pith. sign in

arxiv: 2508.05812 · v2 · pith:6AVZ5ARCnew · submitted 2025-08-07 · 🌊 nlin.PS

Nonlinear Dynamics of Kink Configurations: From Small to Large Kink Collisions

Aliakbar Moradi Marjaneh , Dionisio Bazeia This is my paper

Pith reviewed 2026-05-21 22:42 UTC · model grok-4.3

classification 🌊 nlin.PS
keywords kink collisionsnonlinear wave equationvibrational modesresonance windowstopological defectsparameterized potentialsmall and large kinks
0
0 comments X

The pith

The critical velocity for small kink separation varies non-monotonically with the potential parameter λ.

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

This paper studies collisions of small and large kinks in a nonlinear wave system whose potential is tuned by a continuous parameter λ. For small kinks the minimum speed required for the pair to fly apart after a head-on collision rises and falls as λ changes, linked to the stability of the kinks and the presence of internal vibrational modes. These modes produce resonance windows and temporary bound states at lower λ, but their frequencies drop as λ increases so the scattering becomes simpler. Small-kink collisions create large kinks more often when λ is small, because the mass gap between the two kink types shrinks. Large kinks, which have no vibrational modes, scatter into growing numbers of small-kink pairs as λ and impact speed increase through direct conversion of translational energy into potential energy.

Core claim

In the parameterized model the small-kink solutions support vibrational modes that generate resonance windows and bion formation, whereas the large-kink solutions support none. The critical velocity separating small-kink pairs therefore depends non-monotonically on λ. Small-kink collisions produce large kinks preferentially at lower λ where the mass difference is reduced. Large-kink collisions in turn produce small-kink pairs, with the number of pairs rising with both λ and initial velocity because translational kinetic energy is transferred into the potential energy needed to create the smaller defects.

What carries the argument

The one-parameter family of potentials U_λ(χ) that interpolates between distinct small-kink and large-kink solutions and controls both their mass difference and the spectrum of small-kink vibrational modes.

If this is right

  • Resonance windows and bion formation disappear at higher λ because vibrational frequencies fall.
  • Large kinks emerge from small-kink collisions most readily at low λ where the mass difference is smallest.
  • Large-kink scattering yields more small-kink pairs as both λ and collision speed increase through energy transfer from translational motion.
  • Large kinks exhibit simpler scattering dynamics than small kinks because they lack internal modes.

Where Pith is reading between the lines

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

  • Tuning λ could serve as a practical control knob for selecting which kink species is produced in nonlinear media.
  • The same mass-difference and energy-transfer mechanism may govern defect creation in other field theories that admit multiple topological solutions.
  • The non-monotonic critical-velocity curve implies the existence of discrete λ values at which small-kink stability changes abruptly.

Load-bearing premise

The numerical integration scheme and boundary conditions faithfully reproduce the continuum dynamics of the nonlinear wave equation without introducing artifacts that alter the observed resonance windows or kink-production statistics.

What would settle it

An independent simulation or analytic calculation that finds the critical velocity for small-kink separation to be a strictly monotonic function of λ would falsify the reported non-monotonic dependence.

Figures

Figures reproduced from arXiv: 2508.05812 by Aliakbar Moradi Marjaneh, Dionisio Bazeia.

Figure 1
Figure 1. Figure 1: (a) The deformed potential Uλ(χ) as defined in Eq. (20) for varying λ. (b) Spatial profiles of large kinks, χL,λ(x), from Eq. (22) for different λ values. (c) Spatial profiles of small kinks, χS,λ(x), from Eq. (24) for different λ values. (d) Masses of small (m) and large (M) kinks as functions of λ, calculated using Eqs. (26) and (27), respectively. in Figs. 2(a) and 2(b) for small and large kinks, re￾spe… view at source ↗
Figure 2
Figure 2. Figure 2: (a) The stability potential vS,λ(x) for small kinks, as defined in Eq. (29), for λ = 0, 0.25, 0.50, 0.75, 0.90. With increasing λ, the potential wells become shallower and wider. (b) The stability potential vL,λ(x) for large kinks, as defined in Eq. (30), for the same λ values. For λ from 0 to 0.54, the wells deepen and narrow, while for λ > 0.54, they become shallower and wider. numerically were derived f… view at source ↗
Figure 3
Figure 3. Figure 3: Critical velocity, vibrational mode, and scattering outcomes for small kinks. (a) The critical velocity [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Small kinks scattering for λ = 0.07 and different values of initial velocities, with initial positions X0k = −X0ak = −10 for all cases. (a) At vi = 0.1, the kink and antikink form a bion. (b) At vi = 0.2, the kink and antikink separate after collision. (c) At vi = 0.5, separation occurs with a higher final velocity. (d) At vi = 0.6, large kinks are created. only observed at low λ values (e.g., λ = 0.07), w… view at source ↗
Figure 5
Figure 5. Figure 5: Small kink-antikink collision, (a) and (b) correspond to simulations with parameter [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Large kinks scattering for different values of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
read the original abstract

