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REVIEW 3 major objections 4 minor 34 references

On the Fractional Dynamics of Kinks in sine-Gordon Models

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For the fractional sine-Gordon equation, Caputo order $\beta<2$ damps kinks to rest, while Riesz order $\alpha>2$ turns their tails non-monotonic and makes kink-antikink forces repulsive at long range.

desk verdict A useful numerical map of fractional sine-Gordon dynamics whose main proof appendix is wrong and whose α>2 saddle needs a domain-size check. read the letter →

arxiv 2411.18600 v2 pith:UYPHIKF2 submitted 2024-11-27 nlin.PS

T. Bountis , J. Cantisán , J. Cuevas-Maraver , J.E. Macías-Díaz , P.G. Kevrekidis This is my paper · ORCID
classification nlin.PS MSC 35Q5135R11
keywords sine-GordonequationkinksbreathersfractionalderivativesCaputoderivativeRieszkink-antikinkinteractionsKlein-Gordon
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies the fractional sine-Gordon equation in which the usual second time derivative is replaced by a Caputo derivative of order $\beta$ and the spatial Laplacian by a Riesz derivative of order $\alpha$. It argues that the two fractional orders act as independent knobs: for $\beta<2$ the Caputo derivative injects dissipation, so single kinks slow down exponentially and stop, and kink-antikink pairs either annihilate or separate depending on whether they clear a velocity threshold. For the spatial side, it claims that $\alpha<2$ keeps kink tails monotonic and the kink-antikink force attractive, whereas $\alpha>2$ makes the tails cross zero, creating a saddle equilibrium and long-range repulsion with short-range attraction. When both derivatives are active, the dissipative Caputo effect dominates the decay while the Riesz order sets the interaction landscape, producing breathers that decay and kink-antikink pairs that can settle into permanent separated states. The stakes are that a canonical soliton model gains tunable dissipation and tunable long-range interaction, directly relevant to experimental efforts to engineer dispersion in optical media.

What carries the argument

The machinery is the pair of fractional operators in Eq. (1): the Caputo time derivative $\partial_t^\beta$, whose memory kernel produces effective damping for $1<\beta<2$, and the Riesz space derivative $\partial_x^\alpha$, whose power-law kernel changes the decay and oscillation of kink tails with $\alpha$. The paper also relies on a Mittag-Leffler separation-of-variables argument in the Appendix, intended to show that the static-kink stability spectrum is independent of $\beta$, and on the tail-mediated-force picture, in which the interaction between two solitary waves is carried by their tails, to link the zero crossing of the kink tail to the sign change of the kink-antikink force.

What would settle it

Compute $\partial_t^\beta E_\beta(\lambda^2 t)$ numerically for $\beta=1.5$ and $\lambda^2<0$ and compare it with $\lambda^2 E_\beta(\lambda^2 t)$; if the two differ, the spectral-independence claim is unsupported. Alternatively, solve the linearized fractional equation for a static kink at $\beta=1.5$ and compare its eigenvalues with the $\beta=2$ case: any $\beta$-dependence of the spectrum would falsify the claim.

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Extended reading notes

Core claim

The central claim is that the fractional sine-Gordon equation (1) has a two-parameter phenomenology organized by the orders $\beta$ and $\alpha$. For $\beta<2$, the Caputo time derivative acts like a damping term: kinks launched at finite velocity decelerate exponentially and come to rest; stationary kinks shed radiation that is progressively absorbed; and head-on kink-antikink collisions have a critical velocity below which the pair traps into a breather whose oscillations decay. For the spatial Riesz order, $\alpha<2$ gives monotonic power-law tails and purely attractive kink-antikink forces, while $\alpha>2$ produces non-monotonic tails with a single zero crossing on each side; the zero crossing converts the tail-mediated force to repulsion at large separation, so a saddle (unstable equilibrium) kink-antikink bound state exists for all $\alpha>2$, bifurcating from infinite separation as $\alpha\to 2^+$. With both derivatives active, the Caputo effect dominates the overall decay, but the Riesz order sets the interaction landscape, yielding long-lived or permanent kink-antikink states that settle at fixed positions.

Load-bearing premise

The Appendix's proof that kink stability spectra do not depend on $\beta$ assumes that $\partial_t^\beta E_\beta(\lambda^2 t)=\lambda^2 E_\beta(\lambda^2 t)$, but the standard Caputo identity for the Mittag-Leffler function applies to $E_\beta(\lambda t^\beta)$ rather than $E_\beta(\lambda^2 t)$, so the separation-of-variables step is not a valid derivation as written.

