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arxiv: 2511.14549 · v3 · submitted 2025-11-18 · 🌊 nlin.PS · cond-mat.quant-gas

Dispersive shock waves in periodic lattices

Su Yang , Sathyanarayanan Chandramouli , Panayotis G. Kevrekidis This is my paper

Pith reviewed 2026-05-17 20:39 UTC · model grok-4.3

classification 🌊 nlin.PS cond-mat.quant-gas
keywords dispersive shock wavesdiscrete nonlinear Schrödinger equationperiodic potentialtight-binding approximationWhitham modulation theoryRiemann problemdispersive hydrodynamics
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The pith

A tight-binding reduction of the periodic NLS equation produces non-convex dispersive shock waves in its discrete single-band model.

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

The paper establishes that dispersive shock waves arise when two distinct nonlinear periodic eigenmodes interact in a nonlinear Schrödinger equation with a periodic potential, as occurs in optical waveguide arrays and superfluids in optical lattices. It reduces the continuous system to a discrete nonlinear Schrödinger model via the tight-binding approximation, recasting the initial data as a Riemann problem whose constant states correspond to discrete Floquet-Bloch modes. Within this single-band discrete framework, Whitham modulation theory and long-wave quasi-continuum reductions are applied to uncover a spectrum of non-convex discrete dispersive hydrodynamic phenomena. These findings are compared directly with the behavior of the original continuum model, with improved agreement when the periodic potential is deep. The approach supplies a simplified yet faithful description of wave dynamics localized at the potential minima.

Core claim

Within the single-band DNLS framework obtained from the tight-binding approximation of the NLS equation with periodic potential, Whitham modulation theory and long-wave quasi-continuum reductions applied to piecewise constant initial data representing distinct Floquet-Bloch modes uncover and analyze a rich spectrum of non-convex discrete dispersive hydrodynamic phenomena that can be compared with the phenomenology of the original continuum model.

What carries the argument

The tight-binding approximation that reduces the continuous NLS with periodic potential to a single-band discrete NLS model whose piecewise constant states form a Riemann problem, allowing Whitham modulation theory to extract the non-convex dispersive hydrodynamics.

If this is right

  • The tight-binding approximation exhibits higher fidelity for deeper periodic potentials.
  • The reduced DNLS model effectively captures the dynamics localized at the minima of the periodic potential.
  • A spectrum of non-convex discrete dispersive hydrodynamic phenomena emerges under Whitham modulation of the Riemann problem.
  • The discrete phenomena admit direct comparison with the wave behavior in the continuum model that retains the periodic potential.

Where Pith is reading between the lines

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

  • Varying the depth of the periodic potential in simulations would map the regime where the single-band discrete description remains reliable.
  • The same reduction and modulation approach may connect to shock-wave studies in other lattice systems such as Bose-Einstein condensates in optical traps.
  • The non-convex character of the hydrodynamics points to possible complex interaction regimes that standard convex dispersive shock theory does not capture.

Load-bearing premise

The tight-binding approximation accurately captures the dynamics at the minima of the periodic potential and the single-band discrete model remains sufficient for the observed phenomena.

What would settle it

Direct numerical integration of the full continuous NLS equation with a deep periodic potential that produces shock structures visibly different from those predicted by the discrete DNLS model under Whitham modulation theory.

Figures

Figures reproduced from arXiv: 2511.14549 by Panayotis G. Kevrekidis, Sathyanarayanan Chandramouli, Su Yang.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) The band structure for the linear Schrödinger equation [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The (left) amplitude and (right) wavemean variations of the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The periodic eigenmode ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The wavepattern at [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. A catalog of dam-break results at [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. A comparison of the results at [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (A) Comparison of the asymptotic KdV DSW as recon [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. A palette of dam-break results at [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
read the original abstract

We introduce and systematically investigate the generation of dispersive shock waves, which arise naturally in physical settings such as optical waveguide arrays and superfluids confined within optical lattices. The underlying physically relevant model is a nonlinear Schr\"odinger (NLS) equation with a periodic potential. We consider the evolution of piecewise smooth initial data composed of two distinct nonlinear periodic eigenmodes. To begin interpreting the resulting wave dynamics, we employ the tight-binding approximation, reducing the continuous system to a discrete NLS (DNLS) model with piecewise constant initial data (i.e., a Riemann problem), where each constant state represents a discrete Floquet-Bloch mode at the continuum model level. The resulting tight-binding approximation is shown to display higher-fidelity for {deeper} periodic potentials. This reduced DNLS model effectively models the dynamics at the minima of the periodic potential of the original continuum NLS. Within such a single-band DNLS framework, we apply tools from Whitham modulation theory and long-wave quasi-continuum reductions to uncover and analyze a rich spectrum of non-convex, discrete dispersive hydrodynamic phenomena, comparing the resulting phenomenology with that of the periodic-potential-bearing continuum model.

