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REVIEW 4 major objections 6 minor 54 references

A regularized continuum model for traveling waves and dispersive shocks of the granular chain

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proposes the regularized strain equation $r_{TT} - \frac{\epsilon^2}{12} r_{XXTT} = (r^p)_{XX}$ as a continuum model of the granular chain and shows that its solitary waves, periodic waves, and dispersive shock waves match…

desk verdict A clean, honest regularization of the granular chain that gives usable DSW formulas; the one caveat—unproven hyperbolicity of the modulation system—is admitted up front and partially mitigated by solid numerical validation. read the letter →

arxiv 2411.17874 v2 pith:U3BGOMRM submitted 2024-11-26 nlin.PS

Su Yang , Gino Biondini , Christopher Chong , Panayotis G. Kevrekidis This is my paper · ORCID
classification nlin.PS MSC 35Q5335L6774J35
keywords granularcrystalsdispersiveshockwavesWhithammodulationtheorytravelingcnoidalcontinuumapproximationregularizedPDEDSWfitting
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 proposes a regularized partial differential equation, $$r_{TT} - \frac{\$epsilon^{2}$}{12}\, r_{XXTT} = (r^p)_{XX},$$ as a long-wavelength model for the strain dynamics of a granular chain with power-law contact forces and no precompression ($p=3/2$ for spherical grains). The claim is that this equation plays the same role for the granular chain that the KdV equation plays for the classical atomic chain: a tractable continuum description that reproduces solitary waves, periodic waves, and dispersive shock waves (DSWs) of the discrete lattice. The paper derives exact solitary waves for general $p$, cnoidal-wave solutions for $p=3/2$, $2$, and $3$, conservation laws, and Whitham modulation equations, then uses the DSW fitting method to obtain explicit formulas for leading-edge speed, soliton amplitude, trailing-edge speed, and trailing-edge wavenumber. Systematic comparisons with numerical solutions of both the PDE and the original discrete equations show good agreement in the tested parameter regimes.

What carries the argument

The central object is the regularized strain PDE (2.12), $r_{TT} - \frac{\epsilon^2}{12} r_{XXTT} = (r^p)_{XX}$. It is obtained from a Taylor expansion of the discrete strain equation by the substitution $(r^p)_{XXXX} \to r_{XXTT}$, which converts the unbounded high-wavenumber instability of the earlier model (2.9) into a saturating, real-valued dispersion relation. The argument is carried by the exact solitary wave (3.7) and by the Jacobi-elliptic periodic solutions of Section 4, which are then modulated: the averaged Lagrangian (6.9)-(6.16) produces the four-parameter Whitham modulation system, and at the harmonic and soliton edges of that system the DSW fitting method of [22] reduces everything to the transcendental equations (7.22) and the explicit edge-speed formulas (7.23).

What would settle it

Compute the characteristic speeds of the modulation system of Section 6 near the harmonic and soliton limits; if they are not real and distinct (strict hyperbolicity) or the genuine-nonlinearity condition fails, the DSW fitting edge formulas (7.23) lack their stated justification. Alternatively, measure the trailing-edge speed of the discrete chain over a wider range of jump heights than tested: a non-monotonic dependence would signal exactly the breakdown the paper says is possible elsewhere.

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Extended reading notes

Core claim

The core discovery is that Eq. (2.12) is the right regularized continuum limit of the granular strain equation (2.2), where earlier long-wave models fail. Taylor-expanding the discrete Laplacian gives the non-dispersive wave equation (2.7) at leading order and the fourth-order model (2.9) at the next order; (2.9) has purely imaginary frequencies for wavenumbers $|K|>2\sqrt{3}/\epsilon$, making it ill-posed. Replacing $(r^p)_{XXXX}$ by $r_{XXTT}$ up to $O(\epsilon^4)$ removes that instability and yields a dispersion relation $\Omega^2 = p A^{p-1} K^2/(1+\epsilon^2 K^2/12)$ that is real for all $K$. On this model the paper obtains closed-form solitary waves (3.7), periodic waves for three physically relevant values of $p$, and, through the DSW fitting method, the predictions (7.22)-(7.23) for the edge speeds, soliton amplitude, and trailing wavenumber of a dispersive shock. The numerical comparisons in Section 8 support the conclusion that the model captures the discrete chain's DSWs both qualitatively and quantitatively.

