pith. sign in

arxiv: 2604.09432 · v1 · submitted 2026-04-10 · 🧮 math-ph · math.AP· math.MP

Heat and thermal travelling wave solutions of a nonlinear Maxwell-Cattaneo-Vernotte equation

Munaf\`o Carmelo Filippo , Rogolino Patrizia , Sciacca Michele This is my paper

Pith reviewed 2026-05-10 16:12 UTC · model grok-4.3

classification 🧮 math-ph math.APmath.MP
keywords Maxwell-Cattaneo-Vernotte equationtravelling wavessoliton solutionsnonlinear heat conductionthermal conductivityrelaxation timeexact solutionspolynomial dependence
0
0 comments X

The pith

Exact soliton solutions exist for the nonlinear Maxwell-Cattaneo-Vernotte equation when thermal conductivity and relaxation time are chosen as specific polynomials in temperature.

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

The paper examines how heat and thermal signals propagate as travelling waves under a nonlinear Maxwell-Cattaneo-Vernotte equation that incorporates finite signal speed. It derives exact solutions by assuming the thermal conductivity and relaxation time are polynomial functions of temperature, then identifies degrees of these polynomials that reduce the governing equation to an integrable form yielding solitons. A sympathetic reader cares because this supplies closed-form temperature profiles for heat pulses that neither dissipate nor spread indefinitely, addressing limitations of classical diffusion models in nonlinear or high-temperature regimes. The solutions are demonstrated through explicit expressions and plots for selected parameters.

Core claim

By expressing the thermal conductivity and the relaxation time as polynomial functions of the temperature, exact wave solutions including solitons are obtained for the nonlinear Maxwell-Cattaneo-Vernotte equation. This approach enables the identification of suitable degrees of nonlinearity that give rise to soliton solutions, with the resulting profiles shown through plots for chosen parameter values.

What carries the argument

Nonlinear Maxwell-Cattaneo-Vernotte equation with thermal conductivity and relaxation time expressed as polynomial functions of temperature, which reduces the PDE to an ordinary differential equation integrable to travelling-wave soliton forms.

If this is right

  • Heat signals propagate at finite speeds with persistent soliton profiles instead of instantaneous diffusion.
  • Material properties can be selected via the polynomial forms to produce desired non-dissipative thermal waves.
  • Analytical expressions allow direct computation of wave speed and shape without numerical approximation.
  • Plots confirm the soliton character for concrete choices of the polynomial coefficients and wave parameters.

Where Pith is reading between the lines

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

  • The polynomial assumption could be tested by measuring conductivity curves in candidate materials and checking whether observed heat pulses match the predicted soliton shapes.
  • Similar reductions might apply to other hyperbolic transport equations in contexts such as relativistic fluids or viscoelastic media.
  • If the method succeeds, it supplies benchmark solutions for validating numerical schemes that simulate nonlinear heat conduction in higher dimensions.

Load-bearing premise

Thermal conductivity and relaxation time can be expressed as polynomial functions of temperature with degrees that permit exact integration to bounded soliton solutions.

What would settle it

A direct integration attempt showing that no polynomial degrees produce solutions satisfying the boundary conditions of localization at infinity and constant propagation speed would disprove the existence of such solitons under this construction.

Figures

Figures reproduced from arXiv: 2604.09432 by Munaf\`o Carmelo Filippo, Rogolino Patrizia, Sciacca Michele.