This study explores the scattering dynamics of kinks within a nonlinear system governed by a parameterized potential $U_\lambda(\chi)$, examining the distinct behaviors of small and large kinks across a range of $\lambda$ values and initial velocities. For small kinks, we investigate the critical velocity for separation, the influence of vibrational modes, resonance phenomena, and the conditions under which large kinks emerge from collisions. Our findings reveal that the critical velocity exhibits a non-monotonic dependence on the parameter $\lambda$, reflecting the evolving stability of small kinks, while the decreasing frequency of vibrational modes with increasing $\lambda$ diminishes resonance effects, leading to simpler scattering dynamics at higher $\lambda$. The formation of large kinks from small kink collisions is favored at lower $\lambda$, where the mass difference between small and large kinks is reduced. Conversely, large kink scattering consistently results in the production of small kinks, with the number of small kink pairs growing as both $\lambda$ and initial velocity increase, a process driven by energy transfer from the translational modes of large kinks to the potential energy required for small kink creation. The absence of vibrational modes in large kinks contrasts with their presence in small kinks, where such modes give rise to complex phenomena like bion formation and resonance. These results underscore the pivotal role of $\lambda$ in shaping kink interactions and offer valuable insights into the dynamics of topological defects in nonlinear systems, with potential implications for understanding similar phenomena in condensed matter physics and related fields.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper numerically studies kink scattering in a one-parameter family of nonlinear wave equations with potential U_λ(χ). For small kinks it reports a non-monotonic dependence of the critical separation velocity on λ, resonance windows linked to vibrational-mode frequencies that decrease with λ, and preferential production of large kinks at low λ. Large-kink collisions are found to produce increasing numbers of small-kink pairs with rising λ and velocity, driven by translational-to-potential energy transfer; large kinks lack internal modes while small kinks exhibit bion formation and resonances.

Significance. If the reported collision statistics and λ-dependences are robust, the work supplies concrete, falsifiable numerical predictions for how a single control parameter tunes the balance between resonant and non-resonant kink dynamics, with direct relevance to topological defects in condensed-matter and field-theory models.

major comments (2)
  1. [Numerical Methods] Numerical Methods section: no convergence tests (grid spacing, time-step size, domain size) or error bars are reported on the critical velocities or kink-production counts extracted from the time-dependent solutions. Because the central claims rest on the precise location of resonance windows and the non-monotonicity of the critical velocity, the absence of these diagnostics leaves open the possibility that the reported λ-dependence is influenced by integrator or boundary artifacts.
  2. [§4.1] §4.1 and associated figures: the statement that resonance windows are “tied to vibrational-mode frequencies” is not supported by an explicit tabulation or plot comparing the computed mode frequencies (or their λ-dependence) to the observed resonance velocities; without this comparison the attribution remains qualitative and load-bearing for the interpretation of the scattering data.
minor comments (2)
  1. [Figures] Figure captions should explicitly state the range of initial velocities, the precise definition of “critical velocity,” and the criterion used to count produced kinks.
  2. [Discussion] A short paragraph comparing the observed large-kink production rates to any available analytic estimates (e.g., energy thresholds) would strengthen the discussion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and describe the revisions that will be incorporated to strengthen the numerical validation and interpretive support.

read point-by-point responses

  1. Referee: [Numerical Methods] Numerical Methods section: no convergence tests (grid spacing, time-step size, domain size) or error bars are reported on the critical velocities or kink-production counts extracted from the time-dependent solutions. Because the central claims rest on the precise location of resonance windows and the non-monotonicity of the critical velocity, the absence of these diagnostics leaves open the possibility that the reported λ-dependence is influenced by integrator or boundary artifacts.

