, EPL 138 (2022) 14001.arXiv:2110.13381,doi:10. 1209/0295-5075/ac5b9b

    Pith/arXiv arXiv 2022

  45. [45]

    C. Adam, P. Dorey, A. García Martín-Caro, M. Huidobro, K. Ole ´s, T. Roma ´nczukiewicz, Y . Shnir, A. Wereszczy ´nski, Multikink scatter- ing in theϕ 6 model revisited, Phys. Rev. D 106 (12) (2022) 125003.arXiv:2209.08849, doi:10.1103/PhysRevD.106.125003

    Pith/arXiv arXiv 2022

  46. [46]

    Dorey, T

    P. Dorey, T. Roma ´nczukiewicz, Resonant kink-antikink scattering through quasinormal modes, Phys. Lett. B 779 (2018) 117–123.arXiv:1712.10235,doi:10.1016/ j.physletb.2018.02.003

    Pith/arXiv arXiv 2018

  47. [47]

    J. G. F. Campos, A. Mohammadi, Quasinormal modes in kink excitations and kink–antikink interactions: a toy model, Eur. Phys. J. C 80 (5) (2020) 352.arXiv:1905. 00835,doi:10.1140/epjc/s10052-020-7856-3

    doi:10.1140/epjc/s10052-020-7856-3 2020

  48. [48]

    J. G. F. Campos, A. Mohammadi, Kink-antikink col- lision in the supersymmetricϕ 4 model, JHEP 08 (2022) 180.arXiv:2205.06869,doi:10.1007/ JHEP08(2022)180

    Pith/arXiv arXiv 2022

  49. [49]

    Weigel, Kink-antikink scattering inφ 4 andϕ 6 models, J

    H. Weigel, Kink-antikink scattering inφ 4 andϕ 6 models, J. Phys. Conf. Ser. 482 (2014) 012045.arXiv:1309. 6607,doi:10.1088/1742-6596/482/1/012045

    doi:10.1088/1742-6596/482/1/012045 2014

  50. [50]

    Takyi, H

    I. Takyi, H. Weigel, Collective coordinates in one- dimensional soliton models revisited, Phys. Rev. D 94 (8) (2016) 085008.arXiv:1609.06833,doi:10. 1103/PhysRevD.94.085008

    Pith/arXiv arXiv 2016

  51. [51]

    Weigel, Collective coordinate methods and their ap- plicability toφ 4 models, Springer International Publish- ing, Cham, 2019, Ch

    H. Weigel, Collective coordinate methods and their ap- plicability toφ 4 models, Springer International Publish- ing, Cham, 2019, Ch. 3, pp. 51–74.arXiv:1809. 9 03772,doi:10.1007/978-3-030-11839-6\_3

    doi:10.1007/978-3-030-11839-6 2019

  52. [52]

    N. S. Manton, K. Ole ´s, T. Roma ´nczukiewicz, A. Wereszczy ´nski, Collective coordinate model of kink-antikink collisions inϕ 4 theory, Phys. Rev. Lett. 127 (7) (2021) 071601.arXiv:2106.05153, doi:10.1103/PhysRevLett.127.071601

    Pith/arXiv arXiv 2021

  53. [53]

    N. S. Manton, K. Ole ´s, T. Roma ´nczukiewicz, A. Wereszczy ´nski, Kink moduli spaces: col- lective coordinates reconsidered, Phys. Rev. D 103 (2) (2021) 025024.arXiv:2008.01026, doi:10.1103/PhysRevD.103.025024

    Pith/arXiv arXiv 2021

  54. [54]

    C. Adam, N. S. Manton, K. Ole ´s, T. Roma´nczukiewicz, A. Wereszczy ´nski, Relativistic moduli space for kink collisions, Phys. Rev. D 105 (6) (2022) 065012.arXiv: 2111.06790,doi:10.1103/PhysRevD.105.065012

    Pith/arXiv arXiv 2022

  55. [55]

    C. F. S. Pereira, G. Luchini, T. Tassis, C. P. Constan- tinidis, Some novel considerations about the collective coordinates approximation for the scattering ofϕ 4 kinks, J. Phys. A: Math. Theor. 54 (7) (2021) 075701.arXiv: 2004.00571,doi:10.1088/1751-8121/abd815

