Internal solitary and cnoidal waves of moderate amplitude in a two-layer fluid: the extended KdV equation approximation

Dmitri Tseluiko; Karima Khusnutdinova; Nerijus Sidorovas; Wooyoung Choi

REVIEW 4 major objections 4 minor 12 references

Internal solitary and cnoidal waves of moderate amplitude in a two-layer fluid: the extended KdV equation approximation

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper derives the extended Korteweg–de Vries equation for internal waves in a two-layer fluid with linear shear currents and shows, by comparing with direct simulations of the strongly nonlinear parent model, that this reduced…

desk verdict Solid extension of the eKdV reduction for two-layer internal waves, with genuinely new coefficients and closed-form solutions; the numerical validation is narrower than the claims require, so a serious referee should push for filter-sensitivity and reproducibility. read the letter →

arxiv 2504.21513 v1 pith:R7BMQXNH submitted 2025-04-30 physics.flu-dyn nlin.PS

Nerijus Sidorovas , Dmitri Tseluiko , Wooyoung Choi , Karima Khusnutdinova This is my paper · ORCID

classification physics.flu-dynnlin.PS

keywords internalsolitarywavescnoidalextendedKdVequationGardnernear-identitytransformationtwo-layerfluidlinearshearcurrentsstronglynonlinearmodel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether the extended Korteweg–de Vries (eKdV) equation, derived from a strongly nonlinear two-layer fluid model with linear shear currents, can replace the full model for internal waves of moderate amplitude. Using a Kodama–Fokas–Liu near-identity transformation, the authors map this eKdV equation to an improved Gardner equation whose cubic coefficient and transport term differ from the commonly used truncated Gardner equation. They construct approximate solitary and cnoidal wave solutions through this mapping and test all reduced models against direct numerical simulations of the parent strongly nonlinear two-layer system. In the cases studied, the improved Gardner equation performs best for small and moderate amplitudes, the truncated Gardner equation performs best for large amplitudes near the critical depth ratio, and the eKdV equation tracks the best performer in every case while costing far less than the full model. The analytical approximations also serve as effective initial conditions for generating large-amplitude table-top solitary waves in the parent model.

What carries the argument

The load-bearing object is the nonlocal near-identity transformation (44), ζ = B − ε[aB² + bBξξ + cBξ(∫B dξ + f(T)) + dBTξ], with coefficients fixed at (45); it maps the derived eKdV equation (38) to the Gardner-type equation (46), whose cubic coefficient α₂ is renormalised according to (47), and a final change of variable (50) removes the transport term, yielding the improved Gardner equation (49). This matters because the Gardner equation has known explicit solitary-wave solutions (51) and cnoidal-wave solutions (74), so the inverse transformation turns those known solutions into approximate travelling-wave solutions (58) and (88) of the eKdV equation.

What would settle it

Run the same solitary-wave tests (ε = 0.15, hr = 1/2, ρr = 1.005⁻¹) with two or three different low-pass filter cutoff wavenumbers in the parent-model solver, or with an independent well-posed fully nonlinear solver, and compare the amplitudes and speeds of the evolved waves against eKdV, improved Gardner, and truncated Gardner predictions; if filter choice shifts the errors by more than the reported differences between the models, the claimed ranking is not robust.

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Extended reading notes

Core claim

The central claim is that the derived eKdV equation (38), with coefficients (39)–(42), is a good approximation to the strongly nonlinear parent model for moderate-amplitude internal waves, and that the improved Gardner equation (49), obtained by the near-identity transformation and carrying cubic coefficient (47) plus an extra transport term, is the better reduced model for small and moderate amplitudes, while the truncated Gardner equation performs better for large amplitudes near criticality. The constructed analytical solitary-wave solution (58) and cnoidal-wave solution (88) are close to the parent-model waves in the tested parameter range and, when used as initial conditions, evolve into the corresponding waves of the parent model, including the maximum-amplitude table-top solitary wave.

Load-bearing premise

The whole ranking of reduced models rests on the assumption that the parent model's numerical runs, which are stabilised by a hand-picked low-pass filter because the model is unstable to short waves, faithfully represent the true moderate-amplitude waves; if the filter alters their amplitude or phase, the comparison could favour the wrong equation.

