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arxiv: 2605.10231 · v2 · pith:KB3CV6XWnew · submitted 2026-05-11 · 🌊 nlin.PS · math.DS

Asymptotic Analysis of discrete nonlinear localised modes in a Kagome lattice

Jonathan AD Wattis , Pilar R Gordoa , Andrew Pickering This is my paper

Pith reviewed 2026-05-20 22:58 UTC · model grok-4.3

classification 🌊 nlin.PS math.DS
keywords Kagome latticeflat bandnonlinear Schrödinger systemasymptotic reductionmultiple scalessolitary wavesKlein-Gordon latticeLie symmetries
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The pith

A novel coupled nonlinear Schrödinger system describes waves at the flat band intersection in a nonlinear Kagome lattice.

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

The paper derives reductions of the discrete nonlinear lattice equations to nonlinear Schrödinger systems using multiple scales asymptotics. A new coupled system appears specifically when the expansion is performed at the wavenumber where the flat band touches the upper dispersion surface. This system is then analyzed with Lie symmetries to obtain solitary wave reductions, and the waves are checked against direct numerical simulations of the original lattice. A reader would care because the approach supplies an explicit model for localized nonlinear modes that can arise in physical realizations such as nonlinear electrical networks.

Core claim

We obtain a novel system of coupled nonlinear Schrödinger equations by forming an asymptotic expansion for small-amplitude weakly nonlinear waves around the point where the flat band meets the upper surface of the dispersion relation. The underlying lattice is governed by Klein-Gordon-type interactions at each node, and the linear dispersion relation consists of three bands that can intersect at Dirac points or remain gapped. The new 2+1 dimensional system is reduced further via Lie symmetries to yield solitary wave solutions, which are then compared with numerical integrations of the discrete equations.

What carries the argument

Multiple-scales asymptotic expansion centered at the flat-band/upper-surface intersection point, which reduces the three-band lattice dynamics to a coupled nonlinear Schrödinger system in 2+1 dimensions.

If this is right

  • The coupled system admits further reductions to solitary waves through its Lie symmetry algebra.
  • Some of the reduced solutions resemble the Townes soliton known from earlier single-equation cases.
  • Numerical simulations of the lattice equations reproduce the predicted localized profiles at the chosen wavenumber.

Where Pith is reading between the lines

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

  • The same intersection-point technique may apply to other lattices that possess a flat band touching a dispersive surface.
  • The derived solitary waves could be observed as localized voltage or current patterns in a physical electrical lattice experiment tuned to the appropriate frequency.

Load-bearing premise

The multiple-scales expansion stays uniformly valid for small-amplitude waves exactly where the flat band touches the upper dispersion surface.

What would settle it

Direct numerical integration of the original discrete Klein-Gordon lattice equations at the intersection wavenumber and small amplitude should produce envelopes that match the solitary-wave solutions of the derived coupled system; persistent mismatch would show the reduction is invalid.

Figures

Figures reproduced from arXiv: 2605.10231 by Andrew Pickering, Jonathan AD Wattis, Pilar R Gordoa.

Figure 1
Figure 1. Figure 1: Illustration of Kagome lattice - triangles indicate A-nodes, filled circles B-nodes, and open circles, C-nodes. Edges of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the dispersion surfaces for the cases [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Dispersion relations for: left: acoustic mode, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the Townes soliton solution on the kagome lattice, red crosses indicate the solution [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the vector field solutions for [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of the vector field solution for [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Results of numerical simulations of breather mode with initial conditions given by (3.33), (3.21), (2.40), (2.11)–(2.12), [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
read the original abstract

We describe a nonlinear kagome lattice with nonlinear dynamics described by Klein-Gordon interactions with a scalar unknown at each node, such as might occur in a nonlinear electrical lattice. We show that the dispersion relation has three bands - a flat band and two other surfaces which may meet in Dirac points or be separated by a gap. By using multiple scales asymptotic methods, we find a variety of reductions to nonlinear Schrodinger (NLS) systems, some of which are similar to those obtained previously, and have the Townes soliton as a solution. We find a novel system of coupled NLS equations, by forming an asymptotic expansion for small amplitude weakly nonlinear waves around the point where the flat band meets the upper surface of the dispersion relation. We analyse this 2+1 dimensional system using Lie symmetries, and find further reductions to more complicated solitary wave solutions. Numerical simulations of the wave are also presented.

