{"id":"49816fa8-e85d-48d2-9ce3-259b69889897","arxiv_id":"2504.21513","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"New eKdV coefficients for two-layer internal waves with shear currents, plus Gardner-mapped solitary and cnoidal solutions, are tested against strongly nonlinear simulations, where the improved Gardner equation performs best for small and moderate amplitudes.","lead":"The authors derive an extended KdV equation for internal waves in a two-layer fluid with shear currents, convert it to an improved Gardner equation, and build approximate solitary and cnoidal wave solutions. They test these approximations against numerical simulations of the strongly nonlinear parent model for moderate-amplitude waves.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The numerical validation uses the ill-posed MMCC parent model as ground truth, but the hand-chosen low-pass filter is never tested for distortion, so the ranking of reduced models could be an artifact of regularization.","rationale":"The reader's weakest-assumption analysis already identified the untested reliance on filtered MMCC solutions as the main threat to the validation chain, and the cnoidal truncation as a secondary issue. My independent reading agrees: the paper is transparent about both limitations, but does not quantify their effect. The derivation and near-identity transformation work appear internally consistent and the small set of parameter studies is honestly scoped, so the concern does not justify rejection. It does, however, justify keeping the verdict CONDITIONAL rather than ACCEPT, because a single filter-sensitivity experiment could either confirm or overturn the ranking of improved versus truncated Gardner. My concrete test is deliberately targeted at the most delicate comparison in Figures 4 and 7, where the improved Gardner model is claimed to be better for small/moderate amplitudes and truncated Gardner better near criticality. No additional concern about the coefficient algebra in Appendix A was raised because the current-free coefficients pass the stated limit checks and the numerical speed comparison in Figure 2 provides indirect support; the main unresolved risk remains the empirical ground truth.","tokens_in":26416,"tokens_out":7329,"duration_ms":87716,"concrete_test":"Repeat the M = -0.65 solitary-wave run from Figure 4 (epsilon=0.15, hr=0.5, rho_r=1.005^-1) with the MMCC solver using at least three cutoff wavenumbers k* (for example 0.5, 0.75, and 1.0 times the Nyquist wavenumber) and two spatial resolutions (N=1024 and N=2048). Compare the final profiles at T=200 by L-infinity difference, peak amplitude, and phase shift. If the spread among filtered MMCC runs is small compared with the L-infinity distances from MMCC to eKdV and to both Gardner models, the ranking is robust; if the spread is comparable, the validation is indeterminate and the reported ranking cannot be attributed to the physics of the reduced models.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that eKdV and the improved Gardner equation outperform truncated Gardner for moderate amplitudes rests on comparisons with MMCC simulations (Figures 4-10). Section 7 explicitly states that the MMCC model is 'known to be ill-posed due to the intrinsic short-wave Kelvin-Helmholtz (KH) instability' and that the authors relied on 'a careful choice of a low-pass filter to suppress unstable short waves.' Appendix B says the filter omits all wavenumbers |k| > k* at every RK4 step, but no k* values, resolutions, or convergence tests are reported. If the filter removes short-wave energy that is nonlinearly coupled to the long-wave component, or if it introduces small phase/amplitude shifts, the reported L-infinity differences (roughly 1e-2 to 1e-1) could be comparable to the differences between the reduced models themselves. The paper's ranking of improved Gardner versus truncated Gardner is exactly the kind of delicate comparison that could change under a different filter choice. The secondary cnoidal issue, namely that the approximate solution (84) is only locally periodic and must be truncated after one or a few peaks, also weakens the cnoidal validation, but the filter sensitivity affects all numerical evidence and is therefore the more load-bearing concern.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":26663,"tokens_out":3766,"duration_ms":44960,"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":[{"comment":"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":"Section 7 and Appendix B"},{"comment":"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":"Section 2 and Appendix A"},{"comment":"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.","section":"Section 6 and equations (84)-(89)"},{"comment":"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.","section":"Appendix B, equations (.1)-(.2)"}],"minor_comments":[{"comment":"The caption contains the typo 'red dimonds'; it should read 'red diamonds'.","section":"Section 4, Figure 2 caption"},{"comment":"The text contains the typo 'overvall velocity' and should read 'overall velocity'.","section":"Appendix B, equation (.1)"},{"comment":"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.","section":"References"},{"comment":"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.