    Authors: We agree that explicit convergence diagnostics are essential to substantiate the robustness of the reported critical velocities and resonance windows. In the revised manuscript we will add a new subsection to the Numerical Methods section that reports convergence tests with respect to spatial grid spacing, time-step size, and domain size. We will also include error bars on the extracted critical velocities and kink-production statistics, obtained from ensembles of runs at multiple resolutions. These additions will confirm that the non-monotonic λ-dependence and resonance structure are not numerical artifacts. revision: yes

  2. Referee: [§4.1] §4.1 and associated figures: the statement that resonance windows are “tied to vibrational-mode frequencies” is not supported by an explicit tabulation or plot comparing the computed mode frequencies (or their λ-dependence) to the observed resonance velocities; without this comparison the attribution remains qualitative and load-bearing for the interpretation of the scattering data.

    Authors: We acknowledge that a direct quantitative comparison would make the link between resonance windows and vibrational modes more rigorous. In the revised §4.1 we will insert a new figure (or table) that overlays the computed vibrational-mode frequencies as a function of λ against the observed resonance velocities extracted from the scattering simulations. This explicit comparison will provide quantitative support for the attribution and strengthen the physical interpretation. revision: yes

Circularity Check

0 steps flagged

No circularity: all claims extracted from direct numerical integration of the nonlinear wave equation

full rationale

The paper presents results from numerical simulations of kink scattering in the parameterized potential U_λ(χ). Key observations—the non-monotonic critical velocity versus λ for small-kink separation, resonance windows tied to vibrational frequencies, and preferential large-kink production at low λ—are obtained by integrating the equations of motion forward in time and post-processing the trajectories. No step reduces a reported quantity to a fitted parameter defined in terms of itself, nor does any central claim rest on a self-citation chain whose validity is presupposed by the present work. The derivation chain is therefore self-contained against external benchmarks (the continuum PDE and its numerical solution).

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of two families of kink solutions in the chosen potential family and on the assumption that numerical solutions of the nonlinear wave equation accurately capture their scattering without continuum artifacts.

free parameters (2)
  • λ
    Single tunable parameter that continuously deforms the potential and thereby changes kink masses, vibrational frequencies, and mass gaps.
  • initial velocity
    Continuously varied collision speed used to scan scattering regimes and locate critical velocities.
axioms (2)
  • standard math The system obeys a relativistic nonlinear wave equation derived from the potential U_λ(χ).
    Standard 1+1-dimensional scalar-field setup for kink dynamics.
  • domain assumption Both small and large kink solutions exist and remain stable over the studied range of λ.
    Required to define the two distinct collision partners whose interactions are reported.

pith-pipeline@v0.9.0 · 5807 in / 1550 out tokens · 90553 ms · 2026-05-21T22:42:21.876100+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · 14 internal anchors

  1. [1]

    Rajaraman, Solitons and Instantons: An Introduc- tion to Solitons and Instantons in Quantum Field The- ory, North-Holland (1982)

    R. Rajaraman, Solitons and Instantons: An Introduc- tion to Solitons and Instantons in Quantum Field The- ory, North-Holland (1982)

    work page 1982

  2. [2]

    Vilenkin, and E

    A. Vilenkin, and E. P. S. Shellard, Cosmic Strings and Other Topological Defects , Cambridge University Press (2000)

    work page 2000

  3. [3]

    Manton, and P

    N. Manton, and P. Sutcliffe, Topological Solitons, Cam- bridge University Press, Cambridge University Press (2004)

    work page 2004

  4. [4]

    Y. M. Shnir, Topological and Non-Topological Solitons in Scalar Field Theories , Cambridge University Press (2018)

    work page 2018

  5. [5]

    A. R. Bishop, J. A. Krumhansl, and S. E. Trullinger, Solitons in condensed matter: A paradigm , Physica D 1 (1980) 1

    work page 1980

  6. [6]

    S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Yu. S. Kivshar, Nonlinear switching and solitons in PT-symmetric photonic sys- tems, Laser Photonics Rev. 10 (2016) 2, 177-213

    work page 2016

  7. [7]