    Pith/arXiv arXiv 2021

  56. [56]

    C. F. S. Pereira, E. dos Santos Costa Filho, T. Tassis, Collective coordinates for the hybrid model, Int. J. Mod. Phys. A 38 (1) (2023) 2350006.arXiv:2110.05658, doi:10.1142/S0217751X23500069

    Pith/arXiv arXiv 2023

  57. [57]

    R. H. Goodman, R. Haberman, Kink-antikink collisions in theϕ 4 equation: then-bounce resonance and the sep- aratrix map, SIAM J. Appl. Dynam. Syst. 4 (4) (2005) 1195–1228.doi:10.1137/050632981

    doi:10.1137/050632981 2005

  58. [58]

    S. W. Hawking, I. G. Moss, J. M. Stewart, Bubble colli- sions in the very early universe, Phys. Rev. D 26 (1982) 2681–2693.doi:10.1103/PhysRevD.26.2681. URLhttps://link.aps.org/doi/10.1103/ PhysRevD.26.2681

    doi:10.1103/physrevd.26.2681 1982

  59. [59]

    J. J. Blanco-Pillado, M. Bucher, S. Ghassemi, F. Glanois, When do colliding bubbles produce an expanding universe?, Phys. Rev. D 69 (2004) 103515. doi:10.1103/PhysRevD.69.103515. URLhttps://link.aps.org/doi/10.1103/ PhysRevD.69.103515

    doi:10.1103/physrevd.69.103515 2004

  60. [60]

    Freivogel, G

    B. Freivogel, G. T. Horowitz, S. Shenker, Colliding with a crunching bubble, JHEP 05 (2007) 090.arXiv: hep-th/0703146,doi:10.1088/1126-6708/2007/ 05/090

    Pith/arXiv arXiv 2007

  61. [61]

    Easther, J

    R. Easther, J. T. Giblin, Jr, L. Hui, E. A. Lim, A new mechanism for bubble nucleation: classical transitions, Phys. Rev. D 80 (2009) 123519.arXiv:0907.3234, doi:10.1103/PhysRevD.80.123519

    Pith/arXiv arXiv 2009

  62. [62]

    J. T. Giblin, Jr, L. Hui, E. A. Lim, I.-S. Yang, How to run through walls: dynamics of bubble and soliton collisions, Phys. Rev. D 82 (2010) 045019.arXiv:1005.3493, doi:10.1103/PhysRevD.82.045019

    Pith/arXiv arXiv 2010

  63. [63]

    M. C. Johnson, H. V . Peiris, L. Lehner, Determining the outcome of cosmic bubble collisions in full gen- eral relativity, Phys. Rev. D 85 (2012) 083516.arXiv: 1112.4487,doi:10.1103/PhysRevD.85.083516

    Pith/arXiv arXiv 2012

  64. [64]

    Hwang, B.-H

    D.-i. Hwang, B.-H. Lee, W. Lee, D.-h. Yeom, Bubble collision with gravitation, JCAP 07 (2012) 003.arXiv:1201.6109,doi:10.1088/1475-7516/ 2012/07/003

    Pith/arXiv arXiv 2012

  65. [65]

    J. R. Bond, J. Braden, L. Mersini-Houghton, Cosmic bubble and domain wall instabilities III: the role of oscillons in three-dimensional bubble collisions, JCAP 09 (2015) 004.arXiv:1505.02162,doi:10.1088/ 1475-7516/2015/09/004

    Pith/arXiv arXiv 2015

  66. [66]

    J. C. Aurrekoetxea, K. Clough, E. A. Lim, Cos- mology using numerical relativity, Living Rev. Rel. 28 (1) (2025) 5.arXiv:2409.01939,doi:10.1007/ s41114-025-00058-z

    Pith/arXiv arXiv 2025

  67. [67]

    Takamizu, K.-i

    Y .-i. Takamizu, K.-i. Maeda, Collision of domain walls in asymptotically anti de Sitter spacetime, Phys. Rev. D 73 (2006) 103508.arXiv:hep-th/0603076,doi:10. 1103/PhysRevD.73.103508

    Pith/arXiv arXiv 2006

  68. [68]