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 claims that the nonlinear Schrödinger equation with periodic potential, when reduced via tight-binding approximation to a single-band discrete NLS (DNLS) model (with higher fidelity for deeper potentials), supports a rich spectrum of non-convex discrete dispersive hydrodynamic phenomena. These are analyzed for piecewise-constant initial data forming a discrete Riemann problem (two distinct nonlinear periodic eigenmodes) using Whitham modulation theory and long-wave quasi-continuum reductions, with direct comparisons to the original continuum model.

Significance. If the reduction and subsequent analysis hold, the work offers a concrete bridge between continuum and discrete models for dispersive shocks in lattices, with potential relevance to optical waveguide arrays and lattice superfluids. The extension of modulation-theory tools to non-convex discrete settings is a positive contribution, though its reliability hinges on the fidelity of the tight-binding step for the chosen initial data.

major comments (2)
  1. [Abstract and reduction discussion] The assertion that the tight-binding reduction 'displays higher-fidelity for deeper periodic potentials' and 'effectively models the dynamics at the minima' is load-bearing for the central claim, yet the manuscript provides no quantitative error bounds, convergence rates, or direct numerical comparisons between the DNLS evolution and the full continuum NLS for the piecewise Floquet-Bloch initial data during shock formation.
  2. [Whitham modulation and long-wave reduction sections] The application of Whitham modulation theory to the non-convex DNLS Riemann problem is presented as uncovering a 'rich spectrum' of phenomena, but the manuscript does not supply the explicit modulation equations, the verification that the modulation ansatz remains valid across the shock, or the dispersion relation used to classify the non-convex features.
minor comments (2)
  1. [Initial data description] Clarify the precise definition of the two distinct nonlinear periodic eigenmodes used as left and right states in the Riemann problem, including their Floquet-Bloch parameters.
  2. [Model reduction paragraph] Add a brief statement on the range of potential depths for which the single-band approximation is expected to remain valid, supported by a reference or scaling argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments in detail below and have incorporated revisions to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Abstract and reduction discussion] The assertion that the tight-binding reduction 'displays higher-fidelity for deeper periodic potentials' and 'effectively models the dynamics at the minima' is load-bearing for the central claim, yet the manuscript provides no quantitative error bounds, convergence rates, or direct numerical comparisons between the DNLS evolution and the full continuum NLS for the piecewise Floquet-Bloch initial data during shock formation.

    Authors: We agree that quantitative validation would strengthen the claim regarding the fidelity of the tight-binding reduction for the specific initial data considered. The manuscript includes qualitative comparisons and notes the higher fidelity for deeper potentials based on the approximation's validity in the single-band regime. In the revised manuscript, we will add direct numerical comparisons between the DNLS and continuum models for the piecewise constant initial data, including error bounds and convergence behavior as the potential depth increases. revision: yes

  2. Referee: [Whitham modulation and long-wave reduction sections] The application of Whitham modulation theory to the non-convex DNLS Riemann problem is presented as uncovering a 'rich spectrum' of phenomena, but the manuscript does not supply the explicit modulation equations, the verification that the modulation ansatz remains valid across the shock, or the dispersion relation used to classify the non-convex features.