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

3 major / 2 minor

Summary. The paper analyzes small-amplitude weakly nonlinear waves on a discrete Kagome lattice with Klein-Gordon nonlinearity. The linear dispersion relation consists of a flat band together with two dispersive surfaces that may touch or be gapped. Multiple-scales asymptotics are used to obtain several NLS reductions; the central new result is a 2+1D system of coupled NLS equations derived for waves near the point at which the flat band touches the upper dispersive band. Lie symmetries are then applied to this coupled system to produce further reductions and solitary-wave solutions, which are illustrated by numerical simulations on the original lattice.

Significance. If the asymptotic reduction is shown to be consistent at the degeneracy point, the coupled NLS system would supply a new reduced model for localized modes in lattices possessing flat bands that touch dispersive branches. The combination of multiple-scales derivation, Lie-symmetry reductions, and direct numerical checks is a constructive feature. The absence of error estimates or explicit verification that the derived system satisfies the lattice equations to the stated order limits the immediate impact.

major comments (3)
  1. [Asymptotic analysis section (around the band-intersection point)] The multiple-scales derivation of the novel coupled NLS system (the central claim) does not contain an explicit degeneracy analysis at the flat-band/upper-surface intersection. It is therefore unclear whether the linearized operator admits a two-dimensional kernel and whether the slow-scale modulation remains free of secular terms arising from resonant interactions between the degenerate modes.
  2. [Abstract and the section presenting the novel coupled NLS] No error estimates or residual bounds are supplied for the asymptotic approximation, and there is no direct verification that the derived coupled system satisfies the original lattice equations to the claimed order (e.g., O(ε³)).
  3. [Multiple-scales expansion for the flat-band/upper-surface case] The validity limits of the reduction near the band-touching point are not discussed; standard single-mode NLS reductions are known to break down precisely when group velocities coincide and dispersion surfaces touch, yet the manuscript applies the usual weakly-nonlinear expansion without an adjusted scaling or joint leading-order ansatz.
minor comments (2)
  1. [Dispersion-relation section] Notation for the three dispersion branches and the precise wave-numbers at which the flat band touches the upper surface should be stated explicitly, together with the parameter values that realize this touching.
  2. [Lie-symmetry analysis section] The Lie-symmetry reductions of the coupled NLS system are summarized but the explicit symmetry generators and the resulting ODEs or reduced PDEs are not displayed.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment in detail below, indicating revisions made to strengthen the presentation of the asymptotic derivation and its validity.

read point-by-point responses

  1. Referee: The multiple-scales derivation of the novel coupled NLS system (the central claim) does not contain an explicit degeneracy analysis at the flat-band/upper-surface intersection. It is therefore unclear whether the linearized operator admits a two-dimensional kernel and whether the slow-scale modulation remains free of secular terms arising from resonant interactions between the degenerate modes.

    Authors: At the intersection point the linearized spatial operator possesses a two-dimensional kernel spanned by the flat-band and upper-surface eigenvectors. The multiple-scales ansatz in Section 3.3 incorporates both modes at leading order; the O(ε³) solvability conditions are obtained by projecting onto the adjoint kernel, which automatically removes secular terms and produces the coupled system. We have added an explicit paragraph (new Subsection 3.3.1) that states the kernel dimension, verifies the absence of resonant forcing at this order, and records the projection step. revision: yes

  2. Referee: No error estimates or residual bounds are supplied for the asymptotic approximation, and there is no direct verification that the derived coupled system satisfies the original lattice equations to the claimed order (e.g., O(ε³)).

    Authors: We agree that rigorous a-posteriori error bounds would increase the strength of the result. Deriving such bounds lies outside the present scope, which centers on formal derivation and Lie-symmetry reductions. We have inserted a short remark after the derivation that recalls the formal order of the remainder and notes that the numerical comparisons in Section 5 already provide practical evidence of consistency up to O(ε³). A separate, technically involved error analysis is left for future work. revision: partial

  3. Referee: The validity limits of the reduction near the band-touching point are not discussed; standard single-mode NLS reductions are known to break down precisely when group velocities coincide and dispersion surfaces touch, yet the manuscript applies the usual weakly-nonlinear expansion without an adjusted scaling or joint leading-order ansatz.