    A. E. Miroshnichenko, S. V. Dmitriev, A. A. Vasiliev and T. Shigenari, Inelastic three-soliton collisions in a weakly discrete sine-Gordon system, Nonlinearity. 13 (2000) 837

    work page 2000

  8. [8]

    S. V. Dmitriev, P. G. Kevrekidis, and Y. S. Kivshar, Radiationless energy exchange in three-soliton collisions, Phys. Rev. E 78 (2008) 046604

    work page 2008

  9. [9]

    Extreme values of elastic strain and energy in sine-Gordon multi-kink collisions

    A. Moradi Marjaneh, A. Askari, D. Saadatmand, and S.V. Dmitriev, Extreme values of elastic strain and en- ergy in sine-Gordon multi-kink collisions , Eur. Phys. J. B 91 (2018) 22 [ arXiv:1710.10159]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  10. [10]

    Makhankov, Dynamics of classical solitons (in non- integrable systems), Phys

    V.G. Makhankov, Dynamics of classical solitons (in non- integrable systems), Phys. Rep. 35 (1978) 1

    work page 1978

  11. [11]

    D. K. Campbell, J. F. chonfeld, C. A. Wingate, Reso- nance structure in kink-antikink interactions in φ4 the- ory, Physica D 9 (1980) 1-2

    work page 1980

  12. [12]

    High energy density in the collision of $N$ kinks in the $\phi^4$ model

    A. Moradi Marjaneh, D. Saadatmand, K. Zhou, S.V. Dmitriev, and M.E. Zomorrodian, High energy density in the collision of N kinks in the ϕ4 model, 11 Commun. Nonlinear Sci. Numer. Simulat. 49 (2017) 30 [arXiv:1605.09767]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  13. [13]

    Askari, A

    A. Askari, A. Moradi Marjaneh, Z. G. Rakhmatul- lina, M. Ebrahimi-Loushab, D. Saadatmand, V. A. Gani, P. G. Kevrekidis, and S. V. Dmitriev, Collision of φ4 kinks free of the Peierls-Nabarro barrier in the regime of strong discreteness , Chaos, Solitons and Fractals 138 (2020) 109854 [ arXiv:1912.07953]

    work page arXiv 2020

  14. [14]

    N. S. Manton, K. Ole´ s, T. Roma´ nczukiewicz, and A. Wereszczy´ nski,Collective Coordinate Model of Kink- Antikink Collisions in ϕ4 Theory, Phys. Rev. Lett 127 (2021) 071601 [ arXiv:2106.05153]

    work page arXiv 2021

  15. [15]

    F. C. E. Lima, R. Casana, and C. A. S. Almeida, Kinks and double-kinks in generalized ϕ4-and ϕ4-models, Eur. Phys. J. C 84 (2025) 1266 [ arXiv:2408.04761]

    work page arXiv 2025

  16. [16]

    Multi-kink collisions in the $\phi^6$ model

    A. Moradi Marjaneh, V. A. Gani, D. Saadatmand, S. V. Dmitriev, and K. Javidan, Multi-kink collisions in the ϕ6 model, JHEP 07 (2017) 028 [ arXiv:1704.08353]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  17. [17]

    V. A. Gani, and A. Moradi Marjaneh, Asymmetric kink solutions of hyperbolically deformed model , Journal of Physics: Conference Series 1690 (2020) 012096

    work page 2020

  18. [18]

    Saadatmand, and A

    D. Saadatmand, and A. Moradi Marjaneh, Scattering of the asymmetric φ6 kinks from a PT -symmetric perturba- tion: creating multiple kink–antikink pairs from phonons , Eur. Phys. J. B 95 (2022) 144 [ arXiv:2201.03277]

    work page arXiv 2022

  19. [19]

    C. Adam, K. Oles, T. Romanczukiewicz, and A. Wereszczynski, Spectral walls in antikink-kink scat- tering in the ϕ6 model, Phys. Rev. D 106 (2022) 105027 [arxiv:2209.11479]

    work page arXiv 2022

  20. [20]

    Saadatmand, A

    D. Saadatmand, A. Moradi Marjaneh, A. Askari, and H. Weigel, Phonons scattering off discrete asymmet- ric solitons in the absence of a Peierls-Nabarro poten- tial, Chaos, Solitons and Fractals 180 (2024) 114550 [arXiv:2308.02322]

    work page arXiv 2024

  21. [21]