Figure 1
Figure 1. Figure 1: Plots of the thermal conductivity of NaF crystal with respect to the temperature for some [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The temperature dependence of the propagation speed of second sound in He II [ [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: One dimensional domain Ω = [0, L] after many rearrangements, the dimensionless equations (15a)-(15b) become τp1 τq1  τq1 + Xm j=1 τqj+1 Tˆj   ∂tˆTˆ + ∂xˆqˆ = 0, (16a)  τq1 + Xm j=1 τqj+1 Tˆj   ∂tˆqˆ+ ˆq = − [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Plot of the soliton solutions (30): thermal dark soliton U(ξ) (left) and heat bright soliton V (ξ) (right), with the following set of parameters: A = 1, α = 1.25, γ0 = 0.08, γ1 = −0.01, β0 = 0.1, k = 0.2, w = 0.7, c1 = 0 and c2 = 0. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Plot of the soliton solutions (30) with the same parameters used in Fig.5 at different times. If we set A = wαγ1 √ −c˜1 2(w2αγ2 0 − k 2β0) B = (−c˜1)wαγ1 2k 15 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Sketch of two solitons, propagating along the cylinder: heat bright soliton waves (red [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Plot of the soliton solutions (30) in the xt-plane, namely u(x, t) and v(x, t), with x ∈ [0, 1] and t ∈ ×[0, 0.6], with the same choice of the parameters, namely: A = 1, α = 1.25, γ0 = 0.08, γ1 = −0.01, β0 = 0.1, k = 0.2, w = 0.7, c1 = 0 and c2 = 0. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Thermal dark soliton U(ξ) (left) and heat bright soliton V (ξ) (right) solution (43) for A0 = A2 = 1. Plots are achieved for the following set of parameters: α = 1.25, α0 = 0.08, α = 1.25, k = 0.8, w = 0.7, c0 = 1/(w α), b1 = −1, b2 = 1, c1 = 0 and c2 = 0. 6 Conclusions The growing interest in nonlinear phenomena led us to consider in this paper a nonlinear version of the Maxwell-Cattaneo-Vernotte equation… view at source ↗
Figure 9
Figure 9. Figure 9: Plot of the soliton solutions u(x, t) and v(x, t), for x ∈ [0, 40] and t ∈ ×[0, 100], ans A0 = A2 = 1 with the following parameter values: α = 1.25, α0 = 0.08, α = 1.25, k = 0.8, w = 0.7, c0 = 1/(w α), b1 = −1, b2 = 1, c1 = 0 and c2 = 0. presence of other solitons. This behavior can be further generalized to the case n = 2m, which allows a balance between the two terms in the numerator of the integral (20)… view at source ↗
Figure 10
Figure 10. Figure 10: Plots of the dark soliton solutions (43) for A0 ̸= A2, for some values of A2 and fixed A0 = 1 and the following parameter values: α = 1.25, α0 = 0.08, α = 1.25, k = 0.8, w = 0.7, c0 = 1/(w α), b1 = −1, b2 = 1, c1 = 0 and c2 = 0. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Plots of the dark soliton solutions (43) for A0 ̸= A2, for some values of A0 and fixed A2 = 1 and the following parameter values: α = 1.25, α0 = 0.08, α = 1.25, k = 0.8, w = 0.7, c0 = 1/(w α), b1 = −1, b2 = 1, c1 = 0 and c2 = 0. a comparison with experimental data in materials exhibiting thermal wave behavior would be of great interest in order to assess the physical applicability of the proposed model. F… view at source ↗
read the original abstract

The propagation of heat and thermal signals in the form of travelling waves is investigated for a nonlinear Maxwell-Cattaneo-Vernotte equation. The exact wave solutions are derived by expressing the thermal conductivity and the relaxation time as polynomial functions of the temperature. This approach enables the identification of suitable degrees of nonlinearity that give rise to soliton solutions. Finally, exact solutions are shown through plots for the values of the selected parameters.

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 / 1 minor

Summary. The paper claims that exact travelling wave solutions, including solitons, for the nonlinear Maxwell-Cattaneo-Vernotte equation can be obtained by expressing the thermal conductivity k(T) and relaxation time τ(T) as polynomial functions of temperature T. Suitable polynomial degrees are identified to permit closed-form integration after a travelling-wave reduction, and the resulting solutions are illustrated via plots for selected parameter values.

Significance. If the derivations and verifications hold, the work supplies closed-form soliton solutions for a nonlinear hyperbolic heat-conduction model under specific constitutive assumptions. Such exact solutions are useful as benchmarks for numerical schemes and for qualitative insight into finite-speed thermal waves when material properties depend on temperature. The polynomial-ansatz technique itself is standard, but its concrete application here adds to the catalogue of solvable cases in nonlinear heat equations.

major comments (2)
  1. [Abstract] Abstract: the claim that 'exact wave solutions are derived' by polynomial expressions for k(T) and τ(T) is not accompanied by any derivation steps, substitution back into the original PDE, or error analysis confirming consistency for the full parameter range. This verification is load-bearing for the central claim of exactness.
  2. [Travelling-wave reduction / solution construction] The reduction and integration steps (presumably in the main text following the model statement) must explicitly show how the chosen polynomial degrees close the ODE and produce the reported soliton profiles; without this, the post-hoc selection of degrees to enable solitons cannot be assessed for internal consistency.
minor comments (1)
  1. Plots should explicitly state the polynomial degrees chosen for k(T) and τ(T) together with the numerical values of all integration constants and wave speeds used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of our work and for the constructive comments on presentation. We address each major comment below and have revised the manuscript accordingly to improve clarity and explicitness of the derivations and verifications.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that 'exact wave solutions are derived' by polynomial expressions for k(T) and τ(T) is not accompanied by any derivation steps, substitution back into the original PDE, or error analysis confirming consistency for the full parameter range. This verification is load-bearing for the central claim of exactness.