    Authors: At the touching point both bands have vanishing group velocity, so the standard slow-scale ansatz (X=εx, T=ε²t) remains balanced. We employ a joint two-mode leading-order ansatz precisely to capture the degeneracy; the resulting coupled system is the appropriate reduced model. We have added a paragraph in Section 3.3 that states the validity regime: wave-vectors within O(ε) of the intersection, amplitudes O(ε), and detuning of order ε². This regime avoids the breakdown that occurs for single-mode reductions away from the degeneracy. revision: yes

Circularity Check

0 steps flagged

Asymptotic derivation from lattice equations is self-contained

full rationale

The paper starts from the discrete Klein-Gordon lattice equations on the Kagome lattice, computes the linear dispersion relation showing a flat band touching the upper surface, and applies multiple-scales asymptotic expansion to obtain the coupled NLS system at that intersection point. No parameters are fitted to the target result, no self-citations justify the central reduction, and the steps follow standard perturbation methods without reducing the output to the inputs by construction. The derivation is therefore independent and self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard multiple-scales procedure and the assumed band structure of the linear problem; no new physical entities are introduced and no parameters appear to be fitted to data.

axioms (2)
  • domain assumption The nonlinear dynamics of the Kagome lattice are governed by discrete Klein-Gordon interactions with a scalar unknown at each node.
    Explicitly stated in the opening sentence of the abstract as the model under study.
  • domain assumption The linear dispersion relation possesses a flat band that meets the upper surface at a point permitting a specific asymptotic reduction.
    Invoked when the authors center the expansion at that intersection point.

pith-pipeline@v0.9.0 · 5687 in / 1529 out tokens · 48641 ms · 2026-05-20T22:58:44.106897+00:00 · methodology

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Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [1]

    Ablowitz, N

    M.J. Ablowitz, N. Antar, I. Bakurtas, B. Ilan, V ortex and dipole solitons in complex two-dimensional lattices. Phys Rev A, 86, 033804, (2012)

    work page 2012

  2. [2]

    Int J Solids & Structures, 318, 113387, (2025)

    R Augello, E Carrera, WQ Chen, B Wu, YZ Wang, Nonlinear static analysis of metamaterial structures based on the kagome lattice using beam finite elements and component-wise approach. Int J Solids & Structures, 318, 113387, (2025)

    work page 2025

  3. [3]

    Babicheva, A.S

    R.I. Babicheva, A.S. Semenov, E.G. Soboleva, A.A. Kudreyko, K. Zhou, S.V . Dmitriev. Discrete breathers in a triangular β-Fermi-Pasta-Ulam-Tsingou lattice. Phys Rev E, 103, 052202, (2021)

    work page 2021

  4. [4]

    Bahat-Treidel, O

    O. Bahat-Treidel, O. Peleg, M. Segev, H. Buljan, Breakdown of Dirac dynamics in honeycomb lattices due to nonlinear interactions, Phys Rev A, 82, 013830, (2010)

    work page 2010

  5. [5]

    Benjamin

    T.B. Benjamin. Instability of periodic wavetrains in nonlinear dispersive systems. Proc Roy Soc Lond A , 299, 59-75, (1967)

    work page 1967

  6. [6]

    Bluman, S

    G.W. Bluman, S. Kumei, Symmetries and Di fferential Equations, Springer, Berlin, (1989)

    work page 1989

  7. [7]

    Butt, J.A.D

    I.A. Butt, J.A.D. Wattis, Discrete breathers in a two-dimensional hexagonal Fermi-Pasta Ulam lattice. J Phys A, Math & Theor, 40, 1239–1264, (2007)

    work page 2007

  8. [8]

    Chern, C-C

    G-W. Chern, C-C. Chien, M. Di V entra, Dynamically generated flat-band phases in optical kagome lattices. Phys Rev A , 90, 013609, (2014)

    work page 2014

  9. [9]

    Chiao, E

    R.Y . Chiao, E. Garmire, and C.H. Townes, Self-trapping of optical beams, Phys. Rev. Lett. 13, 479–482, (1964)

    work page 1964

  10. [10]

    Davydova, A.I

    T.A. Davydova, A.I. Y akimenko, Y u.A. Zaliznyak, Two-dimensional solitons and vortices in normal and anomalous disper- sive media, Phys Rev E, 67, 026402, (2003)

    work page 2003

  11. [11]

    Di Sante, T

    D. Di Sante, T. Neupert, G. Sangiovanni, R. Thomale, R. Comin, J.G. Checkelsky, I. Zeljkovic, S.D. Wilson, Kagome metals, Rev Mod Phys, 98, 015002, (2026)

    work page 2026

  12. [12]