    Takamizu, H

    Y .-i. Takamizu, H. Kudoh, K.-i. Maeda, Dynamics of colliding branes and black brane production, Phys. Rev. D 75 (2007) 061304.arXiv:gr-qc/0702138,doi: 10.1103/PhysRevD.75.061304

    Pith/arXiv arXiv 2007

  69. [69]

    Omotani, P

    J. Omotani, P. M. Saffin, J. Louko, Colliding branes and big crunches, Phys. Rev. D 84 (2011) 063526.arXiv: 1107.3938,doi:10.1103/PhysRevD.84.063526

    Pith/arXiv arXiv 2011

  70. [70]

    Maeda, K

    K.-i. Maeda, K. Uzawa, Dynamical brane with an- gles: collision of the universes, Phys. Rev. D 85 (2012) 086004.arXiv:1201.3213,doi:10.1103/ PhysRevD.85.086004

    Pith/arXiv arXiv 2012

  71. [71]

    Tziolas, A

    A. Tziolas, A. Wang, Colliding branes and formation of spacetime singularities, Phys. Lett. B 661 (2008) 5– 10.arXiv:0704.1311,doi:10.1016/j.physletb. 2008.01.058

    Pith/arXiv arXiv 2008

  72. [72]

    Tziolas, A

    A. Tziolas, A. Wang, Z. C. Wu, Colliding branes and for- mation of spacetime singularities in string theory, JHEP 04 (2009) 038.arXiv:0812.1377,doi:10.1088/ 1126-6708/2009/04/038

    Pith/arXiv arXiv 2009

  73. [73]

    C. L. Wainwright, M. C. Johnson, A. Aguirre, H. V . Peiris, Simulating the universe(s) II: phenomenology of cosmic bubble collisions in full general relativity, JCAP 10 (2014) 024.arXiv:1407.2950,doi:10.1088/ 1475-7516/2014/10/024

    Pith/arXiv arXiv 2014

  74. [74]

    C. L. Wainwright, M. C. Johnson, H. V . Peiris, A. Aguirre, L. Lehner, S. L. Liebling, Simulating the uni- verse(s): from cosmic bubble collisions to cosmological observables with numerical relativity, JCAP 03 (2014) 030.arXiv:1312.1357,doi:10.1088/1475-7516/ 2014/03/030

    Pith/arXiv arXiv 2014

  75. [75]

    M. C. Johnson, C. L. Wainwright, A. Aguirre, H. V . Peiris, Simulating the universe(s) III: observables for the full bubble collision spacetime, JCAP 07 (2016) 020.arXiv:1508.03641,doi:10.1088/1475-7516/ 2016/07/020

    Pith/arXiv arXiv 2016

  76. [76]

    M. C. Johnson, W. Lin, Observable signatures of a clas- sical transition, JCAP 03 (2016) 051.arXiv:1508. 03786,doi:10.1088/1475-7516/2016/03/051

    doi:10.1088/1475-7516/2016/03/051 2016

  77. [77]

    Kim, B.-H

    D.-H. Kim, B.-H. Lee, W. Lee, J. Yang, D.-h. Yeom, Gravitational waves from cosmic bubble collisions, Eur. Phys. J. C 75 (3) (2015) 133.arXiv:1410.4648,doi: 10.1140/epjc/s10052-015-3348-2

    Pith/arXiv arXiv 2015

  78. [78]

    Henneaux, Quantum gravity in two dimensions: exact solution of the Jackiw model, Phys

    M. Henneaux, Quantum gravity in two dimensions: exact solution of the Jackiw model, Phys. Rev. Lett. 54 (1985) 959–962.doi:10.1103/PhysRevLett.54. 959

    doi:10.1103/physrevlett.54 1985

  79. [79]

    S. P. de Alwis, Quantization of a theory of 2-d dilaton gravity, Phys. Lett. B 289 (1992) 10 278–282.arXiv:hep-th/9205069,doi: 10.1016/0370-2693(92)91219-Y

    Pith/arXiv arXiv 1992

  80. [80]

    C. Vaz, L. Witten, Formation and evaporation of a naked singularity in 2-d gravity, Phys. Lett. B 325 (1994) 27–32.arXiv:hep-th/9311133,doi:10. 1016/0370-2693(94)90066-3

    Pith/arXiv arXiv 1994

Showing first 80 references.