Load-bearing premise

The DSW edge predictions assume that the underlying modulation equations are strictly hyperbolic and genuinely nonlinear in the parameter regime tested; the paper states this is unproved and checks it only after the fact by numerical agreement.

Editorial extensions

If this is right

  • The regularized PDE (2.12) gives a well-posed, computationally cheap surrogate for the granular chain in the long-wave regime, so DSW formation and evolution can be simulated without integrating the discrete equations grain by grain.
  • For arbitrary exponent $p$, the solitary wave formula (3.7) provides an explicit amplitude-speed-width relation that is independent of the scaling parameter $\epsilon$.
  • For $p=2$ and $p=3$, formulas (4.9) and (4.22) give the leading-edge soliton amplitude in terms of the speed predicted by DSW fitting, yielding closed-form edge characterizations.
  • The DSW fitting predictions (7.23) are independent of $\epsilon$ when translated back to lattice variables via $k=\epsilon K$, so they apply to the discrete chain in the small-$\epsilon$ limit without tuning.
  • As the jump height increases, the trailing-edge wavenumber approaches the Brillouin zone edge, where the PDE approximation is expected to lose accuracy; the paper identifies this as a limitation of the continuum description.

Reading between the lines

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

  • The success of the DSW fitting method here suggests it could be applied directly to other discrete or regularized lattice models where the full Whitham modulation equations are unwieldy, as long as the linear dispersion relation is available; the paper does not pursue that transfer.
  • Because the edge predictions depend on the dispersion relation at the harmonic and soliton limits rather than on the full modulation system, a natural further test is to vary $\epsilon$ over a range (here only $\epsilon=0.1$ is used) to check the claimed $\epsilon$-independence of the lattice-level predictions.
  • For exponents other than $p=3/2, 2, 3$, the absence of explicit periodic solutions blocks the same analytic treatment; numerical interrogation with $p$ near those values could reveal whether the qualitative DSW behaviors persist.
  • The unproven strict hyperbolicity and genuine nonlinearity of the modulation equations mean the monotone trailing-edge speed observed here may not be universal; a wider scan of jump heights could detect a non-monotonic dependence of the type the paper notes can occur in other systems.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. This paper proposes a regularized continuum PDE, Eq. (2.12), r_TT - (epsilon^2/12) r_XXTT = (r^p)_XX, as a model for the strain dynamics of the precompression-free granular chain (2.2). The model is obtained by retaining the next-order term in the long-wave expansion of the discrete Laplacian and replacing (r^p)_XXXX with r_XXTT, which cures the ultraviolet ill-posedness of Eq. (2.9). The authors derive solitary-wave solutions (Sec. 3), periodic cnoidal-wave solutions for p = 3/2, 2, 3 (Sec. 4), conservation laws (Sec. 5), and Whitham modulation equations (Sec. 6). They then apply El's DSW fitting method to obtain closed-form transcendental equations for the trailing-edge wavenumber and leading-edge conjugate wavenumber, Eqs. (7.22)-(7.23), and compare the resulting predictions for leading/trailing edge speeds, soliton amplitude, and wavenumber with direct numerical simulations of both the continuum PDE and the original DDE, reporting generally good agreement (Sec. 8). The paper frames Eq. (2.12) as playing for the granular chain the role that KdV plays for the FPUT problem.

Significance. If the results hold, Eq. (2.12) is a valuable analytically tractable continuum description of the granular chain: it has bounded linear dispersion matching the lattice to O(k^4), admits explicit solitary and periodic waves without a small-amplitude assumption, and yields parameter-free DSW predictions that are independent of the artificial scale epsilon. The manuscript's strengths include a clean asymptotic derivation, explicit formulas for several nonlinearities, and systematic validation against independent numerical solutions of the discrete model rather than against fitted data. The main caveat, acknowledged by the authors themselves in Sections 7.5 and 9, is that the DSW fitting method is applied under an unverified assumption of strict hyperbolicity and genuine nonlinearity of the modulation equations. Because the quantitative DSW claims rest on that assumption, the theoretical status of the central predictions is conditional, and the manuscript would be strengthened by either verifying the assumption or explicitly labeling the predictions as heuristic.