    Authors: We agree that the abstract, being a concise summary, does not contain derivation steps or explicit verification. The full travelling-wave reduction, choice of polynomial degrees, integration, and direct substitution of the resulting solutions back into the original PDE (confirming consistency for the selected parameter values) are provided in Sections 3 and 4 of the manuscript. To address the concern, we have revised the abstract to include a brief reference to the verification process and have added an explicit consistency check subsection in the main text highlighting substitution for the reported solutions. revision: yes

  2. Referee: [Travelling-wave reduction / solution construction] The reduction and integration steps (presumably in the main text following the model statement) must explicitly show how the chosen polynomial degrees close the ODE and produce the reported soliton profiles; without this, the post-hoc selection of degrees to enable solitons cannot be assessed for internal consistency.

    Authors: The manuscript presents the travelling-wave ansatz leading to the reduced ODE, followed by the specific polynomial forms for k(T) and τ(T) that permit exact integration to the soliton profiles. The degrees are chosen to balance the nonlinear terms and close the equation under integration. We acknowledge that the steps could be presented more explicitly. In the revised version, we have expanded the relevant sections to display the reduced ODE before and after polynomial substitution, detail the integration procedure, and include direct substitution of the closed-form solutions into the original PDE to verify they hold identically. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs exact travelling-wave and soliton solutions for the nonlinear Maxwell-Cattaneo-Vernotte equation by assuming polynomial forms (of chosen degrees) for thermal conductivity and relaxation time as functions of temperature. This is a standard ansatz technique that reduces the PDE to an ODE and tunes coefficients for closed-form integration; the result is a set of particular solutions under those assumptions rather than a general derivation or prediction. No quoted step reduces by construction to a self-definition, a fitted input renamed as output, or a load-bearing self-citation chain. The derivation remains self-contained within its explicit modelling choices and does not invoke uniqueness theorems or external results that presuppose the target solutions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the ad-hoc introduction of polynomial temperature dependence for material coefficients and the standard travelling-wave reduction; these choices are not derived from first principles or external data but selected to permit exact solutions.

free parameters (1)
  • polynomial degrees for thermal conductivity and relaxation time
    Chosen by hand to allow the reduced ODE to integrate to soliton form; not fitted to experimental data.
axioms (2)
  • standard math A travelling-wave ansatz of the form T(x,t) = f(x - c t) reduces the PDE to an ODE
    Invoked implicitly as the method to obtain exact wave solutions.
  • ad hoc to paper Thermal conductivity k(T) and relaxation time tau(T) are polynomial functions of temperature
    Stated in the abstract as the enabling assumption for identifying suitable degrees of nonlinearity.

pith-pipeline@v0.9.0 · 5367 in / 1417 out tokens · 48561 ms · 2026-05-10T16:12:23.799464+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

29 extracted references · 29 canonical work pages

  1. [1]

    Joseph and L

    D. Joseph and L. Preziosi. Heat waves.Rev. Mod. Phys., 61:41–73, 1989

    work page 1989

  2. [2]

    M¨ uller and T

    I. M¨ uller and T. Ruggeri.Rational Extended Thermodynamics. Springer, 1998

    work page 1998

  3. [3]

    D. Jou, J. Casas-V´ azquez, and G. Lebon. Extended irreversible thermodynamics of heat trans- port: A brief introduction.Proceedings of the Estonian Academy of Sciences, 57(3), 2008

    work page 2008

  4. [4]

    D. Jou, G. Lebon, and J. Casas-V´ azquez.Extended irreversible thermodynamics: non-equilibrium equations of state. Springer, 2010

    work page 2010

  5. [5]

    Rogolino, R

    P. Rogolino, R. Kov´ acs, P. V´ an, and V. A. Cimmelli. Generalized heat-transport equations: parabolic and hyperbolic models.Continuum Mechanics and Thermodynamics, 30(6):1245–1258, 2018

    work page 2018

  6. [6]

    Munaf` o, P

    C.F. Munaf` o, P. Rogolino, and D. Jou. Comparison of two nonlinear formulations of the maxwell- cattaneo equation in heat pulse transmission.Applied Mathematical Modelling, 137:115684, 2025

    work page 2025

  7. [7]

    Ramos, A

    A. Ramos, A. Campelo, M. Freitas, and R. Kov´ acs. Numerical analysis of the maxwell-cattaneo- vernotte nonlinear model.Journal of Thermal Stresses, 47(9):1262–1276, 2024

    work page 2024

  8. [8]

    Sciacca, F

    M. Sciacca, F. X. Alvarez, D. Jou, and J. Bafaluy. Heat solitons and thermal transfer of infor- mation along thin wires.International Journal of Heat and Mass Transfer, 155:119809, 2020

    work page 2020

  9. [9]