    Successive phase transitions and kink solutions in $\phi^{8}$, $\phi^{10}$, and $\phi^{12}$ field theories

    A. Khare, I. C. Christov, and A. Saxena, Succes- sive phase transitions and kink solutions in ϕ8, ϕ10, and ϕ12 field theories, Phys. Rev. E 90 (2014) 023208 [arXiv:1402.6766]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  22. [22]

    P. A. Blinov, V. A. Gani, and A. Moradi Marjaneh, From thin to thick domain walls: An example of the φ8 model, Journal of Physics: Conference Series 1690 (2020) 012082

    work page 2020

  23. [23]

    Khare, A

    A. Khare, A. Duzgun, and A. Saxena, Explicit Kink Solu- tions in Several One-Parameter Family of Higher Order Field Theory Models , Int. J. Mod. Phys. B 35 (2021) 2150324 [arXiv:2103.05145]

    work page arXiv 2021

  24. [24]

    Bazeia, J

    D. Bazeia, J. G. F. Campos, A. Mohammadi, Kink- antikink collisions in the ϕ8 model: short-range to long- range journey, JHEP 05 (2023) 116 [arXiv:2303.12482]

    work page arXiv 2023

  25. [25]

    Scattering of kinks of the sinh-deformed $\varphi^4$ model

    D. Bazeia, E. Belendryasova, and V. A. Gani, Scattering of kinks of the sinh-deformed ϕ4 model, Eur. Phys. J. C 78 (2018) 340 [ arXiv:1710.04993]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  26. [26]

    Moradi Marjaneh, F

    A. Moradi Marjaneh, F. C. Simas, and D. Bazeia Collisions of kinks in deformed φ4 and φ6 mod- els, Chaos, Solitons and Fractals 164 (2022) 112723 [arXiv:2207.00835]

    work page arXiv 2022

  27. [27]

    Moradi Marjaneh, F

    A. Moradi Marjaneh, F. C. Simas, and D. Bazeia Scattering of kinks in scalar-field models with higher- order self-interactions , Ann. Phys. 470 (2024) 169777 [arXiv:2402.00270]

    work page arXiv 2024

  28. [28]

    D. K. Campbell, M. Peyrard, and P. Sodano, Kink- antikink interactions in the double sine-Gordon equation , Physica D 19 (1986) 165

    work page 1986

  29. [29]

    Interaction Properties of the Periodic and Step-like Solutions of the Double-Sine-Gordon Equation

    M. Peyravi, A. Montakhab, N. Riazi, and A. Gharaati, Interaction Properties of the Periodic and Step-like So- lutions of the Double-Sine-Gordon Equation , Eur. Phys. J. B 72 (2009) 269 [ arXiv:0802.2776]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  30. [30]

    V.A. Gani, A. Moradi Marjaneh, A. Askari, E. Be- lendryasova, and D. Saadatmand, Scattering of the dou- ble sine-Gordon kinks , Eur. Phys. J. C 78 (2018) 345 [arXiv:1711.01918]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  31. [31]

    Zolotaryuk, IO

    Y. Zolotaryuk, IO. Starodub, Moving Embedded Soli- tons in the Discrete Double Sine-Gordon Equation , Non- linear Systems, Vol.2, Understanding Complex Systems. Springer, Cham (2018)

    work page 2018

  32. [32]

    A new look at the double sine-Gordon kink-antikink scattering

    E. Belendryasova, V.A. Gani, A. Moradi Marjaneh, D. Saadatmand, and A. Askari, A new look at the dou- ble sine-Gordon kink-antikink scattering, J. Phys.: Conf. Ser. 1205 (2019) 012007 [ arXiv:1810.00667]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  33. [33]

    Yuriy Yerin, and Stefan-Ludwig Drechsler, Phase soli- tons in a weakly coupled three-component superconductor, Phys. Rev. B 104 (2021) 014518 [ arXiv:2103.17000]

    work page arXiv 2021

  34. [34]