major comments (4)
  1. [Sec. 7.5 and Sec. 9, Eqs. (7.22)-(7.23)] The DSW fitting method of Ref. [22] requires the underlying Whitham modulation equations to be strictly hyperbolic and genuinely nonlinear. The manuscript states in Section 7.5 that this is not known for the modulation equations derived in Section 6, and Section 9 repeats this as an open problem. Since Eqs. (7.22)-(7.23) and their comparisons in Figs. 5-8 constitute the central quantitative claim, this is a load-bearing gap, not a peripheral caveat. The a posteriori agreement is suggestive but does not close the logical gap between the assumptions of the method and its applicability. I ask the authors to either (i) verify the conditions in the tested regime, e.g., by numerically computing the characteristic speeds and the nonlinearity coefficients of the modulation system in Section 6 for p = 2, or (ii) explicitly phrase the DSW fitting results as conditional predictions that are supported only empirically in the stated parameter range.
  2. [Sec. 6 and missing Appendix B] Section 6 states that the derivation of the Whitham modulation equations for p = 3/2 and p = 3 is 'completely analogous, see Appendix B', but the manuscript contains no Appendix B; the appendices end with Appendix A.2. This matters because the numerical validation in Section 8 is performed for p = 3/2 and p = 3, and Section 7.5 concerns the modulation equations for the model generally. Please include the promised Appendix B or, if the DSW fitting computations do not require the explicit modulation equations for those nonlinearities, state this clearly and explain why the unverified hyperbolicity/genuine-nonlinearity assumption can be discussed without them.
  3. [Sec. 7.5, Eqs. (7.22)-(7.23)] The passage from the boundary-value problems in Eq. (7.20) to the transcendental equations in Eq. (7.22) is described only as 'a direct integration'. These equations are the quantitative basis for all DSW comparisons, so the derivation should be shown or at least outlined, including the treatment of integration constants, absolute values, and branches of logarithms. Without this, the reader cannot verify the formulas, check the branch choices for large K or \tilde K, or assess whether the formulas remain valid when the arguments of the logarithms change sign.
  4. [Sec. 8, Eqs. (8.2)-(8.3)] The measured trailing-edge location is defined using the ad hoc window parameters N and nu, with no reported values, no sensitivity study, and no error bars for the resulting speeds and wavenumbers. Since Figs. 7-8 show larger deviations at the trailing edge than at the leading edge, the possibility that these discrepancies are measurement artifacts cannot be excluded. Please report the values of N and nu used, and provide error estimates or a sensitivity analysis, for both the PDE and DDE measurements.
minor comments (6)
  1. [Sec. 7.3, Eq. (7.18b)] The relation between v and the particle velocity is described only as 'related to the particle velocity'. Since v = u_T in Eq. (7.13) and the displacement variable in Eq. (5.1) is defined with a sign opposite to the lattice displacement, the sign convention in Eq. (7.18b) should be stated explicitly.
  2. [Sec. 3.2] The reported errors E in the solitary-wave comparison are not reproducible as stated; please specify the values of the speed c, the number of lattice sites N, and the tolerance used for the 'exact' traveling-wave solution of Eq. (3.15).
  3. [Sec. 6.1] The symbols K (wavenumber) and K_m (complete elliptic integral) are visually similar and appear in the same equations; consider renaming one of them to avoid confusion.
  4. [Fig. 1 caption] The caption should state that the lattice dispersion curve is periodic outside the Brillouin zone [0, pi]; the figure extends beyond the zone specifically to show the ill-posed branch of Eq. (2.9).
  5. [Sec. 7.5] The text contains the typo 'genuinly' instead of 'genuinely'.
  6. [References] Reference [23] appears with an incomplete arXiv identifier 'arXiv:2404.1675'; please provide the full identifier.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the continuum model is derived from the lattice by Taylor expansion, and the DSW predictions are compared to independent lattice simulations without fitted parameters.