Editorial extensions

If this is right

  • For $\beta<2$, any kink launched with finite velocity eventually comes to rest, with exponential velocity decay whose characteristic time diverges as $\beta\to 2$.
  • Kink-antikink collisions in the Caputo case acquire a critical velocity depending on $\beta$ and initial separation: below it the pair forms a damped breather and annihilates; above it they pass and subsequently decelerate.
  • For $\alpha>2$, the kink-antikink interaction is attractive at short range and repulsive at long range, with an unstable stationary bound state at the separatrix; for $\alpha<2$ the interaction is purely attractive.
  • For $\alpha>2$, stationary kinks have non-monotonic tails with a single zero crossing on each side, and the band-edge resonance of the integrable case becomes an internal breathing mode.
  • For $1<\beta<2$ and $\alpha\geq 2$, breather-initiated kink-antikink pairs separate and settle at fixed positions rather than annihilating, with the settling time growing as $\alpha\to 2^+$.
  • For $1<\beta<2$ and $\alpha\geq 2$, breather-initiated kink-antikink pairs separate and settle at fixed positions rather than annihilating, with the settling time growing as $\alpha\to 2^+$.

Reading between the lines

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

  • Beyond the paper's explicit claims, the generic dissipative effect of $\beta<2$ suggests that the Caputo time operator could serve as a minimal phenomenological friction term for any soliton-bearing fractional PDE, without adding a separate damping term to the model.
  • The zero-crossing tail mechanism for $\alpha>2$ points to a design principle: by tuning the spatial derivative order, one can switch solitary-wave interactions in power-law-tailed systems between purely attractive and attract-repel landscapes, which could be probed in engineered dispersive media.
  • The observed settling of kinks at $\pm L/4$ under periodic boundary conditions is likely a finite-domain signature of the long-range interaction; using larger domains or absorbing boundaries should test whether the repulsion is intrinsic or enhanced by image forces.
  • If the claimed $\beta$-independence of the static-kink spectrum holds, the Caputo-induced decay is not a spectral instability but a secular effect of the fractional time evolution, implying that the decay rate should be controlled by $\beta$ alone rather than by the perturbation eigenfunction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the fractional sine-Gordon equation (1) in three regimes: Caputo time-fractional derivative with an ordinary Laplacian, Riesz space-fractional derivative with a second time derivative, and the combined case. Using numerical simulations with the fde_pi12_pc integrator for the Caputo part and a Fourier-spectral implementation for the Riesz part, the authors report that Caputo order 1<β<2 acts as dissipation, slowing and eventually stopping kinks and dampening breathers; that Riesz order α<2 produces monotonically attracting kinks, while α>2 produces non-monotonic kink tails and an unstable kink-antikink saddle equilibrium separating attraction from repulsion; and that the combined case inherits both effects. An appendix attempts to prove that the stability spectrum of static kinks is independent of β.

Significance. If the central claims hold, Eq. (1) offers a tunable dissipation mechanism through the Caputo order and tunable long-range attraction or repulsion through the Riesz order, which is of genuine interest for fractional nonlinear-wave applications. The paper is honest in labeling the 1/τ scaling law as a fit rather than a predictive derivation, and the saddle equilibrium is computed numerically rather than inferred from a fitted model. The manuscript contains extensive numerical explorations, spectral stability computations, and Floquet analysis for breathers. However, the validity of the α>2 saddle-and-repulsion picture is currently weakened by the use of a periodic-domain Riesz implementation without a demonstrated infinite-line limit, and the appendix contains a false Mittag-Leffler identity that undermines the stated β-independence proof.

