Pith. sign in

REVIEW 3 major objections 5 minor 103 references

In two-dimensional dilaton gravity, φ⁴ kink-antikink collisions shift and narrow bounce windows as gravity strengthens, leave a lasting contraction of local spatial scale, and produce no curvature singularity.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 04:38 UTC pith:LCJ6MGG3

load-bearing objection First numerical map of gravitating kink collisions in 2D dilaton gravity; plausible and honest, but the window-shift claim needs a flat-space benchmark with the same κ-dependent potential to be fully convincing. the 3 major comments →

arxiv 2607.13620 v1 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 gravityφ⁴ modelresonance windowsopen-channel resonanceconformal factorspacetime singularities
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 studies what happens when the classic φ⁴ kink-antikink system is made self-gravitating, using a two-dimensional dilaton gravity in which static kinks resemble thick branes. By scanning initial velocities for couplings κ=0.1, 0.3, and 0.5, it finds that gravity is not a passive backdrop: increasing κ shifts the resonance (multi-bounce) windows toward higher velocities and makes them narrower, while the critical escape velocity rises only mildly. It also identifies a long-lived open-channel resonance at weak coupling (Q≈1.2×10³) as the likely heir to the flat-spacetime shape-mode mechanism, since the gravitating kink has no normalizable discrete mode. The collisions permanently lower the conformal factor—contracting the local proper spatial scale—with a stronger effect at larger κ. The Ricci scalar stays finite throughout, so the paper concludes that no spacetime singularity forms in this model, in contrast to higher-dimensional thick-brane collisions.

Core claim

The paper's central claim is that coupling a φ⁴ kink-antikink pair to two-dimensional dilaton gravity changes the scattering problem in three systematic ways. As the gravitational coupling κ increases from 0.1 to 0.5, resonance (bounce) windows shift to higher initial velocities and become narrower, while the critical escape velocity rises mildly; a long-lived open-channel resonance with frequency dω_res≈1.70583+7.05×10⁻⁴ i (Q≈1.2×10³) appears at κ=0.1 and supplies a plausible energy-storage mechanism in place of the flat-space shape mode; and the collision permanently lowers the conformal factor, contracting the local proper spatial scale, with a stronger effect at larger κ. Throughout all

What carries the argument

The conformal gauge ds²=e^{2A}(−dt²+dz²) plus the algebraic reduction φ=2A, which ties the dilaton to the warp factor and yields a two-equation system (for A and φ) with momentum and Hamiltonian constraints. For linear stability, the operator factorizes as Ĥ=† with potential V_eff=f''/f, f=φ'/(2A'); since V_eff→0 at the AdS end, there is no normalizable discrete mode, but a quasi-bound open-channel resonance survives at weak coupling—this complex-frequency mode is the proposed carrier of the bounce-window physics. Curvature is diagnosed directly through R=(κV)², so monitoring the Ricci scalar suffices for singularity searches.

Load-bearing premise

The load-bearing simplification is fixing the dilaton to twice the conformal factor (φ=2A); the paper admits the dilaton equation has other solutions, and if those modes are excited in collisions the window shifts, the permanent contraction, and the no-singularity conclusion could change.

What would settle it

Rerun the κ=0.5, v=0.235 collision (or any high-κ case) with the dilaton evolved independently instead of fixed to φ=2A; if the Ricci scalar then diverges, or if the resonance windows move outside the reported ranges by more than numerical error, the paper's central claims fail.

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

If this is right

  • With κ=0.1, 0.3, and 0.5, the first two-bounce windows sit at higher velocities than in flat spacetime and grow narrower as κ increases; at κ=0.5 they are hard to resolve with the scan step used.
  • The self-gravitating kink has no normalizable discrete vibrational mode, yet a long-lived open-channel resonance at dω_res≈1.70583+7.05×10⁻⁴ i (Q≈1.2×10³) exists for κ=0.1, giving a plausible energy-storage channel for bounce windows.
  • Every collision leaves a permanent imprint on the geometry: the conformal factor e^A decreases after the encounter, i.e., the local proper spatial scale contracts, and the decrease is stronger for larger κ.
  • The Ricci scalar, related to the potential by R=(κV)², develops transient peaks during encounters but remains finite in all runs; the paper concludes there is no evidence of spacetime singularity formation, in contrast to five-dimensional thick-brane collisions.
  • The identified resonance motivates quantum studies of self-gravitating kinks, where its lifetime and scattering signatures could be analysed.