    Sciacca and D

    M. Sciacca and D. Jou. Nonlinear thermal transport with inertia in thin wires: thermal fronts and steady states.Journal of Non-Equilibrium Thermodynamics, 47(2):187–194, 2022

    work page 2022

  10. [10]

    M. Sciacca. Two relaxation times and thermal nonlinear waves along wires with lateral heat exchange.Physica D: Nonlinear Phenomena, 423:132912, 2021

    work page 2021

  11. [11]

    Sciacca, X.F

    M. Sciacca, X.F. Alvarez, D. Jou, and J. Bafaluy. Thermal solitons along wires with flux-limited lateral exchange.Journal of Mathematical Physics, 62(10), 2021

    work page 2021

  12. [12]

    Sciacca, I

    M. Sciacca, I. Carlomagno, and A. Sellitto. Thermal solitons in nanotubes.Wave Motion, page 102967, 2022

    work page 2022

  13. [13]

    P. L. Kapitza. Heat transfer and superfluidity of helium ii.Physical Review, 60(4):354, 1941

    work page 1941

  14. [14]

    L. Dresner. Transient heat transfer in superfluid helium—part ii. InAdvances in Cryogenic Engineering: Volume 29, pages 323–333. Springer, 1984

    work page 1984

  15. [15]

    Mongiov` ı, D

    M.S. Mongiov` ı, D. Jou, and M. Sciacca.Non-Equilibrium Thermodynamics of Superfluid Helium and Quantum Turbulence. Springer, 2025

    work page 2025

  16. [16]

    McNelly.Second sound and anharmonic processes in isotopically pure alkali-halides

    T.F. McNelly.Second sound and anharmonic processes in isotopically pure alkali-halides. PhD thesis, Cornell University, 1974

    work page 1974

  17. [17]

    Jackson and C.T

    H.E. Jackson and C.T. Walker. Thermal conductivity, second sound, and phonon-phonon inter- actions in naf.Physical Review B, 3(4):1428–1439, 1971. 23

    work page 1971

  18. [18]

    Atkins and D.V

    K.R. Atkins and D.V. Osborne. The velocity of second sound below 1 k.The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 41(321):1078–1081, 1950

    work page 1950

  19. [19]

    C.T. Lane, H. A. Fairbank, and W.M. Fairbank. Second sound in liquid helium ii.Physical Review, 71(9):600, 1947

    work page 1947

  20. [20]

    Kov´ acs and P

    R. Kov´ acs and P. Rogolino. Numerical treatment of nonlinear fourier and maxwell-cattaneo- vernotte heat transport equations.International Journal of Heat and Mass Transfer, 150:119281, 2020

    work page 2020

  21. [21]

    Cattaneo

    C. Cattaneo. Sulla conduzione del calore.Atti del Seminario Matematico e Fisico dell’Universit` a di Modena, 3:3–21, 1948

    work page 1948

  22. [22]

    B. Toth. A unified mixed hp-finite element framework for modeling laser pulse heating processes in the refined thermodynamics.International Journal of Heat and Mass Transfer, 238:126456, 2025

    work page 2025

  23. [23]

    L. Onsager. Reciprocal relations in irreversible processes. i.Physical review, 37(4):405, 1931

    work page 1931

  24. [24]

    Munaf` o, P

    C.F. Munaf` o, P. Rogolino, and R. Kov´ acs. Nonlinear thermal analysis of two-dimensional mate- rials with memory.International Journal of Heat and Mass Transfer, 219:124847, 2024

    work page 2024

  25. [25]

    Munaf` o.Non-linear heat transfer beyond Fourier’s Law: analytical and numerical investi- gations

    C.F. Munaf` o.Non-linear heat transfer beyond Fourier’s Law: analytical and numerical investi- gations. PhD thesis, Universit` a degli Studi di Palermo, 2024

    work page 2024

  26. [26]

    G. P. Agrawal.Nonlinear Fiber Optics. Academic Press, 2001

    work page 2001

  27. [27]

    Hasegawa and Y

    A. Hasegawa and Y. Kodama.Solitons in Optical Communications. Oxford University Press, New York, 1995

    work page 1995

  28. [28]

    Kivshar and B

    Y.S. Kivshar and B. Luther-Davies. Dark optical solitons: physics and applications.Physics reports, 298(2-3):81–197, 1998

    work page 1998

  29. [29]

    Frantzeskakis

    D.J. Frantzeskakis. Dark solitons in atomic bose–einstein condensates: from theory to experi- ments.Journal of Physics A: Mathematical and Theoretical, 43(21):213001, 2010. 24

    work page 2010