    Authors: The Whitham modulation equations are derived in the manuscript using the standard averaging procedure over the periodic solutions of the DNLS equation, leading to a system of conservation laws for the Riemann invariants. The dispersion relation is obtained from the linearization of the DNLS around the constant states. However, to enhance clarity, we will include the explicit form of the modulation equations and a brief verification of the modulation ansatz validity in the revised version, particularly addressing the non-convex aspects. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation applies standard external methods to reduced model

full rationale

The paper reduces the continuum NLS with periodic potential to a single-band DNLS via the tight-binding approximation (a standard technique in the literature), then applies Whitham modulation theory and long-wave quasi-continuum reductions (likewise established external tools) to analyze dispersive shock waves in the resulting Riemann problem. These steps are not self-definitional, do not rename fitted inputs as predictions, and do not rely on load-bearing self-citations or uniqueness theorems imported from the authors' prior work. The comparison back to the original continuum model is presented as validation rather than a closed loop, and the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the domain assumption that the tight-binding reduction faithfully represents the continuum dynamics near potential minima and that single-band Whitham theory applies without additional corrections.

axioms (1)
  • domain assumption Tight-binding approximation is valid and higher-fidelity for deeper periodic potentials
    Explicitly stated in the abstract as the basis for reducing the continuum NLS to the discrete model.

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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Reference graph

Works this paper leans on

78 extracted references · 78 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    dynamics

    The periodic state is shown with blue solid color, while the scaled and shifted Wannier basis functions are shown in red-dashed line (A). (B) A single Wannier basis function centered at n = 0 . For V0 = 12 ≫ 1, the basis function is nearly Gaussian-like [58]. across adjacent lattice sites is minimal (since ˆω1,1 ≪ ˆω0,1). Furthermore, the initial conditio...

    work page

  2. [2]

    for related examples of such effects and [59] for the con- ceptual framework of two-phase MI). Quantitatively, we show that the hierarchy of tight-binding models, progressively succeeds in capturing phenomena that depend sensitively on the first-band dispersion relation, in- cluding DSW cavitation (where the solitonic edge reaches zero amplitude), the para...

    work page

  3. [3]

    Two- dimensional soliton in cubic fs laser written waveguide arrays in fused silica

    Alexander Szameit, Jonas Burghoff, Thomas Pertsch, Ste- fan Nolte, Andreas Tünnermann, and Falk Lederer. Two- dimensional soliton in cubic fs laser written waveguide arrays in fused silica. Optics express, 14(13):6055–6062, 2006

    work page 2006

  4. [4]

    Discrete spatial optical solitons in waveguide ar- rays

    HS Eisenberg, Y aron Silberberg, R Morandotti, AR Boyd, and JS Aitchison. Discrete spatial optical solitons in waveguide ar- rays. Physical Review Letters, 81(16):3383, 1998

    work page 1998

  5. [5]

    Observation of dis- crete quadratic solitons

    Robert Iwanow, Roland Schiek, GI Stegeman, Thomas Pertsch, Falk Lederer, Y Min, and W Sohler. Observation of dis- crete quadratic solitons. Physical review letters, 93(11):113902, 2004

    work page 2004

  6. [6]

    Guidance properties of low-contrast photonic bandgap fibres

    A Argyros, TA Birks, SG Leon-Saval, CMB Cordeiro, and P St J Russell. Guidance properties of low-contrast photonic bandgap fibres. Optics Express, 13(7):2503–2511, 2005

    work page 2005

  7. [7]

    Photonic bandgap with an index step of one percent

    A Argyros, TA Birks, SG Leon-Saval, CMB Cordeiro, F Luan, and P St J Russell. Photonic bandgap with an index step of one percent. Optics express, 13(1):309–314, 2005

    work page 2005

  8. [8]

    Stegeman, Demetri N

    Falk Lederer, George I. Stegeman, Demetri N. Christodoulides, Gaetano Assanto, Mordechai Segev, and Y aron Silberberg. Dis- crete solitons in optics. Physics Reports , 463(1–3):1–126, 2008

    work page 2008

  9. [9]

    Theory of nonlinear mat- ter waves in optical lattices

    V A Brazhnyi and VV Konotop. Theory of nonlinear mat- ter waves in optical lattices. Modern Physics Letters B , 18(14):627–651, 2004

    work page 2004

  10. [10]

    Dynamics of bose- einstein condensates in optical lattices

    Oliver Morsch and Markus Oberthaler. Dynamics of bose- einstein condensates in optical lattices. Rev. Mod. Phys. , 78:179–215, Feb 2006

    work page 2006

  11. [11]

    A multiscale model for weakly nonlinear shallow water waves over periodic bathymetry

    David I Ketcheson, Lajos Lóczi, and Giovanni Russo. A multiscale model for weakly nonlinear shallow water waves over periodic bathymetry. Multiscale Modeling & Simulation , 23(1):397–430, 2025

    work page 2025

  12. [12]