major comments (3)
  1. [Appendix, Eqs. (.2)-(.4)] The separation-of-variables step uses the identity ∂_t^β E_β(λ^2 t) = λ^2 E_β(λ^2 t), which is false for the Caputo derivative. The correct Caputo eigenfunction is E_β(λ^2 t^β), satisfying ∂_t^β E_β(λ^2 t^β) = λ^2 E_β(λ^2 t^β); the identity also fails for the argument written in Eq. (.2). In addition, the statement E_2(z) = cosh(z) is incorrect; the correct relation is E_2(z) = cosh(√z). Since Section 5 explicitly invokes this appendix to conclude that the stability spectrum of stationary kinks is independent of β, the proof as written is invalid. The correction is likely local and may preserve the formal conclusion, but the manuscript must be revised to state the correct Mittag-Leffler argument and to discuss the behavior of E_β(λ^2 t^β) for λ^2 < 0 and 1 < β < 2.
  2. [§4.2 and §5.2, Figs. 6, 7, and 10] The central claim of an intrinsic infinite-line saddle equilibrium and long-range repulsion for α > 2 rests on stationary and dynamical kink-antikink computations performed with a spatially periodic implementation of the Riesz derivative. The paper itself notes in §5.2 that kinks equilibrate at x = ±L/4 because each member feels forces from periodic images, and Fig. 6 shows that the α → 2+ bound-state separation appears to tend to L/2, which is precisely the equilibrium separation of the periodic α = 2 sine-Gordon model on a ring. The force zero-crossing inferred from non-monotonic tails may therefore be a finite-size artifact. A domain-size independence study (for example, repeating the stationary K-AK computation and the force curve at L = 200 and L = 400, or using an infinite-line formulation of the Riesz derivative) is required to support the conclusion that the saddle and the repulsive branch survive in the infinite-line limit.
  3. [§3.1 and §4.2] The quantitative claims - the exponential decay of the kink velocity, the logarithmic scaling 1/τ = a log(b|β − 2| + 1), and the α-dependence of the saddle separation - are all numerical, but no grid-convergence or time-step refinement study is reported for either the fde_pi12_pc integrator or the spectral Riesz implementation. Because the long-range tails and the bifurcation near α = 2 are delicate, a resolution study (variation of spatial grid size and time step) should be added to rule out numerical artifacts and to give the reported fit parameters a defensible accuracy.
minor comments (4)
  1. [§3.2, Eq. (4)] The initial condition in Eq. (4) appears to have an unbalanced parenthesis: the expression for (u0, u˙0) is missing a closing parenthesis after the second component.
  2. [Fig. 8 caption] The caption states 'β = 2.15 (right)', but this section fixes β = 2 and varies α; the right panel is presumably α = 2.15 and the caption should be corrected.
  3. [§3.2, text before Fig. 4] There is a typo in 'ifv0 is above a certain threshold'; a space is missing after 'if'.
  4. [References] Reference [4] is incomplete: the journal name and volume are missing from the citation for the review on fractional calculus in biological modeling.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main claims rest on direct numerical simulation, the one fitted scaling law is explicitly labeled a fit, and self-citations are contextual rather than load-bearing; the Appendix identity concern is a mathematical error, not a circular reduction.

full rationale

No circular step was found in the paper's derivation chain. In Sec. 3.1, the relation 1/tau = a*log(b*|beta-2|+1) is presented explicitly as a fit ("we included a logarithmic fit of the form... with a = 0.0581, b = 2.6410 and r = 0.9998"), with fit parameters reported and no claim that the formula derives from the model, so no fitted quantity is renamed as a prediction. The kink-antikink saddle equilibrium for alpha > 2 in Sec. 4.2 is obtained by solving the stationary eigenproblem numerically, and its instability is assessed from the computed eigenvalues; the interpretation via attractive/repulsive tail-mediated forces cites the standard solitary-wave tail argument [27] and does not reduce to the paper's own fitted parameters. Self-citations [20,23] are used to state prior numerical observations about damped breathers in Caputo-fractional sine-Gordon models and to motivate longer-time revisits; these citations are contextual and not the load-bearing justification for the present simulations, which are run directly from Eq. (1). A genuine mathematical concern is present in the Appendix: the separation ansatz w(x,t)=E_beta(lambda^2 t)v(x) uses the identity D^beta E_beta(lambda^2 t)=lambda^2 E_beta(lambda^2 t), whereas the standard Caputo Mittag-Leffler identity applies to E_beta(lambda t^beta), and E_2(z)=cosh(sqrt(z)), not cosh(z). This makes the beta-independence proof suspect as written, but that is a correctness or derivation-error issue, not a circular equivalence between input and output. Similarly, the finite-domain periodic implementation of the Riesz derivative noted in Sec. 5 (kinks equilibrating at x=±L/4) raises a possible finite-size artifact for the infinite-line 'long-range repulsion' claim, but again this is an external-validity concern rather than a circular reduction. Overall, the central results are numerically computed phenomena, the one scaling relation is honestly labeled a fit, and the self-citations do not carry the argument.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central results rest on two fitted constants in the scaling law, on standard fractional calculus identities that are misstated in the Appendix, and on implicit assumptions about well-posedness, numerical convergence, and periodic boundary treatment of the Riesz derivative. No new physical entities are introduced.