    Hadipour, D

    F. Hadipour, D. Saadatmand, M. Ashhadi, A. Moradi Marjaneh, I. Evazzade, A. Askari, and S. V. Dmitriev, Interaction of phonons with discrete breathers in one-dimensional chain with tunable type of anharmonicity, Phys. Lett. A 384 (2020) 126100

    work page 2020

  35. [35]

    Simas, K

    Fabiano C. Simas, K. Z. Nobrega, and D. Bazeia, Bi- furcation and chaos in one dimensional chains of small particles, Chaos, Solitons and Fractals 161 (2022) 112387 [arXiv:2206.12734]

    work page arXiv 2022

  36. [36]

    F. C. Simas, F. C. Lima, K. Z. Nobrega, and A. R. Gomes, Solitary oscillations and multiple antikink- kink pairs in the double sine-Gordon model , JHEP 12 (2020) 143 [ arXiv:2007.12318]

    work page arXiv 2020

  37. [37]

    From sine-Gordon to vacuumless systems in flat and curved spacetimes

    D. Bazeia, and D. C. Moreira, From sine-Gordon to vacu- umless systems in flat and curved spacetimes , Eur. Phys. J. C 77 (2017) 884 [ arXiv:1703.06363]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  38. [38]

    A discrete phi^4 system without Peierls-Nabarro barrier

    J.M. Speight, A discrete ϕ4 system without a Peierls–Nabarro barrier , Nonlinearity 10, 1615 (1997) [arXiv:patt-sol/9703005]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  39. [39]

    Kevrekidis, On a class of discretizations of Hamil- tonian nonlinear partial differential equations, Physica D 183, 68 (2003)

    P.G. Kevrekidis, On a class of discretizations of Hamil- tonian nonlinear partial differential equations, Physica D 183, 68 (2003)

    work page 2003

  40. [40]

    Flach, A.V

    S. Flach, A.V. Gorbach, Discrete breathers — Advances in theory and applications , Phys. Rep. 467, 1 (2008)

    work page 2008

  41. [41]

    Zh. G. Rakhmatullina, P. G. Kevrekidis and S. V. Dmitriev, Non-symmetric kinks in Klein-Gordon chains free of the Peierls-Nabarro potential, IOP Conf. Series: Materials Science and Engineering. 447 (2018) 012057

    work page 2018

  42. [42]

    Discrete Klein-Gordon models with static kinks free of the Peierls-Nabarro potential

    S.V. Dmitriev, P.G. Kevrekidis, N. Yoshikawa, Discrete Klein–Gordon models with static kinks free of the Peierls– Nabarro potential , J. Phys. A: Math. Gen. 38, 7617 (2005) [arXiv:nlin/0506001]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  43. [43]

    Deformed Defects

    D. Bazeia, L. Losano, and J.M.C. Malbouisson, De- formed defects , Phys. Rev. D 66 (2002) 101701(R) [arXiv:hep-th/0209027]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  44. [44]

    Deformed defects with applications to braneworlds

    D. Bazeia and L. Losano, Deformed defects with appli- cations to braneworlds , Phys. Rev. D 73 (2006) 025016 [arXiv:hep-th/0511193]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  45. [45]

    Khare and A

    A. Khare and A. Saxena, Novel Deformation Func- tion Creating or Destroying any Number of Even Kink Solutions, Phys. Lett. A 424 (2022) 127830 [arXiv:2107.07048]

    work page arXiv 2022

  46. [46]

    P. A. Blinov, T. V. Gani, V. A. Gani, Deforma- tions of Kink Tails, Ann. Phys. 437 (2022) 168739 [arXiv:2008.13159]. 12

    work page arXiv 2022

  47. [47]

    Sampaio, J

    J. Sampaio, J. Grollier, P. J. Metaxas, Domain Wall Mo- tion in Nanostructures. Chapter 8. In Magnetism of Sur- faces, Interfaces, and Nanoscale Materials. Edited by R. E. Camley, Z. Celinski, R. L. Stamps. Handbook of Sur- face Science, 5 (2015) 335

    work page 2015

  48. [48]

    M. D. DeJong and K. L. Livesey, Analytic theory for the switch from Bloch to N´ eel domain wall in nanowires with perpendicular anisotropy, Phys. Rev. B 92 (2015) 214420 [arXiv:1510.07673]

    work page internal anchor Pith review Pith/arXiv arXiv 2015