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of Physica D and the analytical derivation appears sound, with convincing limit checks. My main concern is the numerical validation: the ill-posed parent model is regularized by an undocumented filter, and the reported differences between reduced models are comparable to the likely regularization error. If the authors provide the missing filter data, convergence tests, and delta v values, and make the O(epsilon^2) derivation or its computer-algebra verification available, I would be willing to accept the paper. The duplicate reference [10]/[30] suggests a minor lapse in the reference list. No concerns about novelty disclosure: the relation to the earlier thesis [37] is acknowledged."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nI 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.","headline":"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.","tokens_in":27238,"tokens_out":1720,"would_cite":true,"duration_ms":18763,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["internal solitary waves","cnoidal waves","extended KdV equation","Gardner equation","near-identity transformation","two-layer fluid","linear shear currents","strongly nonlinear model"],"falsifier":"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.","tokens_in":26127,"feed_emoji":"🌊","tokens_out":7947,"duration_ms":76372,"temperature":0.7,"pith_summary":"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.","feed_headline":"eKdV reproduces moderate internal waves at a fraction of the cost","feed_subtitle":"Tests against the strongly nonlinear two-layer model show the improved Gardner equation is best for small and moderate amplitudes.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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*."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the strongly nonlinear parent (MMCC) model in current-free form and the maximum-amplitude solitary wave used for comparisons.","marker":"[23]"},{"why":"Establishes the strongly nonlinear two-layer model with linear shear currents from which the eKdV equation is derived.","marker":"[24]"},{"why":"Introduces the approach of using near-identity transformations to map an eKdV-type equation to the Gardner equation, which the paper streamlines.","marker":"[27]"},{"why":"Provides the numerical method (spectral discretisation, RK4 time stepping, sponge layers) and prior evidence that extended KdV models capture moderate-amplitude waves.","marker":"[32]"},{"why":"Gives the Kodama near-identity transformation framework for reducing higher-order KdV-type equations.","marker":"[34]"},{"why":"Complements the normal-form analysis with the near-identity transformation used here.","marker":"[35]"},{"why":"Supplies the Fokas–Liu full near-identity transformation including nonlocal terms.","marker":"[36]"},{"why":"Provides the four-root cnoidal wave solution of the Gardner equation in canonical form, adapted in the paper.","marker":"[38]"},{"why":"Documents the short-wave Kelvin–Helmholtz instability of the strongly nonlinear model, motivating the low-pass filter.","marker":"[39]"},{"why":"Describes how to stabilise the strongly nonlinear internal wave model, the basis for the filter used in the numerics.","marker":"[40]"}],"fun_headline_variants":["Improved Gardner equation tops eKdV for moderate internal waves","New mapping makes eKdV waves match parent model","Better Gardner equation from near-identity transform","eKdV and improved Gardner: moderate waves got it right","Truncated Gardner loses to improved for moderate amplitude"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Improved Gardner equation tops eKdV for moderate internal waves","New mapping makes eKdV waves match parent model","Better Gardner equation from near-identity transform","eKdV and improved Gardner: moderate waves got it right","Truncated Gardner loses to improved for moderate amplitude"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000584,"raw_usage":{"total_tokens":2692,"prompt_tokens":836,"completion_tokens":1856,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":452,"completion_tokens_details":{"reasoning_tokens":1779}},"tokens_in":452,"tokens_out":1856,"duration_ms":12785,"temperature":1.0,"reasoning_tokens":1779,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:03:17.867630+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Sidorovas, D","cited_arxiv_id":null,"evidence_quote":"Provides the numerical method (spectral discretisation, RK4 time stepping, sponge layers) and prior evidence that extended KdV models capture moderate-amplitude waves."},{"cited_title":"Kodama, On integrable systems with higher order corrections , Phys","cited_arxiv_id":null,"evidence_quote":"Gives the Kodama near-identity transformation framework for reducing higher-order KdV-type equations."},{"cited_title":"Kodama, Normal forms for weakly dispersive wave equations , Phys","cited_arxiv_id":null,"evidence_quote":"Complements the normal-form analysis with the near-identity transformation used here."},{"cited_title":"Fokas, Q.M","cited_arxiv_id":null,"evidence_quote":"Supplies the Fokas–Liu full near-identity transformation including nonlocal terms."},{"cited_title":"Kamchatnov, Y.-H","cited_arxiv_id":null,"evidence_quote":"Provides the four-root cnoidal wave solution of the Gardner equation in canonical form, adapted in the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Documents the short-wave Kelvin–Helmholtz instability of the strongly nonlinear model, motivating the low-pass filter."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Describes how to stabilise the strongly nonlinear internal wave model, the basis for the filter used in the numerics."}],"review_version":1}