Editorial extensions

If this is right

For moderate-amplitude internal waves (ε about 0.10–0.15) in a two-layer fluid away from the critical depth ratio, the eKdV equation reproduces the parent model's solitary and cnoidal wave evolution with errors clearly smaller than KdV and comparable to the best Gardner model, at a fraction of the computational cost.
The improved Gardner equation is the asymptotically correct reduced model for small and moderate amplitudes, while the truncated Gardner equation should be used for large amplitudes near the critical depth ratio hr = sqrt(ρr).
The analytical solitary and cnoidal solutions (58) and (88) can be used as initial conditions that evolve into the corresponding parent-model waves, including the maximum-amplitude table-top solitary wave, and the same construction is available for cases with linear shear currents since the derivation includes them.
In the tested regime the eKdV equation's range of validity is wider than that of its analytical approximations obtained via near-identity transformations, which impose additional smallness requirements on the amplitude parameter.

Reading between the lines

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

If the ranking holds, the improved Gardner equation should also outperform the truncated Gardner equation for other density ratios and weak shear currents at the same amplitude parameters, since the derivation includes shear but the numerical tests are current-free.
The same near-identity construction applied to the surface-wave limit (ρr → 0) would yield approximate solitary and cnoidal waves for surface waves with an underlying current, a regime the paper derives but does not test.
A stricter test of the ranking would repeat the comparisons with several low-pass filter cutoffs in the parent solver, or with a well-posed fully nonlinear solver; if the ordering of eKdV, improved Gardner, and truncated Gardner errors changes with the filter, the claim that eKdV is a reasonable universal model would need qualification.
The table-top soliton formation from the horned approximate solution suggests a controlled way to generate extreme-amplitude internal waves in laboratory experiments by initialising the fluid with the analytical eKdV solution at M very near M*.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Worth a serious look. The genuinely new pieces are real: the eKdV coefficients for two-layer flow with linear shear currents (Appendix A), the near-identity transformation to an improved Gardner equation with a modified cubic coefficient and phase-corrected transport term, and the closed-form cnoidal solutions (88)-(89). The current-free reduction checks out against Choi-Camassa, and the limits rho_r -> 0 and U_i -> 0 hit the surface-wave eKdV. That part is careful, honestly scoped asymptotics.

The numerical tests are where I would push. The MMCC parent is ill-posed and stabilized by a low-pass filter, but no k*, no resolution studies, and no filter-sensitivity analysis are reported. The L-infinity differences are of order 1e-2 to 1e-1, and the ranking between improved and truncated Gardner is exactly the kind of marginal comparison that can flip under regularization. So the stress-test note is right, though I would not call it fatal: the larger claim, that eKdV beats KdV and works at epsilon ~0.1-0.15, is supported by the scale of the differences and the two depth ratios tested. But the sharper claim, that improved Gardner wins at moderate amplitude while truncated wins near the critical depth ratio, needs better evidence before I would pass it to an oceanographer as a rule of thumb.

The O(epsilon^2) derivation is not shown ('too cumbersome'), and Appendix A is not independently checked. That is normal for this literature, but it makes the coefficient table hard to trust without code or a symbolic verification notebook. The cnoidal validation is also narrow: one or two peaks of an only-approximately-periodic solution, with a truncation choice that is itself an extra degree of freedom. They disclose all of this in Section 7, which I respect.

I would send it to peer review. The derivation and the closed forms are a real contribution, and the paper is honest about its boundaries. I would ask for release of the numerical code, a convergence/filter-sensitivity section, and preferably a symbolic check of Appendix A. Then the qualitative ranking claims would be on solid ground.

Referee Report

4 major / 4 minor

Summary. The paper derives an extended Korteweg-de Vries (eKdV) equation from the strongly nonlinear Miyata-Maltseva-Choi-Camassa (MMCC) two-layer model with linear shear currents, using a multiple-scale asymptotic expansion. It then applies a Kodama-Fokas-Liu near-identity transformation to map this eKdV equation to a Gardner equation with a modified cubic nonlinearity coefficient and an additional transport term, called the improved Gardner equation. Approximate solitary-wave and cnoidal-wave solutions of the eKdV equation are constructed from the corresponding Gardner solutions via the inverse transformation. The accuracy of the eKdV, improved Gardner, truncated Gardner, and KdV models is assessed by comparing their numerical solutions with direct simulations of the current-free MMCC parent system for depth ratios hr=1/2 and hr=1/3 (solitary waves) and hr=2 and hr=1/2 (cnoidal waves). The authors conclude that the improved Gardner equation is better for small and moderate amplitude waves, the truncated Gardner equation is better for large-amplitude waves near the critical depth ratio, and the eKdV equation is a robust compromise across amplitudes.

