pith. sign in

arxiv: 2606.20821 · v1 · pith:T4UDLZMXnew · submitted 2026-06-18 · 🧮 math.AP · math-ph· math.MP

Perturbative Approach to Nonlinear Capacitance Matrix Formulations

Habib Ammari , Clemens Thalhammer This is my paper

Pith reviewed 2026-06-26 16:14 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP
keywords nonlinear Helmholtz systemhigh-contrast inclusionsDirichlet-to-Neumann operatorcapacitance matrixperturbative expansionsolitonssubwavelength scalingsymmetry-breaking bifurcation
0
0 comments X

The pith

A rigorous two-way correspondence exists between continuous nonlinear Helmholtz solitons and discrete nonlinear capacitance systems via expansions in √δ.

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

The paper develops a perturbative method for a nonlinear Helmholtz system with cubic nonlinearity on high-contrast inclusions in three dimensions. It expands the resonant frequency and field in powers of the square root of the contrast δ using the Dirichlet-to-Neumann operator and capacitance formalism. This produces an algorithmic construction that gives a convergent analytic series for every discrete solution lifted to a continuous soliton, and reduces every continuous subwavelength family back to the discrete system. The same framework supplies higher-order corrections in both subwavelength and non-subwavelength regimes. Numerical examples illustrate the approach by tracking a symmetry-breaking bifurcation in a symmetric dimer.

Core claim

Using the Dirichlet-to-Neumann operator and a capacitance formalism, we develop a perturbative cascade that expands the resonant frequency and field in powers of √δ. Our main result is a rigorous two-way correspondence with a finite discrete nonlinear capacitance system: every discrete solution lifts to a continuous soliton (a convergent expansion, analytic in √δ), and every continuous family with the natural subwavelength scaling reduces to a discrete one. The construction is algorithmic, giving higher-order corrections in both the subwavelength and non-subwavelength regimes, the latter via a frequency-dependent capacitance matrix.

What carries the argument

The perturbative cascade in powers of √δ via the Dirichlet-to-Neumann operator and nonlinear capacitance formalism, which produces the exact lifting from discrete solutions to continuous solitons and the reduction from continuous families to discrete ones.

If this is right

  • Higher-order corrections to frequency and field can be computed algorithmically for any given discrete solution.
  • A frequency-dependent capacitance matrix yields the corresponding expansions outside the strict subwavelength regime.
  • Symmetry-breaking bifurcations in symmetric configurations such as dimers can be tracked through the discrete system and then lifted.
  • The correspondence applies uniformly to any finite collection of high-contrast inclusions supporting cubic nonlinearity.

Where Pith is reading between the lines

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

  • The discrete capacitance system may serve as a reduced-order model for designing nonlinear metamaterial responses at low computational cost.
  • The same perturbative structure could be tested on other nonlinearities or on inclusions with different shapes to check whether the lifting remains analytic.
  • If the correspondence holds, then stability properties proven for the discrete system would transfer directly to the continuous solitons for small δ.

Load-bearing premise

The high-contrast limit δ → 0 combined with subwavelength scaling permits a convergent perturbative cascade in √δ that establishes the exact lifting and reduction between the continuous nonlinear Helmholtz system and the discrete capacitance system.

What would settle it

A direct numerical solution of the continuous nonlinear Helmholtz equation for small but positive δ that deviates from the field predicted by lifting a discrete capacitance solution beyond the remainder bound of the √δ series would falsify the two-way correspondence.

Figures

Figures reproduced from arXiv: 2606.20821 by Clemens Thalhammer, Habib Ammari.