full rationale

The paper's central claim is that Eq. (2.12) is a regularized continuum model derived from the granular lattice (2.2). The derivation is a Taylor expansion: substituting r_n(t)=r(X,T) into (2.2) gives (2.6), truncating gives (2.7), adding the next-order term gives (2.9), and the regularization r_XXXX -> r_TTXX + O(eps^2) gives (2.12). No quantity used later is fitted to the benchmark phenomena. The DSW predictions in Eqs. (7.22)-(7.23) follow from the model's linear dispersion relation (2.13), the dispersionless limit (7.2), and El's DSW fitting method [22]; the only input is the jump height r_-/r_+ and the exponent p, and the predictions are independent of epsilon. Validation is against direct numerical simulations of the discrete model (2.2), which are external to the PDE model. Several cited works involve the current authors (e.g., Refs. [23], [25], [26]) but they are contextual and not load-bearing; in particular, the DSW fitting method is cited to the external work [22]. The explicit caveat in Sec. 7.5 and Sec. 9 that strict hyperbolicity and genuine nonlinearity of the modulation equations are unproven is an acknowledged correctness risk, not a circularity, because the paper validates a posteriori and no parameter is adjusted to force agreement. Hence no circular derivation step is exhibited.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The model is derived from a standard Taylor expansion; the only substantive unproved ingredient is the hyperbolicity and genuine nonlinearity of the modulation equations. No new physical entities are postulated, and no theoretical parameter is fitted to the validation data.

free parameters (1)
  • Trailing-edge measurement window parameters N and nu = N unspecified positive integer; nu = |r_- - r_+|/5
    Chosen by hand in Section 8 (Eqs. 8.2 and 8.3) to define the sets of local maxima/minima used to estimate the trailing-edge location; the reported numerical values of s_- and k_- depend on these choices. They are not model parameters but measurement free parameters.
assumptions (6)
  • standard math Taylor expansion of the slow modulations is valid; the shifted strain terms in Eq. (2.6) are expanded and truncated at O(epsilon^2).
    Standard continuum limit; all terms are assumed smooth on the X, T scale.
  • domain assumption Strains remain nonnegative for the considered dynamics, so the positive-part subscript can be dropped.
    The Riemann data have r_+ and r_- positive and the DSW keeps strains positive in the tested runs, but the model would differ if contacts separated.
  • ad hoc to paper The Whitham modulation equations are strictly hyperbolic and genuinely nonlinear, as required by the DSW fitting method.
    Stated as unknown in Section 7.5 and Section 9; the paper proceeds under this assumption and validates a posteriori.
  • domain assumption The continuum dispersion (2.13) is a sufficient proxy for the lattice dispersion (2.3) across the DSW wavenumber range.
    The two match to O(k^4), but the lattice wavenumber is bounded by the Brillouin zone, while the PDE wavenumber is unbounded; the paper notes larger errors for larger jumps.
  • standard math The averaged Lagrangian procedure (Whitham's method) is valid for slowly modulated periodic waves.
    Standard in the field.
  • domain assumption The Riemann data for the discrete chain are created from the continuum initial conditions via Eq. (7.18), with the velocity field v filtered through the operator (1 + epsilon^2 K^2/12)^-1.
    The mapping connects PDE and DDE initial data, but a pure discrete step would differ; this affects how directly the discrete DSW can be compared to theory.

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Pith. "Pith review of A regularized continuum model for traveling waves and dispersive shocks of the granular chain." pith.science (2026). https://pith.science/paper/U3BGOMRM

@misc{pith2026241117874,
  author       = {Pith},
  title        = {Pith review of: A regularized continuum model for traveling waves and dispersive shocks of the granular chain},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U3BGOMRM}},
  note         = {Machine review of arXiv:2411.17874}
}
read the original abstract

In this paper we focus on a discrete physical model describing granular crystals, whose equations of motion can be described by a system of differential difference equations (DDEs). After revisiting earlier continuum approximations, we propose a regularized continuum model variant to approximate the discrete granular crystal model through a suitable partial differential equation (PDE). We then compute, both analytically and numerically, its traveling wave and periodic traveling wave solutions, in addition to its conservation laws. Next, using the periodic solutions, we describe quantitatively various features of the dispersive shock wave (DSW) by applying Whitham modulation theory and the DSW fitting method. Finally, we perform several sets of systematic numerical simulations to compare the corresponding DSW results with the theoretical predictions and illustrate that the continuum model provides a good approximation of the underlying discrete one.