    Ferrari, F

    F. Ferrari, F. Becca, R. V alenti, Charge density waves in kagome-lattice extended Hubbard model at the van Hove filling. Phys Rev B, 106, L081107, (2022)

    work page 2022

  13. [13]

    Flach, A.V

    S. Flach, A.V . Gorbach, Discrete breathers-advances in theory and applications, Phys. Rep. 467, 1–116, (2008)

    work page 2008

  14. [14]

    R. Fu, J. Zhan, M. Durnagel, H. Hohmann, R. Thomale, J. Hu, Z. Wang, S. Zhou, X. Wu. Exotic charge-density waves and superconductivity on the kagome lattice. National Science Review, 12, nwaf414, (2025)

    work page 2025

  15. [15]

    Hofstrand

    A. Hofstrand. Families of discrete breathers on a nonlinear kagome lattice. Phys Rev E, 111, 064212, (2025)

    work page 2025

  16. [16]

    Hofstrand

    A. Hofstrand. Discrete breathers in a honeycomb lattice near a semi =Dirac point. (2026). https://arxiv.org/pdf/2602.07655

    work page arXiv 2026

  17. [17]

    Hofstrand, H

    A. Hofstrand, H. Li, M.I. Weinstein, Discrete breathers of nonlinear dimer lattices: bridging the anti-continuous and contin- uous limits, Journal of Nonlinear Science 33, 59, (2023)

    work page 2023

  18. [18]

    J-H. Im, C-J. Kang, Electronic structure and physical properties of the monolayer kagome lattice system A V3Sb5 (A = K, Rb, Cs). Curr Appl Phys 54, 26–30, (2023)

    work page 2023

  19. [19]

    Jacobsen, C.R

    J.L. Jacobsen, C.R. Scullard, Critical manifold of the kagome-lattice Potts model. J. Phys. A: Math. Theor ., 45, 494003 (2012)

    work page 2012

  20. [20]

    V .I. Karpman. Stabilization of soliton instabilities by higher-order dispersion: fourth-order nonlinear Schrödinger-type equa- tions. Phys Rev E, 53, R1336-R1339, (1996)

    work page 1996

  21. [21]

    Kartashov, B.A

    Y .V . Kartashov, B.A. Malomed, V .A. Vysloukh, L. Turner. Two-dimensional solitons in nonlinear lattices.Optics Letters, 34, 770–772, (2009). 21

    work page 2009

  22. [22]

    Kerner, M

    J. Kerner, M. Taufer, J. Wintermayr. Robustness of the flat bands on the perturbed kagome and the perturbed super-kagome lattice. Ann Henri Poincaré, 25, 3831–3857, (2024)

    work page 2024

  23. [23]

    Kirsch, Y

    M.S. Kirsch, Y . Zhang, M. Kremer, L.J. Maczewsky, S.K Ivanov, Y .V . Kartashov, L. Torner, D. Bauer, A. Szameit, M. Hein- rich. Nonlinear second-order photonic topological insulators, Nature Physics, 17, 995–1000, (2021)

    work page 2021

  24. [24]

    Kolokolov

    A.A. Kolokolov. Stability of the dominant mode of the nonlinear wave equation in a cubic medium. J Applied Mechanics and Technical Physics, 14, 426–428, (1973)

    work page 1973

  25. [25]

    K.J.H. Law, A. Saxena, P .G. Kevrekidis, A.R. Bishop. Localized structures in kagome lattices. Phys Rev A , 79, 053818, (2009)

    work page 2009

  26. [26]

    Lee, Dirac Cones for Point Scatterers on a Honeycomb Lattice, SIAM J Math Anal , 48, 1459–1488, (2016)

    M. Lee, Dirac Cones for Point Scatterers on a Honeycomb Lattice, SIAM J Math Anal , 48, 1459–1488, (2016)

    work page 2016

  27. [27]

    F.W.S. Lima, N. Crokidakis. Nonequilibrium opinion dynamics on triangular, honeycomb, and kagome lattices. Int J Mod Phys C, 28, 1750123, (2017)

    work page 2017

  28. [28]

    X. Liu, J. Lai, J.Z. Zhan, T. Y u, P . Liu, S. Y unoki, X-Q. Chen, Nonlinear optical response in kagome lattice with inversion symmetry breaking. ACS Nano 19, 19018–19026, (2025)

    work page 2025

  29. [29]

    MacKay, S

    R.S. MacKay, S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators., Nonlinearity, 7, 1623–1644, (1994)

    work page 1994

  30. [30]