Figure 1
Figure 1. Figure 1: Comparison of perturbative approach versus exact solution for a single resonator with radius r = 0.1, interior wave speed v1 = 1/3.48 and exterior wave speed v = 1. (a) K-convergence of ω at δ = 0.002 (b) δ-convergence [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of perturbative approach versus multipole solution for a two sphere system with radii r = 0.1, distance 0.3, interior wave speed v1 = v2 = 1/3.48 and exterior wave speed v = 1. 7.1. Symmetry Breaking Bifurcations in Symmetric Dimers In this subsection, we study the nonlinear problem (2.2) for a symmetric dimer setup of two identical spherical resonators of radius r = 0.1, separated by a distance… view at source ↗
Figure 3
Figure 3. Figure 3: Bifurcation diagram and error convergence for a single resonator with radius r = 0.1, interior wave speed v1 = 1/3.48 and exterior wave speed v = 1. matrix has the form C =  a b b a , (7.1) with eigenvalues λ± = v 2 0 (a ± b)/|D0| corresponding to the symmetric and antisymmetric eigenmodes. Moreover, the entries a and b admit closed-form series expressions in bispherical coordinates [26], valid for all s… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of perturbative approach versus exact solution of the second resonance frequency for a single resonator with radius r = 0.1, interior wave speed v1 = 1/3.48 and exterior wave speed v = 1. (a) K-convergence of ω at δ = 0.01 (b) δ-convergence [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of perturbative approach versus multipole solution of the second resonance for a two sphere system with radii r = 0.1, distance 0.3, interior wave speed v1 = v2 = 1/3.48 and exterior wave speed v = 1. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Small amplitude soliton bifurcating from the second resonance frequency for a single resonator with radius r = 0.1, interior wave speed v1 = 1/3.48 and exterior wave speed v = 1. Antisymmetric branch (x = −y). The second factor vanishes, yielding σx2 = a − b − λ λ . This branch bifurcates from the trivial solution at λ = a − b along the antisymmetric eigendirection. Asymmetric branch (x ̸= ±y). Both factor… view at source ↗
Figure 7
Figure 7. Figure 7: Bifurcation diagram for discrete dimer model, the markers de￾note the snapshots for the subsequent plots. (a) K-convergence of ω at δ = 0.005 (b) δ-convergence at K = 9 (c) Plot of ℜ(u) at δ = 0.005 [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Error and field plots for the antisymmetric mode. Note the that in figure 8a the rate of convergence does not match the reference slope. Numerical evidence suggests that this is due to the exponential growth of the coefficient ωn, un. 8. Conclusion In this paper, we have studied the nonlinear Helmholtz problem with outgoing radiation condition and cubic nonlinearity in the high-contrast regime. Using pertu… view at source ↗
Figure 9
Figure 9. Figure 9: Error and field plots for the symmetric mode. As in Figure 3b, the error plateau in 9b is most likely due to truncation error. model: under a non-degeneracy assumption on the discrete data, every solution of the nonlinear capacitance system lifts to a continuous soliton, and every continuous soliton with the natural subwavelength scaling reduces to a discrete one. The capacitance matrix therefore faithfull… view at source ↗
Figure 10
Figure 10. Figure 10: Error and field plots for the asymmetric mode. One again observes the familiar error plateau in Figure 10b. References [1] Ambrosetti Antonio and Malchiodi Andrea, Nonlinear analysis and semilinear elliptic problems, vol. 104, Cambridge university press, 2007. [2] Ammari H. and Nédélec J.-C., “Full low-frequency asymptotics for the reduced wave equation”, in: Applied Mathematics Letters 12.1 (1999), pp. 1… view at source ↗
read the original abstract

We study a nonlinear Helmholtz system with cubic nonlinearity on high-contrast inclusions in three dimensions, and the solitons that emerge as the contrast $\delta$ tends to zero. Using the Dirichlet-to-Neumann operator and a capacitance formalism, we develop a perturbative cascade that expands the resonant frequency and field in powers of $\sqrt{\delta}$. Our main result is a rigorous two-way correspondence with a finite discrete nonlinear capacitance system: every discrete solution lifts to a continuous soliton (a convergent expansion, analytic in $\sqrt{\delta}$), and every continuous family with the natural subwavelength scaling reduces to a discrete one. The construction is algorithmic, giving higher-order corrections in both the subwavelength and non-subwavelength regimes, the latter via a frequency-dependent capacitance matrix. We illustrate the theory numerically and characterise a symmetry-breaking bifurcation in a symmetric dimer.

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

1 major / 2 minor

Summary. The manuscript develops a perturbative approach to a nonlinear Helmholtz system with cubic nonlinearity on high-contrast inclusions in three dimensions. Using the Dirichlet-to-Neumann operator and a capacitance formalism, it constructs an expansion of the resonant frequency and field in powers of √δ. The central claim is a rigorous two-way correspondence with a finite discrete nonlinear capacitance system: every discrete solution lifts to a continuous soliton via a convergent expansion analytic in √δ, and every continuous family with subwavelength scaling reduces to a discrete one. The construction is algorithmic, providing higher-order corrections in both subwavelength and non-subwavelength regimes (via a frequency-dependent capacitance matrix), and the theory is illustrated numerically by characterizing a symmetry-breaking bifurcation in a symmetric dimer.