Where Pith is reading between the lines

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

  • Editorial inference: The no-singularity result likely depends on the φ=2A truncation; if the full dilaton dynamics were evolved, extra channels could drain energy and prevent singularities, or conversely focus energy and create one—a testable extension of the paper's numerics.
  • Editorial inference: The lasting decrease of e^A is a form of gravitational memory in the conformal frame; in two dimensions the energy lost to the AdS boundary may be what keeps curvature finite, which would explain the contrast with five-dimensional thick-brane collisions.
  • Editorial inference: The resonance Q≈1.2×10³ at κ=0.1 suggests a sharp threshold: if Q drops quickly with κ, the window width should shrink correspondingly; a systematic scan of window width versus κ could test the resonance-mediated mechanism without needing full flux extraction.

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 numerically kink-antikink collisions in a two-dimensional dilaton-gravity model (MMSS gravity) with a self-gravitating φ^4 scalar field. The authors work in a conformal gauge, impose the special relation φ(t,z)=2A(t,z) for the dilaton, and evolve the resulting coupled scalar-metric equations with a finite-difference/ode78 scheme. Their main claims are: (i) as the gravitational coupling κ increases, the resonance (multi-bounce) windows shift to higher initial velocities and become narrower, while the critical escape velocity increases mildly; (ii) a linear perturbation analysis reveals a long-lived open-channel resonance for κ=0.1 with quality factor Q≈1.2×10^3; and (iii) collisions produce a lasting decrease of the conformal factor e^A (interpreted as contraction of the local conformal spatial scale) but no curvature singularity is observed in any simulation. The paper compares the window structure only with literature values for flat-space φ^4, and acknowledges that the resonance mechanism is not proven.

Significance. If the numerical claims are robust, this is a valuable first exploration of soliton collisions in a two-dimensional dilaton-gravity setting, with potential analogies to higher-dimensional thick-brane collisions. The absence of singularity formation, if confirmed, contrasts with the 5D thick-brane results of Takamizu–Maeda, and the open-channel resonance with Q≈1200 is an interesting and falsifiable prediction. The manuscript includes several good practices: the resonance eigenvalue is checked against five cutoff values; constraint violations are monitored throughout evolution; the paper explicitly identifies the truncation φ=2A and the finite-time limitation. However, the central quantitative claims rest on a numerical code that is not validated with a flat-space benchmark, and the observables (window positions, critical velocities) are not given with uncertainties. These gaps currently prevent the paper from fully supporting its headline conclusions.