Significance. If the results hold, the paper makes a useful contribution: it provides parameter-free eKdV coefficients for two-layer flows with linear shear, clarifies the relation between eKdV and Gardner-type reductions, and offers explicit analytical approximations that can serve as initial conditions for nonlinear simulations. The derivation is strengthened by the stated limit checks: the current-free reduction to the Choi-Camassa eKdV equation, the rho_r -> 0 surface-wave reduction, and the simultaneous U_i -> 0, rho_r -> 0 reduction to the Green-Naghdi eKdV equation. The numerical study also includes an energy-conservation check for the MMCC simulation. The main value is in the practical recommendation that eKdV can replace the strongly nonlinear parent model for moderate amplitudes, and in the construction of pre-conditioners for generating large-amplitude MMCC solitary waves. The central uncertainty is the numerical validation, because the MMCC model is ill-posed and is regularized by a low-pass filter whose parameters are not documented; this affects the ranking of the reduced models.

major comments (4)

[Section 7 and Appendix B] The numerical validation is the load-bearing evidence for the ranking of the reduced models, but the regularization of the ill-posed MMCC parent model is not documented quantitatively. Section 7 states that the MMCC model is ill-posed due to the Kelvin-Helmholtz instability and that a careful choice of a low-pass filter is required, while Appendix B states only that wavenumbers |k| > k* are omitted at every RK4 step. No values of k*, grid resolutions, filter shapes, or convergence tests are reported. The reported L-infinity differences between models are on the order of 1e-2 to 1e-1, which is comparable to the differences between the improved and truncated Gardner models in several figures (e.g., Figures 4, 7, 9, 10). The ranking could therefore change under a different filter choice, and the conclusions in Section 7 are not yet robust. Please report the filter cutoff(s), resolution, and a systematic study showing that the computed MMCC solutions and the resulting model rankings are insensitive to the filter parameters within a reasonable range.
[Section 2 and Appendix A] The O(epsilon^2) computation leading to the eKdV equation (37) and the coefficients in Appendix A is not shown; the text says the equations are too cumbersome to write down. The limit checks provided are valuable but do not uniquely verify the coefficients, especially the current-dependent coefficients (A.1)-(A.19). Since the central claim that equations (38)-(42) and (47) are parameter-free approximations depends on the correctness of these coefficients, the derivation should be made available, for instance as a supplementary computer-algebra script or an explicit statement of the two O(epsilon^2) equations before the linear combination is taken. At minimum, please state the verification method used to obtain the coefficients and confirm whether they have been checked independently, e.g., by symbolic computation.
[Section 6 and equations (84)-(89)] The cnoidal-wave validation uses approximate solutions (84) that are only approximately periodic, as the text acknowledges, and the simulations require truncating after one or several peaks. The paper compares one-peak and seven-peak truncations in Figures 8 and 9 and reports qualitative differences, but does not provide a systematic convergence check with respect to the truncation length, the number of peaks, or the domain size. Since the cnoidal-wave conclusions depend on this truncation choice, please add a test showing that the L-infinity differences between the reduced models and MMCC stabilize as the truncation length and domain size are varied.
[Appendix B, equations (.1)-(.2)] The frame-boost parameter delta v is introduced in Appendix B but its values are not reported for any of the numerical experiments. The L-infinity differences used in the figures depend on the relative phase alignment of the profiles, which is affected by the frame velocity. Please report the delta v values used for each run and, ideally, show that the qualitative rankings are insensitive to delta v over a reasonable interval.

minor comments (4)

[Section 4, Figure 2 caption] The caption contains the typo 'red dimonds'; it should read 'red diamonds'.
[Appendix B, equation (.1)] The text contains the typo 'overvall velocity' and should read 'overall velocity'.
[References] References [10] and [30] appear to be the same paper (Horikis, Frantzeskakis and Smyth, Wave Motion 112 (2022) 102934); please consolidate or replace the duplicate.
[Appendix A] The notation in Appendix A is compact and difficult to verify; a short paragraph defining the structure of the expressions (e.g., which hatted coefficients are independent of v and which are auxiliary) would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: eKdV and Gardner coefficients are parameter-free asymptotic results and the MMCC comparisons are external validation, not fitted inputs.