    Maple, a division of Waterloo Maple Inc, Waterloo, Ontario, (2020)

    work page 2020

  31. [31]

    Marin, F.M

    J.L. Marin, F.M. Russell, and J.C. Eilbeck. Breathers in cuprate-like lattices, Phys. Lett. A 281, 21–25, (2001)

    work page 2001

  32. [32]

    Marín, J.C

    J.L. Marín, J.C. Eilbeck, and F.M. Russell. Localized moving breathers in a 2D hexagonal lattice, Phys. Lett. A, 248, 225–229, (1998)

    work page 1998

  33. [33]

    Olver, Applications of Lie Groups to Differential Equations, 2nd edn., Springer, New Y ork, (1993)

    P .J. Olver, Applications of Lie Groups to Differential Equations, 2nd edn., Springer, New Y ork, (1993)

    work page 1993

  34. [34]

    Olver, D.W

    F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark, Handbook of Mathematical Functions, NIST, CUP , Cambridge, (2010), dlmf.nist.gov [§16.17]

    work page 2010

  35. [35]

    Prabith, G

    K. Prabith, G. Theocharis, R. Chaunsali. Nonlinearity-induced corner states in a kagome lattice. New Journal of Physics, 27, 083501, (2025)

    work page 2025

  36. [36]

    C. Shi, R. Parker, P .G. Kevrekidis, M.I. Weinstein. Stability of flat-band solitons in nonlinear wave systems. (2025) https://arxiv.org/pdf/2511.05671v2

    work page arXiv 2025

  37. [37]

    Singh, A.C

    S. Singh, A.C. Garcia-Castro. Kagome KMn 3Sb5 metal: magnetism, lattice dynamics, and anomalous Hall conductivity. Phys Rev B, 108, 245143, (2023)

    work page 2023

  38. [38]

    Stephani, Differential Equations, Their Solution Using Symmetries, CUP , Cambridge, (1989)

    H. Stephani, Differential Equations, Their Solution Using Symmetries, CUP , Cambridge, (1989)

    work page 1989

  39. [39]

    Uzunok, S

    H.Y . Uzunok, S. Bagci, I. Sumer, K. Bashan, H.M. Tutuncu, Role of transition metal based kagome lattice in the physical properties of LaIr 5, LaIr3B2, LaIr3Ga2, LaRh3B2, and YRu 3B2 superconductors. Phys. Rev. B, 112, 214520, (2025)

    work page 2025

  40. [40]

    V akhitov, A.A

    N.G. V akhitov, A.A. Kolokolov. Stationary solutions of the wave equation in a medium with nonlinearity saturation. Radio- physics and Quantum Electronics , 16, 783–789, (1973)

    work page 1973

  41. [41]

    Vicencio, M

    R.A. Vicencio, M. Johansson, Discrete flat-band solitons in the kagome lattice, Phys Rev A, 87, 061803, (2013)

    work page 2013

  42. [42]

    Wattis, A.S.M

    J.A.D. Wattis, A.S.M. Alzaidi. Asymptotic analysis of breather modes in a two-dimensional mechanical lattice, Physica D, 401, 132207, (2020)

    work page 2020

  43. [43]

    J.A.D. Wattis. Asymptotic approximation of discrete breather modes in two-dimensional lattices. in Ed J.F.R. Archilla, et al (eds), Quodons in Mica, Springer series in materials Science, vol 221, Springer, Cham. (2015)

    work page 2015

  44. [44]

    Wattis, L.M

    J.A.D. Wattis, L.M. James. Discrete Breathers in honeycomb Fermi-Pasta-Ulam lattices. J Phys A, Math Theor , 47, 345101, (23pp), (2014)

    work page 2014

  45. [45]

    Whitham, Linear and Nonlinear Waves, Wiley, New Y ork, (1974), [Section 15.6]

    G.B. Whitham, Linear and Nonlinear Waves, Wiley, New Y ork, (1974), [Section 15.6]

    work page 1974

  46. [46]

    Proc Roy Soc Lond A, 479, 20220713, (2023)

    Z-J Y ang, Y -Z Wang, Non-Abelian mechanics of elastic waves in kagome metamaterials with internal structures. Proc Roy Soc Lond A, 479, 20220713, (2023). 22 -40 -20 0 20 40 n -0.5 0 0.5 qA 0,n, qB 0,n, qC 0,n Figure 7: Results of numerical simulations of breather mode with initial conditions given by (3.33), (3.21), (2.40), (2.11)–(2.12), then numericall...

    work page 2023