major comments (3)
  1. [Sec. 3.1, Eq. (5)] The central claim that 'gravity shifts the resonance windows toward higher velocities and narrows them' is not isolated from a pure matter-potential effect. The scalar potential (5) explicitly contains κ through the term −(κ/8d^2)(φ−φ^3/3−2/3)^2, so increasing κ simultaneously changes the self-interaction potential. All comparisons in Sec. 3.1 are to the standard flat φ^4 windows from Ref. [15]; there is no κ=0 simulation with the same code, the same initial-data construction, and the same boundary conditions. Without this control, the observed shifts could be partly or entirely due to the κ-dependent potential deformation. I ask for a κ=0 flat-space run of the same code (setting A=0 in the scalar equation, or taking κ→0 in the full equations) and, ideally, runs with the same potential (5) at fixed κ but no metric backreaction. This is needed to attribute the effect to gravity.
  2. [Sec. 3.1, Fig. 3 and §2.3] The quantitative claims—window positions, window widths, critical velocity—are not quantified with any uncertainty, and the velocity scan has step Δv=0.002. For κ=0.5 the text states that the windows are 'difficult to resolve clearly', yet the conclusion that they 'become progressively narrower' is drawn from this same figure. Moreover, no convergence study of the scattering outcome with grid spacing or time tolerance is presented; the initial-data convergence of the momentum constraint (Sec. 2.3) and the constraint-violation snapshots (Fig. 2) are useful diagnostics but do not establish that a particular velocity lies inside or outside a window. Please provide window boundaries with estimated uncertainties (e.g., by bisection or by runs at multiple resolutions) and a convergence test for a representative set of (κ,v) outcomes.
  3. [Sec. 3.3, Sec. 2.2] The no-singularity statement is conditional on the truncation φ=2A and on finite-time, finite-resolution simulations. While the text does use the careful phrase 'no evidence' and notes that the result 'warrants further investigation', the abstract and conclusions could be read as a stronger statement. The omitted dilaton modes ψ=φ−2A obey a linear wave equation (13) and are not sourced if initially zero, so the truncation is dynamically consistent; nevertheless, this is a sector condition that should be stated explicitly wherever the no-singularity claim is summarized. I recommend adding one sentence in Sec. 4 making explicit that the result applies to the φ=2A sector and to the finite time interval simulated.
minor comments (5)
  1. [Sec. 2.4] The boundary condition A'(±L)=∓kγ e^{A(±L)} and the asymptotic form (22) rely on the assumption that the kink profile has reached its asymptotic form at the boundaries. It would be useful to state the values of L and z0 used in the actual production runs (the text only says L/d≫1 and z0=5), and to comment on how the results depend on L.
  2. [Sec. 3.2] The resonance eigenvalue computation is described only schematically ('we solve this boundary-value problem'). The five-cutoff stability test is a good check, but the numerical method (shooting, finite-difference, spectral?) and the accuracy of the reported digits in dω_res≃1.70583+7.05×10^{-4}i should be specified. A relative error estimate would strengthen the claim.
  3. [Sec. 3.3] The statement 'in the conformal gauge, the proper distance is dl=e^A dz' and the interpretation of the decrease of e^A as 'contraction of the local conformal spatial scale' is correct but gauge-dependent. The paper already acknowledges this (line after Eq. (31)), but a brief caveat in Sec. 4 would avoid overstatement.
  4. [Throughout] There are several typographical and formatting issues: 'MA TLABsolverode78' in Sec. 2.2; some references in the bibliography use inconsistent author accents (e.g., 'Roma´nczukiewicz'); the abstract line 'self-gravitatingϕ 4 model' is missing a space. These are cosmetic and do not affect the science.
  5. [Acknowledgments] The acknowledgment of a 'Xiaomi MiMo Orbit 100T Token Grant' and the statement that part of the work was a bachelor's thesis are unusual but not problematic; however, the authors should ensure that the funding body is correctly identified and that no conflict of interest arises from the computational grant.

Circularity Check

0 steps flagged

No significant circularity: the central claims are direct PDE and eigenvalue-problem outputs, with self-citations serving as model inputs rather than predictions.

full rationale

The paper's new results are numerical scans and a linear-stability computation, none of which is fitted to its own conclusions. The bounce-window shifts (Sec. 3.1, Fig. 3) are obtained by integrating the coupled PDEs (15)-(16) over a v scan with delta_v=0.002; no parameter is tuned to produce 'windows shift with kappa'. The resonance frequency (29) is the solution of the Schrodinger-type boundary-value problem (25)-(28) with outgoing/evanescent BCs and is converged over five cutoffs; it is therefore a computed output, not an input. The paper explicitly disclaims the causal link to bounces ('Establishing that it is directly responsible for the observed bounce windows would, however, require projecting the nonlinear collision dynamics onto the resonant profile, which we leave for future work'), so it does not dress the eigenvalue as a prediction of the windows. The no-singularity claim is explicitly a finite-time numerical observation ('no evidence ... in all simulations considered'), not a theorem derived from an ansatz. The phi=2A truncation (14) is stated to be a special case of a more general dilaton equation, which makes the scope conditional but not circular. The static solution and stability factorization taken from the authors' Ref. [90] are parameter-free analytic inputs that define the model; they are not the target results and are independently checkable. The lack of a kappa=0 code benchmark and the kappa-dependence of the matter potential (5) are validation/attribution concerns, not instances where a prediction reduces by construction to a fit or to a self-citation.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

No parameters are fitted to the collision output: the central results are direct numerical scans. However, the φ=2A truncation and approximate initial data are ad hoc restrictions; they carry most of the validity burden. No new particles or forces are postulated.