free parameters (2)
  • a (logarithmic fit prefactor) = 0.0581
    Fitted to the numerically measured halving times in the scaling law 1/τ = a·log(b|β-2|+1) reported in Section 3.1.
  • b (logarithmic fit scale) = 2.6410
    Second fitted constant in the same empirical logarithmic scaling law; the fit is reported with r=0.9998 but no error bars.
assumptions (3)
  • standard math The Caputo derivative of the Mittag-Leffler function satisfies ∂^β_t E_β(λ^2 t) = λ^2 E_β(λ^2 t).
    Invoked in the Appendix (Eq. (.4)) to prove β-independence of the static kink stability spectrum. As written this identity is false for 1<β<2; the standard Caputo identity uses E_β(λ t^β).
  • domain assumption The fractional PDE (1) with Caputo time order 1<β<2 and Riesz space order 1<α<3 has well-posed initial-value solutions on bounded domains, and the fde_pi12_pc scheme converges to them.
    All numerical interpretations of the simulations depend on this; no well-posedness or convergence proof is given in the paper.
  • domain assumption The Riesz derivative on the finite computational domain is implemented with periodic boundary conditions and the Fourier symbol |k|^α.
    This boundary treatment is used in Section 5.2 to explain kinks settling at x=±L/4, and it shapes the long-range interaction forces the paper reports.

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Pith. "Pith review of On the Fractional Dynamics of Kinks in sine-Gordon Models." pith.science (2026). https://pith.science/paper/UYPHIKF2

@misc{pith2026241118600,
  author       = {Pith},
  title        = {Pith review of: On the Fractional Dynamics of Kinks in sine-Gordon Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UYPHIKF2}},
  note         = {Machine review of arXiv:2411.18600}
}
abstract

In the present work we explore the dynamics of single kinks, kink-anti-kink pairs and bound states in the prototypical fractional Klein-Gordon example of the sine-Gordon equation. In particular, we modify the order $\beta$ of the temporal derivative to that of a Caputo fractional type and find that, for $1<\beta<2$, this imposes a dissipative dynamical behavior on the coherent structures. We also examine the variation of a fractional Riesz order $\alpha$ on the spatial derivative. Here, depending on whether this order is below or above the harmonic value $\alpha=2$, we find, respectively, monotonically attracting kinks, or non-monotonic and potentially attracting or repelling kinks, with a saddle equilibrium separating the two. Finally, we also explore the interplay of the two derivatives, when both Caputo temporal and Riesz spatial derivatives are involved.

Figures

Figures reproduced from arXiv: 2411.18600 by the authors.

Figure 1
Figure 1. Time evolution of the kink with initial velocity [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Left panel: Kink velocity decay with the order of the Caputo derivative. For [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Time evolution of the pulse-like derivative of a stationary kink with weak (numerical scheme induced) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: The evolution of a kink-anti-kink pair initially separated by a distance 2 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Single kink, Riesz Derivative of fractional order [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Left panel: Kink-anti-kink stationary configuration for a Riesz Derivative of fractional order [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Dynamics of the kink-anti-kink bound state for fractional order [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: Breathers with β = 2 and α = 1.85 (left) and β = 2.15 (right). Their frequency is ω = 0.8 and their domain is [−50, 50]. Insets show a zoom on the breather wings (right) and the Floquet multipliers spectrum (left). 5. Caputo derivative of order β ̸= 2 and Riesz derivat…
Figure 9
Figure 9. Figure 9: Kink velocity decay with time for different values of the order of the Caputo derivative for [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: Dynamics of the kink-anti-kink bound state for fractional order [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: Characteristic decay time τ for β = 1.99 (left panel) and β = 1.9 (right panel) and different values of the Riesz derivative order α. Here, we consider ω = 0.5. Finally, it is interesting to consider here a connection with the earlier work of [20], where a different r…
Figure 12
Figure 12. Figure 12: Evolution of u(x, t) for β = 1.4, α = 1.8 and u0 = 0. for β = 1.4. We observe that as α approaches 2, T increases exponentially. Hence, the relevant state may, depending on the combination of α and β, become extremely long-lived, yet it always has a finite lifetime […
Figure 13
Figure 13. Figure 13: Left panel: Duration of the transient bound state dependence on the value of the Riesz derivative [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]