full rationale

The derivation chain is self-contained at the level claimed. The eKdV coefficients (39)-(42) are explicit algebraic functions of hr and rho_r obtained by multiple-scale expansion of the parent system (20)-(21); no coefficient is fitted to MMCC output. The near-identity transformation (44) with coefficients (45) and the improved Gardner cubic coefficient (47) are fixed expressions derived from the requirement to remove the gamma_1, gamma_2 and beta_1 terms, and the free parameter M in (51) and the roots B_i in (74) parametrize solution families without tuning to the numerical comparisons. The validation in Sections 4 and 6 uses independently computed MMCC numerical solutions as the benchmark; although the MMCC model is also the starting point of the asymptotic derivation, solving it numerically is an external check and not an input to the reduction. The self-citations [27] (NIT method), [32] (numerical scheme) and [37] (preliminary comparisons) supply tools and context, but the load-bearing coefficients and solutions are re-derived in this paper. The admitted ill-posedness of MMCC and the hand-chosen low-pass filter (Section 7, Appendix B) is a numerical-robustness limitation that could affect the quantitative ranking, but it does not make any prediction equivalent to its input by construction. No circular step is found.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The derivation itself is parameter-free: the eKdV coefficients (39)-(42) and Appendix A, and the improved Gardner coefficient (47), follow from the multiple-scale expansion and the NIT without fitting to the target data. The free choices are the solution parametrization (M, B1-B3) and the numerical controls (filter cutoff, frame boost). The axioms are: the MMCC model as ground truth, the rigid-lid approximation, smallness of omitted O(epsilon^2) terms, negligibility of the NIT remainder, validity of truncating the approximately periodic cnoidal solution, and fidelity of the filtering procedure. No ad hoc physical entities are introduced; all new objects are derived models. The largest charges against the contribution are the unverified heavy algebra of Appendix A (only limit checks) and the by-hand numerical controls.

free parameters (4)

M (Gardner soliton amplitude parameter) = -0.25, -0.65, and truncations of M* about -1.307 (for epsilon = 0.15, hr = 0.5, rho_r = 1.005^-1)
One-parameter family of Gardner solitary waves (51); test values chosen by hand; M* = -alpha/(epsilon alpha2) is the table-top limit fixed by the coefficients, not fitted.
Cnoidal roots B1, B2, B3 = (-0.001, 0, 1), (0.001, 0, -1), (-1, 0, 1), with B4 from the root-sum condition (81)
Parametrize the approximate cnoidal waves (84); the elliptic modulus m and wavelength follow from these choices. The values are chosen by hand for each run.
Low-pass filter cutoff k* for MMCC simulations = not reported
Needed to suppress the Kelvin-Helmholtz instability of the ill-posed MMCC model; the paper only says the filter is chosen carefully and that runs fail for hr < 1/3.
Frame boost velocity delta v = 0.03515 quoted for one run
Keeps the evolving wave inside the moving computational frame; chosen per run, with no sensitivity study.

assumptions (6)

domain assumption The MMCC strongly nonlinear model [24], equations (8)-(11), is a faithful parent model at O(delta^4) accuracy.
All model comparisons take filtered MMCC solutions as ground truth; the fidelity of the parent model to real fluids is inherited from [23, 24] and is not tested here.
domain assumption The rigid-lid approximation filters the barotropic (surface) mode with negligible effect on the baroclinic (internal) mode.
Stated in Section 1 and used throughout the derivation.
domain assumption The multiple-scale expansions (24)-(25) with epsilon = delta^2, truncated at O(epsilon), capture the next-order wave dynamics at epsilon ~ 0.1-0.15.
Section 2: the eKdV model omits O(epsilon^2) terms, whose smallness at the tested amplitudes is asserted rather than demonstrated.
standard math The near-identity transformation (43)-(45) reduces the eKdV equation to the Gardner equation to O(epsilon^2) accuracy.
Kodama-Fokas-Liu normal-form theory [34-36], as streamlined in [27]; requires the remainder term (48) to vanish or be small.
ad hoc to paper For cnoidal waves the remainder term (48) is small when waves are sufficiently long, and a finite truncation of the only-approximately-periodic solution (84) is a valid initial condition.
Section 6 admits that the solution's amplitude grows away from the central peak; the truncation width (1, 3, or 7 peaks) is a hand choice that affects the results.
ad hoc to paper The low-pass filtering of the ill-posed MMCC model does not distort the moderate-amplitude long-wave dynamics being compared.
Section 7 admits the ill-posedness and the need for filtering; no filter-sensitivity study is provided, so this fidelity is assumed.