Figures

Figures reproduced from arXiv: 2411.17874 by the authors.

Figure 1
Figure 1. Comparison of the linearized dispersion relations of different models in terms of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Comparison of the traveling solitary waves in different continuum models for different values of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The initial conditions in Eqs. (7.15). The background and parameter values are r − = 1, r + = 0.9, p = 2, and ϵ = 0.1. 7.4 Numerical simulation and DSWs [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Numerical simulation of the Riemann problem: The left panel shows the profile of the DSW of the [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: Comparison of the DSW profiles for the cases of [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: The leading-edge quantities comparison: the left and the right panel refer to the comparison of [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: The trailing-edge speeds comparison: Note that the panels (a), (b), and (c) refer to the leading [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: The trailing-edge wavenumbers comparison: Notice that the red solid curves, from the top to the [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

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Works this paper leans on

54 extracted references · 40 canonical work pages

  1. [22]

    G. A. El. Resolution of a shock in hyperbolic systems modified by weak dispersion. Chaos: An Interdis- ciplinary Journal of Nonlinear Science , 15(3):037103, 10 2005

    2005

  2. [1]

    M. J. Ablowitz and M. Hoefer. Dispersive shock waves. Scholarpedia, 4(11):5562, 2009

    2009

  3. [2]

    El and M.A

    G.A. El and M.A. Hoefer. Dispersive shock waves and modulation theory . Physica D: Nonlinear Phe- nomena, 333:11, 2016

    work page 2016

  4. [3]

    G.B. Whitham. Linear and Nonlinear Waves . Wiley , New York, 1974

    work page 1974

  5. [4]

    D. H. Tsai and C. W. Beckett. Shock wave propagation in cubic lattices. J. of Geophys. Res. , 71(10):2601, 1966

    work page 1966

  6. [5]

    Nesterenko

    V.F. Nesterenko. Dynamics of Heterogeneous Materials . Springer-Verlag, New York, 2001

    work page 2001

  7. [6]

    Hascoet and H

    E. Hascoet and H. J. Herrmann. Shocks in non-loaded bead chains with impurities. Eur . Phys. J. B, 14:183, 2000

    work page 2000

  8. [7]

    E. B. Herbold and V. F. Nesterenko. Solitary and shock waves in discrete strongly nonlinear double power-law materials. Appl. Phys. Lett. , 90(26):261902, 2007

    work page 2007

Show all 54 references
  1. [8]

    E. B. Herbold and V. F. Nesterenko. Shock wave structure in a strongly nonlinear lattice with viscous dissipation. Phys. Rev. E, 75:021304, 2007

    2007

  2. [9]

    Molinari and C

    A. Molinari and C. Daraio. Stationary shocks in periodic highly nonlinear granular chains. Phys. Rev. E, 80:056602, 2009

    2009

  3. [10]

    H. Kim, E. Kim, C. Chong, P. G. Kevrekidis, and J. Yang. Demonstration of dispersive rarefaction shocks in hollow elliptical cylinder chains. Phys. Rev. Lett., 120:194101, 2018

    2018

  4. [11]

    Chong and P

    C. Chong and P. G. Kevrekidis. Coherent Structures in Granular Crystals: From Experiment and Mod- elling to Computation and Mathematical Analysis . Springer, New York, 2018

    2018

  5. [12]

    Starosvetsky , K.R

    Yu. Starosvetsky , K.R. Jayaprakash, M. Arif Hasan, and A.F. Vakakis.Dynamics and Acoustics of Ordered Granular Media. World Scientific, Singapore, 2017

    2017

  6. [13]

    Chong, Mason A

    C. Chong, Mason A. Porter, P. G. Kevrekidis, and C. Daraio. Nonlinear coherent structures in granular crystals. J. Phys.: Condens. Matter , 29:413003, 2017

    2017

  7. [14]

    S. Sen, J. Hong, J. Bang, E. Avalos, and R. Doney . Solitary waves in the granular chain. Phys. Rep., 462:21, 2008

    2008

  8. [15]

    Fleischer

    Shu Jia, Wenjie Wan, and Jason W. Fleischer. Dispersive shock waves in nonlinear arrays. Phys. Rev. Lett., 99:223901, Nov 2007