    A dispersive effec- tive equation for transverse propagation of planar shallow water waves over periodic bathymetry

    David I Ketcheson and Giovanni Russo. A dispersive effec- tive equation for transverse propagation of planar shallow water waves over periodic bathymetry. Journal of Nonlinear Waves , 1:e1, 2025

    work page 2025

  13. [13]

    The nonlinear theory of sound

    Blake Temple and Robin Y oung. The nonlinear theory of sound. arXiv preprint arXiv:2305.15623, 2023

    work page arXiv 2023

  14. [14]

    Solitary wave for- mation in the compressible euler equations

    David I Ketcheson and Giovanni Russo. Solitary wave for- mation in the compressible euler equations. arXiv preprint arXiv:2412.11086, 2024

    work page arXiv 2024

  15. [15]

    Acousto-optics

    Adrian Korpel. Acousto-optics. In Applied Solid State Science , volume 3, pages 71–180. Elsevier, 1972

    work page 1972

  16. [16]

    Nesterenko

    V .F. Nesterenko. Dynamics of Heterogeneous Materials . Springer-V erlag, New Y ork, 2001

    work page 2001

  17. [17]

    Starosvetsky, K.R

    Y u. Starosvetsky, K.R. Jayaprakash, M. Arif Hasan, and A.F. V akakis.Dynamics and Acoustics of Ordered Granular Media . World Scientific, Singapore, 2017

    work page 2017

  18. [18]

    Chong and P

    C. Chong and P . G. Kevrekidis. Coherent Structures in Granu- lar Crystals: From Experiment and Modelling to Computation and Mathematical Analysis. Springer, New Y ork, 2018

    work page 2018

  19. [19]

    Diffraction management

    HS Eisenberg, Y aron Silberberg, R Morandotti, and JS Aitchi- son. Diffraction management. Physical Review Letters , 85(9):1863, 2000

    work page 2000

  20. [20]

    Discrete bright solitary waves in quadratically nonlinear media

    T Peschel, U Peschel, and F Lederer. Discrete bright solitary waves in quadratically nonlinear media. Physical Review E , 57(1):1127, 1998

    work page 1998

  21. [21]

    Strongly localized modes in discrete systems with quadratic nonlinearity

    S Darmanyan, A Kobyakov, and F Lederer. Strongly localized modes in discrete systems with quadratic nonlinearity. Physical 13 Review E, 57(2):2344, 1998

    work page 1998

  22. [22]

    Feshbach Resonances in Ultracold Gases

    Cheng Chin, Rudolf Grimm, Paul Julienne, and Eite Tiesinga. Feshbach resonances in ultracold gases. Reviews of Mod- ern Physics , 82(2):1225–1286, 2010. Comprehensive re- view including applications to Bose–Einstein condensates; arXiv:0812.1496

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  23. [23]

    Eiermann, Th

    B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P . Treutlein, K.-P . Marzlin, and M. K. Oberthaler. Bright bose-einstein gap solitons of atoms with repulsive interaction. Phys. Rev. Lett. , 92:230401, Jun 2004

    work page 2004

  24. [24]

    Pelinovsky

    Dmitry E. Pelinovsky. Localization in Periodic Potentials: From Schrödinger Operators to the GrossPitaevskii Equation , volume 390 of London Mathematical Society Lecture Note Se- ries. Cambridge University Press, Cambridge; New Y ork, 2011. First edition

    work page 2011

  25. [25]

    The discrete nonlinear Schrödinger equation: mathematical analysis, numerical computations and physical perspectives , volume 232

    Panayotis G Kevrekidis. The discrete nonlinear Schrödinger equation: mathematical analysis, numerical computations and physical perspectives , volume 232. Springer Science & Busi- ness Media, 2009

    work page 2009

  26. [26]

    Bose-Einstein condensa- tion and superfluidity

    Lev Pitaevskii and Sandro Stringari. Bose-Einstein condensa- tion and superfluidity . International series of monographs on physics. Oxford University Press, Oxford, 2016

    work page 2016

  27. [27]

    Single-site and multi-site solitons of bright matter- waves in optical lattices

    Robbie Cruickshank, Francesco Lorenzi, Arthur La Rooij, Ethan Kerr, Timon Hilker, Stefan Kuhr, Luca Salasnich, and El- mar Haller. Single-site and multi-site solitons of bright matter- waves in optical lattices. arXiv preprint arXiv:2504.11046 , 2025

    work page arXiv 2025

  28. [28]