free parameters (4)
  • d (kink thickness) = 1 (normalized)
    All lengths are normalized by d. Results are reported in units of d, so d is a scale choice, not a fit.
  • κ (gravitational coupling) = 0.1, 0.3, 0.5
    Chosen scan values of the model's gravitational coupling; the paper's main result is how collision outcomes depend on κ. Not fitted to data.
  • v (initial velocity) = 0.002–0.3, step 0.002
    Scan grid over initial velocity used to locate resonance windows and critical velocity; the finite step limits the precision of window boundaries.
  • z0 (initial half-separation) = 5d
    Initial kink-antikink separation chosen because larger separations did not reduce constraint spikes; dependence of the window structure on z0 is not systematically reported.
axioms (6)
  • standard math The action (1) and the derived field equations (15)-(18) correctly describe the MMSS dilaton-gravity model.
    All dynamics follow from this action; standard variational principle.
  • domain assumption The conformal gauge ds^2=e^{2A}(−dt^2+dz^2) covers the relevant spacetime without coordinate singularities during the collision.
    Gauge choice in Sec. 2.2; if the gauge degenerates, the interpretation of coordinate velocities and contraction could be affected.
  • ad hoc to paper The consistent truncation φ=2A (Eq. (14)) captures the gravitational degrees of freedom relevant to kink-antikink collisions.
    The dilaton equation has more general solutions; this is imposed 'for simplicity' and is not justified as the only physical sector. Central claims may not extend to the full theory.
  • domain assumption The approximate superposed boosted-solution initial data (Eqs. (20)-(21)) with constraint violations ≤10^-6 (and ≤10^-3 during evolution in the region of interest) are accurate enough to determine resonance windows and curvature behavior.
    They do not solve the constraint equations; violations are monitored but not eliminated. The no-singularity result is only as good as this numerical approximation.
  • domain assumption The boundary conditions (Sec. 2.4) and the restriction |z|<L−t keep boundary-induced constraint violations from contaminating reported results.
    They claim violations in the causal region are ~10^-7 away from initial spikes; no independent convergence study of scattering outcomes with L or N is given.
  • domain assumption The linear stability analysis of Ref. [90] (factorization of Ĥ, absence of normalizable discrete mode, Veff asymptotics) is correct and the open-channel resonance computation is a valid linear diagnostic.
    Used to interpret bounce windows; the leap from resonance to nonlinear bounce windows is explicitly left unproven.

pith-pipeline@v1.3.0-alltime-deepseek · 15657 in / 21360 out tokens · 187235 ms · 2026-08-02T04:38:41.758623+00:00 · methodology

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 possible long-lived resonant excitations in the weak-gravity regime, which may be associated with the persistence of bounce windows. 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 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 ↗

discussion (0)

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

Reference graph

Works this paper leans on

103 extracted references · 10 canonical work pages

  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- 7 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. arXiv:1707.08541,doi:10.1142/9789813237278_ 0008

    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

    work page 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

    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

    work page 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

    work page 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 8 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

    work page 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. 03772,doi:10.1007/978-3-030-11839-6\_3

    work page 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]

    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

  56. [56]

    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

  57. [57]

    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

    work page doi:10.1103/physrevd.69.103515 2004

  58. [58]

    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

  59. [59]

    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

  60. [60]

    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

  61. [61]

    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

  62. [62]

    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

  63. [63]

    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. 9

    Pith/arXiv arXiv 2015

  64. [64]

    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

  65. [65]

    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

  66. [66]

    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

  67. [67]

    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

  68. [68]

    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

  69. [69]

    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

  70. [70]

    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

  71. [71]

    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

  72. [72]

    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

  73. [73]

    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

  74. [74]

    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

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

  75. [75]

    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

  76. [76]

    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

    work page doi:10.1103/physrevlett.54 1985

  77. [77]

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

    Pith/arXiv arXiv 1992

  78. [78]

    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

  79. [79]

    C. G. Callan, Jr., S. B. Giddings, J. A. Harvey, A. Stro- minger, Evanescent black holes, Phys. Rev. D 45 (4) (1992) R1005.arXiv:hep-th/9111056,doi:10. 1103/PhysRevD.45.R1005

    Pith/arXiv arXiv 1992

  80. [80]

    Bilal, C

    A. Bilal, C. G. Callan, Jr., Liouville models of black hole evaporation, Nucl. Phys. B 394 (1993) 73–100.arXiv:hep-th/9205089,doi:10.1016/ 0550-3213(93)90102-U

    Pith/arXiv arXiv 1993

Showing first 80 references.