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Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [14]

    R. J. Decker, A. Demirkaya, T. J. Alexander, G. A. Tsolias, P. G. Kevrekidis, Fractional solitons: A homotopic continuation from the biharmonic to the harmonic ϕ4 model (2024). arXiv:2410.18426

    work page Pith review arXiv 2024

  2. [1]

    Malomed,Optical Solitons and Vortices in Fractional Media: A Mini-Review of Recent Results

    B.A. Malomed,Optical Solitons and Vortices in Fractional Media: A Mini-Review of Recent Results. Photonics 8 (2021) 353

    work page 2021

  3. [2]

    Longhi, Fractional schr¨ odinger equation in optics, Opt

    S. Longhi, Fractional schr¨ odinger equation in optics, Opt. Lett. 40 (6) (2015) 1117–1120

    work page 2015

  4. [3]

    Qureshi, Real life application of caputo fractional derivative for measles epidemiological autonomous dynamical system, Chaos, Solitons & Fractals 134 (2020) 109744

    S. Qureshi, Real life application of caputo fractional derivative for measles epidemiological autonomous dynamical system, Chaos, Solitons & Fractals 134 (2020) 109744

    work page 2020

  5. [4]

    Ionescu, A

    C. Ionescu, A. Lopes, D. Copot, J. A. T. Machado, J. H. T. Bates, The role of fractional calculus in modeling biological phenomena: A review 51 141–159

    work page

  6. [5]

    H. Ming, J. Wang, M. Feˇ ckan, The application of fractional calculus in Chinese economic growth models, Mathematics 7 (8) (2019) 665. 15

    work page 2019

  7. [6]

    Kevrekidis, J

    P. Kevrekidis, J. Cuevas-Maraver, Fractional Dispersive Models and Applications, Springer Nature, Berlin, 2024

    work page 2024

  8. [7]

    I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Vol. 198, Elsevier, 1998

    work page 1998

Show all 34 references
  1. [8]

    Samko, A.A Kilbas, O

    S. Samko, A.A Kilbas, O. Marichev, Fractional integrals and derivatives. Theory and applications, Gordon and Breach, Amsterdam, 1993

    1993

  2. [9]

    Blanco-Redondo, C

    A. Blanco-Redondo, C. M. de Sterke, J. Sipe, T. F. Krauss, B. J. Eggleton, C. Husko, Pure-quartic solitons, Nature Communications 7 (2016) 10427

    2016

  3. [10]

    A. F. J. Runge, D. D. Hudson, K. K. K. Tam, C. M. de Sterke, A. Blanco-Redondo, The pure-quartic soliton laser, Nature Photonics 14 (8) (2020) 492–497

    2020

  4. [11]

    K. K. K. Tam, T. J. Alexander, A. Blanco-Redondo, C. M. de Sterke, Stationary and dynamical properties of pure-quartic solitons, Opt. Lett. 44 (13) (2019) 3306–3309

    2019

  5. [12]

    K. K. K. Tam, T. J. Alexander, A. Blanco-Redondo, C. M. de Sterke, Generalized disper- sion Kerr solitons, Phys. Rev. A 101 (2020) 043822

    2020

  6. [13]

    V. T. Hoang, J. Widjaja, Y. L. Qiang, M. Liu, T. J. Alexander, A. F. J. Runge, C. M. de Sterke, Observation of fractional evolution in nonlinear optics (2024)

    2024

  7. [15]

    Cuevas-Maraver, P

    J. Cuevas-Maraver, P. Kevrekidis, F. Williams, The sine-Gordon Model and its Applica- tions, Nonlinear Systems and Complexity, Springer Cham, 2014

    2014

  8. [16]

    McLachlan, A gallery of constant-negative-curvature surfaces, Math

    R. McLachlan, A gallery of constant-negative-curvature surfaces, Math. Intelligencer 16 (1994) 31–37

    1994

  9. [17]

    Frenkel, T

    J. Frenkel, T. Kontorova, On the theory of plastic deformation and twinning, J. Phys. Acad. Sci. USSR 1 (1939) 137

    1939

  10. [18]