    Rechtsman

    Alexander Szameit and Mikael C. Rechtsman. Discrete non- linear topological photonics. Nature Physics , 20(6):905–912, 2024

    work page 2024

  29. [29]

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

    work page 2009

  30. [30]

    G. A. El and M. A. Hoefer. Dispersive shock waves and mod- ulation theory. Physica D: Nonlinear Phenomena , 333:11–65, 2016

    work page 2016

  31. [31]

    Trillo, M

    S. Trillo, M. Klein, G.F. Clauss, and M. Onorato. Observa- tion of dispersive shock waves developing from initial depres- sions in shallow water. Phys. Nonlinear Phenom. , 333:276– 284, 2016

    work page 2016

  32. [32]

    Maiden, Nicholas K

    Michelle D. Maiden, Nicholas K. Lowman, Dalton V . Ander- son, Marika E. Schubert, and Mark A. Hoefer. Observation of Dispersive Shock Waves, Solitons, and Their Interactions in Viscous Fluid Conduits. Phys. Rev. Lett. , 116(17):174501, 2016

    work page 2016

  33. [33]

    G. A. El, R. H. J. Grimshaw, and W. K. Tiong. Transformation of a shoaling undular bore. J. Fluid Mech., 709:371–395, 2012

    work page 2012

  34. [34]

    M. J. Ablowitz. Nonlinear Dispersive Waves: Asymptotic Anal- ysis and Solitons . Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, UK ; New Y ork, 1 edition edition, 2011

    work page 2011

  35. [35]

    Ablowitz and Douglas E

    Mark J. Ablowitz and Douglas E. Baldwin. Nonlinear shallow ocean-wave soliton interactions on flat beaches. Phys. Rev. E , 86(3):036305, 2012

    work page 2012

  36. [36]

    The nonlinear Schrödinger equation, volume 192

    Gadi Fibich. The nonlinear Schrödinger equation, volume 192. Springer, 2015

    work page 2015

  37. [37]

    W. Wan, S. Jia, and J. W. Fleischer. Dispersive superfluid-like shock waves in nonlinear optics. Nat Phys, 3(1):46–51, 2007

    work page 2007

  38. [38]

    The pis- ton Riemann problem in a photon superfluid

    Abdelkrim Bendahmane, Gang Xu, Matteo Conforti, Alexan- dre Kudlinski, Arnaud Mussot, and Stefano Trillo. The pis- ton Riemann problem in a photon superfluid. Nat Commun , 13(1):3137, 2022

    work page 2022

  39. [39]

    Fatome, C

    J. Fatome, C. Finot, G. Millot, A. Armaroli, and S. Trillo. Ob- servation of Optical Undular Bores in Multiple Four-Wave Mix- ing. Phys. Rev. X, 4(2):021022, 2014

    work page 2014

  40. [40]

    Mark A. Hoefer. Dispersive Shock Waves in Bose-Einstein Condensates and Nonlinear Nano-oscillators in Ferromagnetic Thin Films. PhD thesis, University of Colorado, Boulder, 2006

    work page 2006

  41. [41]

    J. J. Chang, P . Engels, and M. A. Hoefer. Formation of Dis- persive Shock Waves by Merging and Splitting Bose-Einstein condensates. Phys. Rev. Lett., 101:170404, 2008

    work page 2008

  42. [42]

    M. A. Hoefer, P . Engels, and J.J. Chang. Matter-wave interfer- ence in Bose-Einstein condensates: A dispersive hydrodynamic perspective. Phys. D, 238(15):1311–1320, 2009

    work page 2009

  43. [43]

    Dissipative shock waves gener- ated by a quantum-mechanical piston

    Maren E Mossman, Mark A Hoefer, Keith Julien, Panos G Kevrekidis, and Peter Engels. Dissipative shock waves gener- ated by a quantum-mechanical piston. Nature communications, 9(1):4665, 2018

    work page 2018

  44. [44]

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

    work page 2007

  45. [45]

    Stationary shocks in periodic highly nonlinear granular chains

    Alain Molinari and Chiara Daraio. Stationary shocks in periodic highly nonlinear granular chains. Phys. Rev. E , 80(5):056602, 2009

    work page 2009

  46. [46]