    2007

  9. [16]

    Observation of ultraslow shock waves in a tunable magnetic lattice

    Jian Li, S Chockalingam, and Tal Cohen. Observation of ultraslow shock waves in a tunable magnetic lattice. Phys. Rev. Lett., 127:014302, Jun 2021

    2021

  10. [17]

    A. M. Filip and S. Venakides. Existence and modulation of traveling waves in particles chains. Comm. Pure and Appl. Math. , 52(6):693, 1999

    1999

  11. [18]

    Dreyer, M

    W. Dreyer, M. Herrmann, and A. Mielke. Micro-macro transition in the atomic chain via Whitham’s modulation equation. Nonlinearity, 19(2):471, 2005

    2005

  12. [19]

    A. M. Bloch and Y. Kodama. Dispersive regularization of the whitham equation for the toda lattice. SIAM Journal on Applied Mathematics , 52(4):909–928, 1992

    1992

  13. [20]

    Biondini, C

    G. Biondini, C. Chong, and P. G. Kevrekidis. On the Whitham modulation equations for the Toda lattice and the quantitative description of its dispersive shocks. Physica D, 469:134315, 2024

    2024

  14. [21]

    T. R. Marchant and Noel F. Smyth. Approximate techniques for dispersive shock waves in nonlinear media. Journal of Nonlinear Optical Physics and Materials , 21(03):1250035, 2012

    2012

  15. [23]

    Dispersive hydrodynamics of a discrete conservation law

    Patrick Sprenger, Christopher Chong, Emmanuel Okyere, Michael Herrmann, Panayotis Kevrekidis, and Mark Hoefer. Dispersive hydrodynamics of a discrete conservation law. arXiv:2404.1675, 2024

    2024

  16. [24]

    Kevrekidis

    Christopher Chong, Michael Herrmann, and P.G. Kevrekidis. Dispersive shock waves in lattices: A dimension reduction approach. Physica D: Nonlinear Phenomena , 442:133533, 2022

    2022

  17. [25]

    Kevrekidis, and Gino Biondini

    Christopher Chong, Ari Geisler, Panayotis G. Kevrekidis, and Gino Biondini. Integrable approximations of dispersive shock waves of the granular chain. Wave Motion, 130:103352, 2024. 25

    2024

  18. [26]

    Yasuda, C

    H. Yasuda, C. Chong, J. Yang, and P. G. Kevrekidis. Emergence of dispersive shocks and rarefaction waves in power-law contact models. Phys. Rev. E, 95:062216, 2017

    2017

  19. [27]

    K. L. Johnson. Contact Mechanics. Cambridge University Press, Cambridge, UK, 1985

    1985

  20. [28]

    Yuli Starosvetsky and Alexander F. Vakakis. Traveling waves and localized modes in one-dimensional homogeneous granular chains with no precompression. Phys. Rev. E, 82:026603, 2010

    2010

  21. [29]

    Chong, P

    C. Chong, P. G. Kevrekidis, and G. Schneider. Justification of leading order quasicontinuum approxi- mations of strongly nonlinear lattices. Disc. Cont. Dyn. Sys. A , 34:3403, 2014

    2014

  22. [30]

    Compactons and chaos in strongly nonlinear lattices

    Karsten Ahnert and Arkady Pikovsky . Compactons and chaos in strongly nonlinear lattices. Physical Review E, 79(2), February 2009

    2009

  23. [31]

    P. Rosenau. Dynamics of nonlinear mass-spring chains near the continuum limit. Phys. Lett. A , 118:222, 1986

    1986

  24. [32]

    P. Rosenau. Hamiltonian dynamics of dense chains and lattices: Or how to correct the continuum. Phys. Lett. A , 311:39, 2003

    2003

  25. [33]

    N. J. Zabusky and M. D. Kruskal. Interactions of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett., 15:240–243, 1965

    1965

  26. [34]

    Gallavotti

    G. Gallavotti. The Fermi–Pasta–Ulam Problem: A Status Report . Springer-Verlag, Berlin, Germany , 2008

    2008

  27. [35]