Significance. If the two-way correspondence and convergence hold, the work supplies a rigorous bridge between continuous nonlinear PDE models and discrete capacitance systems for subwavelength solitons. The algorithmic construction of corrections and the numerical bifurcation example are concrete strengths that could support efficient analysis in nonlinear metamaterials or photonics.

major comments (1)
  1. [Main result (Theorem on two-way correspondence, likely §4)] The proof that the √δ perturbative cascade converges and remains analytic under the cubic nonlinearity (invoked via the Dirichlet-to-Neumann operator) requires explicit a-priori estimates that close the induction uniformly with respect to inclusion geometry. The current argument appears to extend linear-case bounds without fully controlling secular terms or remainder growth at each order, which is load-bearing for both the lifting and reduction directions of the main correspondence.
minor comments (2)
  1. The abstract states the setting is three-dimensional, but the title does not specify dimension; add a clarifying sentence in the introduction.
  2. [Numerical illustration] In the numerical section, state the mesh resolution and solver tolerance used for the dimer bifurcation computation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive summary and for identifying a point where the proof of the main two-way correspondence can be strengthened for clarity. We address the concern regarding a-priori estimates below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The proof that the √δ perturbative cascade converges and remains analytic under the cubic nonlinearity (invoked via the Dirichlet-to-Neumann operator) requires explicit a-priori estimates that close the induction uniformly with respect to inclusion geometry. The current argument appears to extend linear-case bounds without fully controlling secular terms or remainder growth at each order, which is load-bearing for both the lifting and reduction directions of the main correspondence.

    Authors: We appreciate the referee drawing attention to the need for explicit uniform estimates. In the proof of the main theorem (Theorem 4.1), the induction proceeds by constructing successive corrections to the field and frequency. Secular terms arising from the cubic nonlinearity are removed at each order by an algebraic correction to the resonant frequency, which is always solvable because the leading-order capacitance matrix is positive definite. The remainder after N orders is controlled in the appropriate Sobolev space by combining the boundedness of the Dirichlet-to-Neumann map (uniform under the fixed-geometry, high-contrast assumptions) with the Lipschitz continuity of the cubic term. These bounds are independent of δ and of the particular inclusion shapes provided the inclusions remain separated and have C^2 boundaries. Nevertheless, we agree that the current write-up compresses the induction closure and does not isolate the uniform-in-geometry constants. In the revision we will insert a new auxiliary lemma (after Lemma 4.3) that states the precise a-priori estimate on the remainder and verifies that the induction constant remains bounded uniformly with respect to the inclusion geometry. This will also make the control of remainder growth explicit at each order. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit perturbative construction

full rationale

The paper constructs an explicit perturbative cascade in powers of √δ using the Dirichlet-to-Neumann operator and capacitance formalism, then proves a two-way correspondence between discrete nonlinear capacitance solutions and continuous solitons. No step reduces a claimed prediction to a fitted input by construction, nor does any load-bearing premise rest solely on self-citation of an unverified uniqueness result. The analyticity and convergence claims are presented as theorems derived from the cascade, not presupposed. The numerical illustrations are post-hoc verification rather than the source of the result. This is the normal case of a self-contained mathematical derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no specific free parameters, axioms, or invented entities can be identified from the text.

pith-pipeline@v0.9.1-grok · 5666 in / 1099 out tokens · 36012 ms · 2026-06-26T16:14:45.144801+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

31 extracted references · 2 canonical work pages

  1. [1]

    104, Cambridge university press, 2007

    AmbrosettiAntonio andMalchiodiAndrea,Nonlinear analysis and semilinear elliptic problems, vol. 104, Cambridge university press, 2007

    2007

  2. [2]

    Full low-frequency asymptotics for the reduced wave equation

    AmmariH. andNédélecJ.-C., “Full low-frequency asymptotics for the reduced wave equation”, in:Applied Mathematics Letters12.1 (1999), pp. 127–131

    1999

  3. [3]

    Functional Analytic Methods for Discrete Approximations of Subwavelength Resonator Systems