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Cite this review

Pith. "Pith review of Internal solitary and cnoidal waves of moderate amplitude in a two-layer fluid: the extended KdV equation approximation." pith.science (2026). https://pith.science/paper/R7BMQXNH

@misc{pith2026250421513,
  author       = {Pith},
  title        = {Pith review of: Internal solitary and cnoidal waves of moderate amplitude in a two-layer fluid: the extended KdV equation approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R7BMQXNH}},
  note         = {Machine review of arXiv:2504.21513}
}

read the original abstract

We consider travelling internal waves in a two-layer fluid with linear shear currents from the viewpoint of the extended Korteweg-de Vries (eKdV) equation derived from a strongly-nonlinear long-wave model. Using an asymptotic Kodama-Fokas-Liu near-identity transformation, we map the eKdV equation to the Gardner equation. This improved Gardner equation has a different cubic nonlinearity coefficient and an additional transport term compared to the frequently used truncated Gardner equation. We then construct approximate solitary and cnoidal wave solutions of the eKdV equation using this mapping and test validity and performance of these approximations, as well as performance of the truncated and improved Gardner and eKdV equations, by comparison with direct numerical simulations of the strongly-nonlinear two-layer long-wave parent system in the absence of currents.

Figures

Figures reproduced from arXiv: 2504.21513 by the authors.

**Figure 2.** The velocity c ∗ m of the maximum amplitude MMCC soliton, the linear long wave velocity c ∗ 0 , and the maximum velocity c ∗ 1 of the eKdV soliton corresponding to M = M∗ over hr when ρr = 1.005−1 . Note that c ∗ 1 is plotted for ε = 0.15. The vertical dotted line represents the critical depth ratio hr = √ ρr. The purple dots correspond to the parameter values at which approximate eKdV solitary waves were selected a… view at source ↗

**Figure 3.** Approximate solutions (58) of the eKdV equation fo [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Initial (top panels) and final (middle panels) solu [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: Initial (T = 0), intermediate (T = 5 and T = 10), and final (T = 50) solution profiles of the MMCC, truncated Gardner, improved Gardner, eKdV and KdV models for the M value corresponding to 10 significant figures of M∗ (for reference, M∗ ≈ −1.307), when ε = 0.15, hr = …

**Figure 6.** Figure 6: Left panel demonstrates the time evolution of the c [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: Comparison of MMCC, Gardner, and eKdV models under [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

**Figure 8.** Figure 8: Indeed, while the seven-peak truncation used in Fi [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]

**Figure 8.** Figure 8: Comparison of numerical results from the MMCC, eKd [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]

**Figure 9.** Figure 9: Comparison of numerical results from the MMCC, eKd [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]

**Figure 10.** Figure 10: Comparison of numerical results from the MMCC, Ga [PITH_FULL_IMAGE:figures/full_fig_p028_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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Horikis, D.J

T.P. Horikis, D.J. Frantzeskakis, T.R. Marchant, N.F. Smyth, Higher-dimensional ex- tended shallow water equations and resonant soliton radiat ion, Phys. Fluids 6 (2021) 104401

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T.P. Horikis, D.J. Frantzeskakis, T.R. Marchant, N.F. Smyth, Extended shallow water wave equations, Wave Motion 112 (2022) 102934

work page 2022

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S. Bager, N. Smyth, Whitham shocks and resonant dispersive shock waves governe d by the higher order Korteweg-de Vries equation , Proc. R. Soc. A 479 (2023) 20220580

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Fokas, Q.M

A.S. Fokas, Q.M. Liu: Asymptotic integrability of water waves , Phys. Rev. Lett. 77 (1996) 2347 - 2351

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