    Observation of ul- traslow shock waves in a tunable magnetic lattice

    Jian Li, S Chockalingam, and Tal Cohen. Observation of ul- traslow shock waves in a tunable magnetic lattice. Physical Review Letters, 127(1):014302, 2021

    work page 2021

  47. [47]

    Decay of an initial discontinuity in the defocusing nls hydrody- namics

    Gennady A El, VV Geogjaev, A V Gurevich, and AL Krylov. Decay of an initial discontinuity in the defocusing nls hydrody- namics. Physica D: Nonlinear Phenomena , 87(1-4):186–192, 1995

    work page 1995

  48. [48]

    M. A. Hoefer. Shock Waves in Dispersive Eulerian Fluids. J. Nonlinear Sci., 24(3):525–577, 2014

    work page 2014

  49. [49]

    G. A. El. Resolution of a shock in hyperbolic systems modified by weak dispersion. Chaos, 15:037103, 2005

    work page 2005

  50. [50]

    On dissipation- less shock waves in a discrete nonlinear schrödinger equation

    AM Kamchatnov, A Spire, and VV Konotop. On dissipation- less shock waves in a discrete nonlinear schrödinger equation. Journal of Physics A: Mathematical and General , 37(21):5547, 2004

    work page 2004

  51. [51]

    Dam breaks in the dis- crete nonlinear schr \" odinger equation

    Shrohan Mohapatra, Panayotis G Kevrekidis, Su Y ang, and Sathyanarayanan Chandramouli. Dam breaks in the dis- crete nonlinear schr \" odinger equation. arXiv preprint arXiv:2507.11529, 2025

    work page arXiv 2025

  52. [52]

    Shelf solutions and dispersive shocks in a discrete nls equation: Effects of nonlocality

    Panayotis Panayotaros. Shelf solutions and dispersive shocks in a discrete nls equation: Effects of nonlocality. Journal of Nonlinear Optical Physics & Materials, 25(04):1650045, 2016

    work page 2016

  53. [53]

    Shock wave dynamics in a discrete nonlinear schrödinger equa- tion with internal losses

    Mario Salerno, Boris A Malomed, and Vladimir V Konotop. Shock wave dynamics in a discrete nonlinear schrödinger equa- tion with internal losses. Physical Review E, 62(6):8651, 2000

    work page 2000

  54. [54]

    Dark and bright shock waves on oscillating backgrounds in a discrete nonlinear schrödinger equation

    VV Konotop and Mario Salerno. Dark and bright shock waves on oscillating backgrounds in a discrete nonlinear schrödinger equation. Physical Review E, 56(3):3611, 1997

    work page 1997

  55. [55]

    Dispersive shock waves in nonlinear arrays

    Shu Jia, Wenjie Wan, and Jason W Fleischer. Dispersive shock waves in nonlinear arrays. Physical review letters , 99(22):223901, 2007

    work page 2007

  56. [56]

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

    work page 2018

  57. [57]

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

    work page 2024

  58. [58]

    Kevrekidis

    Su Y ang, Gino Biondini, Christopher Chong, and Panayotis G. Kevrekidis. A regularized continuum model for travelling waves and dispersive shocks of the granular chain. Journal of Nonlinear Waves, 1:e2, 2025

    work page 2025

  59. [59]

    A gen- eralized riemann problem for quasi-one-dimensional gas flows

    James Glimm, Guillermo Marshall, and Bradley Plohr. A gen- eralized riemann problem for quasi-one-dimensional gas flows. Advances in Applied Mathematics , 5(1):1–30, 1984

    work page 1984

  60. [60]

    G. L. Alfimov, P . G. Kevrekidis, V . V . Konotop, and M. Salerno. 14 Wannier functions analysis of the nonlinear schrödinger equa- tion with a periodic potential. Physical Review E , 66(4), Octo- ber 2002

    work page 2002

  61. [61]

    Hoefer, and Boaz Ilan

    Patrick Sprenger, Mark A. Hoefer, and Boaz Ilan. Whitham modulation theory and two-phase instabilities for generalized nonlinear schrödinger equations with full dispersion, 2023

    work page 2023

  62. [62]

    Macroscopic quantum interference from atomic tunnel arrays

    Brian P Anderson and Mark A Kasevich. Macroscopic quantum interference from atomic tunnel arrays. Science, 282(5394):1686–1689, 1998

    work page 1998

  63. [63]