    Barone, G

    A. Barone, G. Patern` o, Physics and Applications of the Josephson Effect, Wiley, 1982

    1982

  11. [19]

    B. A. Malomed, The sine-Gordon Model: General Background, Physical Motivations, Inverse Scattering, and Solitons, Springer International Publishing, Cham, 2014, pp. 1–30

    2014

  12. [20]

    J. E. Mac ´ ıas-Diaz, T. Bountis, An efficient dissipation-preserving numerical scheme to solve a caputo–riesz time-space-fractional nonlinear wave equation, Fractal/Fractional 6 (9) (2022) 500–525

    2022

  13. [21]

    Birnir, H

    B. Birnir, H. P. McKean, A. Weinstein, The rigidity of sine-gordon breathers, Communi- cations on Pure and Applied Mathematics 47 (8) (1994) 1043–1051

    1994

  14. [22]

    Garrappa, Numerical solution of fractional differential equations: A survey and a soft- ware tutorial, Mathematics 6 (2018) 16

    R. Garrappa, Numerical solution of fractional differential equations: A survey and a soft- ware tutorial, Mathematics 6 (2018) 16

    2018

  15. [23]

    Bountis, J

    T. Bountis, J. Cantis´ an, J. Cuevas-Maraver, J. Mac ´ ıas-D ´ ıaz, P. G. Kevrekidis, The dissi- pative effect of Caputo–time-fractional derivatives and its implications for the solutions of nonlinear wave equations, Partial Differential Equations in Applied Mathematics 11 (20...

    2024

  16. [24]

    Carretero-Gonz´ alez, L

    R. Carretero-Gonz´ alez, L. Cisneros-Ake, R. Decker, G. Koutsokostas, D. Frantzeskakis, P. Kevrekidis, D. Ratliff, Kink–antikink stripe interactions in the two-dimensional sine–gordon equation, Communications in Nonlinear Science and Numerical Simulation 109 (2022) 106123

    2022

  17. [25]

    Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, M. Peyrard, Internal modes of solitary waves, Phys. Rev. Lett. 80 (1998) 5032–5035

    1998

  18. [26]

    D. K. Campbell, J. F. Schonfeld, C. A. Wingate, Resonance structure in kink-antikink interactions in ϕ4 theory, Physica D 9 (1) (1983) 1–32

    1983

  19. [27]

    N. S. Manton, An effective Lagrangian for solitons, Nuclear Physics B 150 (1979) 397–412

    1979

  20. [28]

    Alfimov, T

    G. Alfimov, T. Pierantozzy, L. V´ azquez, Numerical study of a fractional sine-gordon equa- tion, in: A. L. M´ ehaut´ e, J. T. Machado, J. Trigeassou, J. Sabatier (Eds.), Fractional Differentiation and its Applications, U-Books Verlag, Augsburg, 2005

    2005

  21. [29]

    Chirilus-Bruckner, P

    M. Chirilus-Bruckner, P. Kevrekidis, J. Cuevas-Maraver, Stability of breathers for a peri- odic Klein–Gordon equation, Entropy 26 (2024) 756

    2024

  22. [30]

    Kevrekidis, R

    P. Kevrekidis, R. Carretero-Gonz´ alez, J. Cuevas-Maraver, D. Frantzeskakis, J.-G. Caputo, B. Malomed, Breather stripes and radial breathers of the two-dimensional sine-gordon equation, Communications in Nonlinear Science and Numerical Simulation 94 (2021) 105596

    2021

  23. [31]

    Flach, A

    S. Flach, A. V. Gorbach, Discrete breathers — advances in theory and applications, Physics Reports 467 (1) (2008) 1–116

    2008

  24. [32]

    Sulem, P

    C. Sulem, P. Sulem, The nonlinear Schr¨ odinger equation: self-focusing and wave collapse, Springer, New York, 1999

    1999

  25. [33]

    Ablowitz, B

    M. Ablowitz, B. Prinari, A. Trubatch, Discrete and Continuous Nonlinear Schr¨ odinger Systems, Cambridge University Press, Cambridge, 2004

    2004

  26. [34]

    Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solitons & Fractals 7 (9) (1996) 1461–1477

    F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solitons & Fractals 7 (9) (1996) 1461–1477. 17

    1996

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