    Sulem and P.L

    C. Sulem and P.L. Sulem. The nonlinear Schrödinger equation: self-focusing and wave collapse . Springer, New York, 1999

    1999

  28. [36]

    Hochstrasser, F.G

    D. Hochstrasser, F.G. Mertens, and H. Büttner. An iterative method for the calculation of narrow solitary excitations on atomic chains. Physica D: Nonlinear Phenomena , 35(1):259–266, 1989

    1989

  29. [37]

    Chatterjee

    A. Chatterjee. Asymptotic solution for solitary waves in a chain of elastic spheres. Phys. Rev. E , 59:5912, 1999

    1999

  30. [38]

    English and Robert L

    Jared M. English and Robert L. Pego. On the solitary wave pulse in a chain of beads. Proceedings of the AMS, 133:1763, 2005

    2005

  31. [39]

    G. A. El, M. A. Hoefer, and M. Shearer. Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws. SIAM Review, 59(1):3–61, January 2017

    2017

  32. [40]

    Partial differential equations: An introduction

    Walter A Strauss. Partial differential equations: An introduction . John Wiley & Sons, 2007

    2007

  33. [41]

    G. Whitham. Non-linear dispersive waves. Proc. Roy. Soc. Ser . A, 283:238–261, 1965

    1965

  34. [42]

    A. V. Gurevich and L. P. Pitaevskii. Nonstationary structure of a collisionless shock wave. Zh. Eksp. T eor . Fiz., 65:590–605, 1974

    1974

  35. [43]

    Nonlinear Periodic Waves and Their Modulations

    A M Kamchatnov. Nonlinear Periodic Waves and Their Modulations . World Scientific, 2000

    2000

  36. [44]

    Florian Theil and Valery I. Levitas. A study of a hamiltonian model for martensitic phase fransforma- tions including microkinetic energy .Mathematics and Mechanics of Solids , 5(3):337–368, 2000

    2000

  37. [45]

    N. K. Lowman and M. A. Hoefer. Dispersive hydrodynamics in viscous fluid conduits. Phys. Rev. E , 88:023016, Aug 2013

    2013

  38. [46]

    M.A. Hoefer. Shock waves in dispersive Eulerian fluids. J. Nonlinear Sci. , 24:525–577, 2014

    2014

  39. [47]

    Uniform asymptotic expansions for the fundamental solution of infinite harmonic chains

    Mielke Alexander and Patz Carsten. Uniform asymptotic expansions for the fundamental solution of infinite harmonic chains. Z. Anal. Anwend. , 36:437–475, 2017

    2017

  40. [48]

    G. A. El, R. H. J. Grimshaw, and N. F. Smyth. Unsteady undular bores in fully nonlinear shallow-water theory .Physics of Fluids , 18(2):027104, 02 2006

    2006

  41. [49]

    Venakides, D

    S. Venakides, D. Percy , and O. Roger. The Toda shock problem. Comm. on Pure and Appl. Math. , 44(8):1171, 1991

    1991

  42. [50]

    Chong, Y

    C. Chong, Y. Wang, D. Marechal, E. G. Charalampidis, Miguel Molerón, Alejandro J. Martínez, Ma- son A. Porter, P. G. Kevrekidis, and Chiara Daraio. Nonlinear localized modes in two-dimensional hexagonally-packed magnetic lattices. New Journal of Physics , 23:043008, 2021

    2021

  43. [51]

    Leonard, F

    A. Leonard, F. Fraternali, and C. Daraio. Directional wave propagation in a highly nonlinear square packing of spheres. Exp. Mech., 53:327, 2013. 26

    2013

  44. [52]

    Leonard, C

    A. Leonard, C. Daraio, A. Awasthi, and P. Geubelle. Effects of weak disorder on stress wave anisotropy in centered square nonlinear granular crystals. Phys. Rev. E, 86:031305, 2012

    2012

  45. [53]

    Leonard, C

    A. Leonard, C. Chong, P. G. Kevrekidis, and C. Daraio. Traveling waves in 2D hexagonal granular crystal lattices. Granular Matter, 16(4):531, 2014

    2014

  46. [54]

    M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard. Dispersive and classical shock waves in bose-einstein condensates and gas dynamics. Phys. Rev. A , 74:023623, Aug 2006. 27

    2006

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.