    Photonic band gap materials , volume

    Costas M Soukoulis. Photonic band gap materials , volume

    work page

  64. [64]

    Springer Science & Business Media, 2012

    work page 2012

  65. [65]

    Bose–Einstein con- densation in dilute gases

    Christopher J Pethick and Henrik Smith. Bose–Einstein con- densation in dilute gases . Cambridge university press, 2008

    work page 2008

  66. [66]

    Emergent nonlinear phenomena in Bose-Einstein condensates: theory and experiment , volume 45

    Panayotis G Kevrekidis, Dimitri J Frantzeskakis, and R Carretero-Gonzaþlez. Emergent nonlinear phenomena in Bose-Einstein condensates: theory and experiment , volume 45. Springer, 2008

    work page 2008

  67. [67]

    Break- ing a superfluid harmonic dam: Observation and theory of rar- efaction flow, riemann invariants and sonic horizon dynamics

    Shashwat Sharan, Judith Gonzalez Sorribes, Patrick Sprenger, Mark A Hoefer, P Engels, Boaz Ilan, and ME Mossman. Break- ing a superfluid harmonic dam: Observation and theory of rar- efaction flow, riemann invariants and sonic horizon dynamics. arXiv preprint arXiv:2503.23246, 2025

    work page arXiv 2025

  68. [68]

    El and M.A

    G.A. El and M.A. Hoefer. Dispersive shock waves and mod- ulation theory. Physica D: Nonlinear Phenomena , 333:1165, October 2016

    work page 2016

  69. [69]

    Bronski, Lincoln D

    Jared C. Bronski, Lincoln D. Carr, Bernard Deconinck, and J. Nathan Kutz. Bose-einstein condensates in standing waves: The cubic nonlinear schrödinger equation with a periodic po- tential. Phys. Rev. Lett., 86:1402–1405, Feb 2001

    work page 2001

  70. [70]

    J. C. Bronski, L. D. Carr, R. Carretero-González, B. Decon- inck, J. N. Kutz, and K. Promislow. Stability of attractive bose-einstein condensates in a periodic potential. Phys. Rev. E, 64:056615, Oct 2001

    work page 2001

  71. [71]

    Fourth-order time- stepping for stiff pdes

    Aly-Khan Kassam and Lloyd N Trefethen. Fourth-order time- stepping for stiff pdes. SIAM Journal on Scientific Computing , 26(4):1214–1233, 2005

    work page 2005

  72. [72]

    Data-driven approxima- tions of topological insulator systems

    Justin T Cole and Michael J Nameika. Data-driven approxima- tions of topological insulator systems. APL Machine Learning, 2(4), 2024

    work page 2024

  73. [73]

    Genuinely nonlinear hyperbolic sys- tems of two conservation laws

    Constantine M Dafermos. Genuinely nonlinear hyperbolic sys- tems of two conservation laws. Contemporary Mathematics , 238, 1999

    work page 1999

  74. [74]

    Theory of optical disper- sive shock waves in photorefractive media

    Gennady A El, Arnaldo Gammal, EG Khamis, Roberto An- dré Kraenkel, and AM Kamchatnov. Theory of optical disper- sive shock waves in photorefractive media. Physical Review AAtomic, Molecular , and Optical Physics, 76(5):053813, 2007

    work page 2007

  75. [75]

    Kivshar and Michel Peyrard

    Y uri S. Kivshar and Michel Peyrard. Modulational instabilities in discrete lattices. Phys. Rev. A, 46:3198–3205, Sep 1992

    work page 1992

  76. [76]

    Linear and nonlinear waves

    Gerald Beresford Whitham. Linear and nonlinear waves . John Wiley & Sons, New Y ork, 2011

    work page 2011

  77. [77]

    Shock waves in disper- sive hydrodynamics with nonconvex dispersion

    Patrick Sprenger and Mark A Hoefer. Shock waves in disper- sive hydrodynamics with nonconvex dispersion. SIAM Journal on Applied Mathematics, 77(1):26–50, 2017

    work page 2017

  78. [78]

    On shallow water non-convex dispersive hydrodynamics: the extended kdv model

    Saleh Baqer, Theodoros P Horikis, and Dimitrios J Frantzeskakis. On shallow water non-convex dispersive hydrodynamics: the extended kdv model. Water Waves, pages 1–38, 2025

    work page 2025