    AmmariHabib,DaviesBryn, andHiltunenErik Orvehed, “Functional Analytic Methods for Discrete Approximations of Subwavelength Resonator Systems”, in:Pure and Applied Analysis6.3 (2024)

    2024

  4. [4]

    136, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 2026

    AmmariHabib,DaviesBryn, andHiltunenErik Orvehed,Mathematical Theories for Metamaterials: From Condensed Matter Theory to Subwavelength Physics, vol. 136, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 2026

    2026

  5. [5]

    Minnaert resonances for acoustic waves in bubbly media

    AmmariHabib,FitzpatrickBrian,GontierDavid,LeeHyundae, andZhangHai, “Minnaert resonances for acoustic waves in bubbly media”, in:Annales de l’Institut Henri Poincaré C, Analyse non linéaire35.7 (2018), pp. 1975–1998

    2018

  6. [6]

    Nonlinear Subwavelength Resonances in Three Dimensions

    AmmariHabib andKoscheThea, “Nonlinear Subwavelength Resonances in Three Dimensions”, in:Studies in Applied Mathematics154.3 (2025)

    2025

  7. [7]

    Dielectric scattering resonances for high-refractive resonators with cubic nonlinearity

    AmmariHabib andLiBowen, “Dielectric scattering resonances for high-refractive resonators with cubic nonlinearity”, in:Journal of Differential Equations475 (2026), p. 114459. 26

    2026

  8. [8]

    Frequency-dependent capac- itance matrix formulation for Fabry-P\’erot resonances. Part I: One-dimensional finite systems

    AmmariHabib,LiBowen,LiuPing, andShaoYingjie, “Frequency-dependent capac- itance matrix formulation for Fabry-P\’erot resonances. Part I: One-dimensional finite systems”, in:arXiv preprint arXiv:2604.01159(2026)

    arXiv 2026

  9. [9]

    AmmariHabib,LiBowen,LiuPing,ShaoYingjie, andUhlmannAlexander, Frequency-dependent capacitance matrix formulation for Fabry-Perot resonances in two and three dimensional systems, 2026, arXiv:2605.27572 [math.AP]

    Pith/arXiv arXiv 2026

  10. [10]

    Analysis of nonlinear resonances in resonator crystals: Tight-binding approximation and existence of subwavelength soliton-like solutions

    AmmariHabib andQiuJiayu, “Analysis of nonlinear resonances in resonator crystals: Tight-binding approximation and existence of subwavelength soliton-like solutions”, in: arXiv preprint arXiv:2509.04184(2025)

    arXiv 2025

  11. [11]

    A radiation and propagation problem for a Helmholtz equation with a compactly supported nonlinearity

    AngermannLutz, “A radiation and propagation problem for a Helmholtz equation with a compactly supported nonlinearity”, in:Communications in Nonlinear Science and Numerical Simulation126 (2023), p. 107422

    2023

  12. [12]

    Symmetry breaking regime in the nonlinear Hartree equation

    AschbacherWH,FröhlichJ,GrafGM,SchneeK, andTroyerM, “Symmetry breaking regime in the nonlinear Hartree equation”, in:Journal of Mathematical Physics 43.8 (2002), pp. 3879–3891

    2002

  13. [13]

    Gaps in the spectrum of two-dimensional square packing of stiff disks

    D’EliaL andNazarovSA, “Gaps in the spectrum of two-dimensional square packing of stiff disks”, in:Applicable Analysis102.9 (2023), pp. 2611–2627

    2023

  14. [14]

    Roadmap on metamaterial theory, modelling and design

    DaviesBryn,SzyniszewskiStefan,DiasMarcelo A,W aalLeo de,KisilAnastasia, P SmyshlyaevValery,CooperShane,KamotskiIlia V,TouboulMarie,Craster Richard V, et al., “Roadmap on metamaterial theory, modelling and design”, in:Journal of Physics D: Applied Physics58.20 (2025), p. 203002

    2025

  15. [15]

    200, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2019

    DyatlovSemyon andZworskiMaciej,Mathematical theory of scattering resonances, vol. 200, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2019

    2019

  16. [16]

    The discrete self-trapping equation

    EilbeckJ Ch,LomdahlPS, andScottAlwyn C, “The discrete self-trapping equation”, in:Physica D: Nonlinear Phenomena16.3 (1985), pp. 318–338

    1985

  17. [17]

    Dual variational methods and nonvanishing for the nonlinear Helmholtz equation

    EvequozGilles andWethTobias, “Dual variational methods and nonvanishing for the nonlinear Helmholtz equation”, in:Advances in Mathematics280 (2015), pp. 690–728

    2015

  18. [18]

    Subwavelength resonant acoustic scattering in fast time-modulated media

    FepponF. andAmmariH., “Subwavelength resonant acoustic scattering in fast time-modulated media”, in:Journal de Mathématiques Pures et Appliquées187 (2024), pp. 233–293

    2024

  19. [19]

    Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies

    FepponFlorian andAmmariHabib, “Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies”, in:Studies in Applied Mathematics149.1 (2022), pp. 164–229

    2022

  20. [20]

    Subwavelength Resonances in One-Dimensional High-Contrast Acoustic Media

    FepponFlorian,ChengZijian, andAmmariHabib, “Subwavelength Resonances in One-Dimensional High-Contrast Acoustic Media”, in:SIAM J. Appl. Math.83.2 (2023), pp. 625–665

    2023

  21. [21]

    SingularperturbationDirichletproblemin a double-periodic perforated plane

    FerraressoFrancescoandTaskinenJari,“SingularperturbationDirichletproblemin a double-periodic perforated plane”, in:ANNALI DELL’UNIVERSITA’DI FERRARA 61.2 (2015), pp. 277–290

    2015

  22. [22]

    Spatial solitons in periodic nanostruc- tures

    GorbachAndriy V andSkryabinDmitry V, “Spatial solitons in periodic nanostruc- tures”, in:Physical Review A—Atomic, Molecular, and Optical Physics79.5 (2009), p. 053812

    2009

  23. [23]

    Inverse medium scat- tering for a nonlinear Helmholtz equation

    GriesmaierRoland,KnöllerMarvin, andMandelRainer, “Inverse medium scat- tering for a nonlinear Helmholtz equation”, in:Journal of Mathematical Analysis and Applications515.1 (2022), p. 126356

    2022

  24. [24]

    KielhoferHansjorg,Bifurcation Theory, en, 2nd ed., Applied Mathematical Sciences (Springer), Springer Science+Business Media, Jan. 2012

    2012

  25. [25]

    Symmetry- breaking bifurcation in nonlinear Schrödinger/Gross–Pitaevskii equations

    KirrEW,KevrekidisPG,ShlizermanEli, andWeinsteinMichael I, “Symmetry- breaking bifurcation in nonlinear Schrödinger/Gross–Pitaevskii equations”, in:SIAM journal on mathematical analysis40.2 (2008), pp. 566–604. 27

    2008

  26. [26]

    Electrostatics of two charged conducting spheres

    LeknerJohn, “Electrostatics of two charged conducting spheres”, in:Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences468.2145 (2012), pp. 2829–2848

    2012

  27. [27]

    XVI. On Musical Air-Bubbles and the Sounds of Running Water

    MinnaertM., “XVI. On Musical Air-Bubbles and the Sounds of Running Water”, in: The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 16.104 (1933), pp. 235–248,doi:10.1080/14786443309462277

    work page doi:10.1080/14786443309462277 1933

  28. [28]

    NečasJindřich,Direct Methods in the Theory of Elliptic Equations, Springer Berlin Heidelberg, 2012,doi:10.1007/978-3-642-10455-8

    work page doi:10.1007/978-3-642-10455-8 2012

  29. [29]

    Gap solitons in periodic discrete nonlinear Schrödinger equations

    PankovA, “Gap solitons in periodic discrete nonlinear Schrödinger equations”, in: Nonlinearity19.1 (2005), p. 27

    2005

  30. [30]

    Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities

    PankovAlexander, “Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities”, in:Journal of mathematical analysis and applications 371.1 (2010), pp. 254–265

    2010

  31. [31]

    Perturbation theory for nonlinear Schrödinger equations

    SacchettiAndrea, “Perturbation theory for nonlinear Schrödinger equations”, in: Nonlinearity36.11 (2023), pp. 6048–6070. Habib Ammari ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland, orcid.org/0000-0001-7278-4877 . Email address:habib.ammari@math.ethz.ch Clemens Thalhammer ETH Zürich, Department of Mathematics, Rämistrasse...

    2023