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arxiv: 2604.09646 · v1 · submitted 2026-03-26 · 🧮 math.AP

The Kadomtsev-Petviashvili equation in conformal variables for waves over topography

David Andrade , Marcelo V. Flamarion This is my paper

Pith reviewed 2026-05-14 23:51 UTC · model grok-4.3

classification 🧮 math.AP
keywords Kadomtsev-Petviashvili equationconformal mappingwater wavestopographyasymptotic expansionweakly nonlinear wavesdispersive waveseffective depth
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The pith

A Kadomtsev-Petviashvili equation formulated in conformal variables models weakly transversal surface waves over topography.

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

The paper extends the conformal mapping technique from one-dimensional potential flows to cases with weak transverse dependence. Through asymptotic expansions it derives a Kadomtsev-Petviashvili-type equation whose coefficients incorporate the bottom via the Jacobian of the conformal map. This Jacobian supplies an effective depth that is treated as slowly varying, so the physical topography need not be smooth or even classically differentiable. A reader cares because the resulting model unifies several earlier weakly nonlinear dispersive equations while remaining numerically tractable for irregular seabeds. The derivation therefore supplies a consistent bridge between conformal-variable methods and transverse wave propagation over realistic bottoms.

Core claim

By means of asymptotic expansions, a Kadomtsev-Petviashvili type equation is obtained in conformal variables for weakly transversal surface waves propagating over topography. The topography enters the equation only through the slowly varying Jacobian of the conformal map, which defines an effective depth; this Jacobian need not correspond to a smooth or even classical function in the physical plane.

What carries the argument

Conformal mapping of the fluid domain to a fixed strip, with the Jacobian of the map supplying the effective depth as a slowly varying coefficient in the derived KP equation.

If this is right

  • The equation reduces to known one-dimensional models when transverse variation vanishes.
  • Irregular or non-smooth topography is admissible provided the effective depth remains slowly varying.
  • Numerical schemes already developed for conformal-variable water-wave problems can be reused for the new equation.
  • The model supplies a consistent extension of several existing weakly nonlinear dispersive wave equations reported in the literature.

Where Pith is reading between the lines

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

  • The same conformal-variable approach could be applied to derive transverse corrections for other asymptotic models such as the nonlinear Schrödinger equation.
  • If the slow-variation assumption is relaxed, higher-order corrections might be needed to capture sharper topographic features.
  • The formulation suggests that coastal-wave forecasting codes could incorporate realistic bathymetry data directly through precomputed conformal maps.

Load-bearing premise

The Jacobian of the conformal map is a slowly varying function of the horizontal coordinates.

What would settle it

A direct numerical comparison, for a chosen irregular topography and initial wave packet, between solutions of the derived equation and solutions of the full Euler equations in the same conformal coordinates.

Figures

Figures reproduced from arXiv: 2604.09646 by David Andrade, Marcelo V. Flamarion.

Figure 1
Figure 1. Figure 1: Blue line: Shape of the submerged topography [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Snapshots of the time evolution of an incoming wave at times [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
read the original abstract

The conformal mapping approach is a well established technique for solving the Euler equations for potential flows with one spatial dimension. In this work, we extend this framework to problems with a weakly transversal dependence and, by means of asymptotic expansions, obtain a Kadomtsev-Petviashvili type equation formulated in conformal variables as a model for weakly transversal surface waves propagating over topography. A key advantage of this formulation is that the topography, defined in the physical domain, does not need to be a smooth function, or even a function in the classical sense because, our asymptotic analysis relies on the effective depth, which comes through the Jacobian of the conformal map which is assumed to be a slowly varying function. The resulting equation provides a consistent extension of several well known weakly nonlinear dispersive wave models previously reported in the literature. Numerical simulations are performed to illustrate the newly derived equation.

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 / 2 minor

Summary. The manuscript extends the conformal mapping technique for the Euler equations of potential flow to include weak transversal dependence. By means of asymptotic expansions, it derives a Kadomtsev-Petviashvili-type equation formulated directly in conformal variables, intended as a model for weakly nonlinear, dispersive surface waves propagating over topography. Topography enters the model solely through the Jacobian of the conformal map, which is assumed to be a slowly varying function representing an effective depth; this allows the bottom to be non-smooth. The resulting equation is claimed to provide a consistent extension of several known weakly nonlinear dispersive models, and the work includes numerical simulations for illustration.

Significance. If the asymptotic reduction can be made rigorous with explicit parameter ordering, the conformal-variable KP model would supply a useful intermediate description between fully 2D Euler simulations and simpler 1D models, particularly for waves over irregular or non-differentiable topography where classical depth-averaged approaches may fail. The formulation preserves the advantage of conformal coordinates (exact free-surface boundary conditions) while incorporating weak transverse effects and variable effective depth, potentially simplifying numerical implementation. Credit is due for attempting to unify several existing models under one framework and for providing numerical illustrations.

major comments (2)
  1. [Asymptotic analysis and derivation of the KP-type equation] The slow-variation assumption on the Jacobian is load-bearing for the central claim yet lacks an explicit scaling regime. The abstract states that the Jacobian is 'assumed to be a slowly varying function,' but the derivation supplies no multiple-scale ansatz, no ordering relations among the small parameters (nonlinearity, dispersion, transversality, and bottom variation), and no truncation argument showing that derivatives of the Jacobian remain higher-order. Without this, it is unclear whether the claimed consistency with prior models survives when the bottom varies on scales comparable to the wave scales.
  2. [Consistency checks and limiting cases] The statement that the new equation 'provides a consistent extension of several well known weakly nonlinear dispersive wave models' is not supported by explicit reductions. The manuscript should demonstrate, by setting the Jacobian to a constant or to specific slowly varying profiles, that the standard KP, KdV, or other referenced models are recovered at leading order, including any necessary rescalings of variables and coefficients.
minor comments (2)
  1. [Abstract and numerical section] The abstract refers to 'numerical simulations' performed to illustrate the equation but gives no information on the test cases, initial conditions, domain sizes, or quantitative comparisons with known solutions or other models. These details should be added to the main text or a dedicated section.
  2. [Notation and setup] Notation for the conformal map, its Jacobian, and the slow scale should be introduced with explicit dependence on the slow variable made visible in the governing equations, to avoid ambiguity when the Jacobian varies.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We appreciate the positive remarks on the potential significance of the conformal-variable KP model. Below, we address each major comment point by point and outline the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: The slow-variation assumption on the Jacobian is load-bearing for the central claim yet lacks an explicit scaling regime. The abstract states that the Jacobian is 'assumed to be a slowly varying function,' but the derivation supplies no multiple-scale ansatz, no ordering relations among the small parameters (nonlinearity, dispersion, transversality, and bottom variation), and no truncation argument showing that derivatives of the Jacobian remain higher-order. Without this, it is unclear whether the claimed consistency with prior models survives when the bottom varies on scales comparable to the wave scales.

    Authors: We agree that the scaling regime should be made explicit to strengthen the derivation. In the revised manuscript, we will introduce a new subsection detailing the asymptotic analysis. We will define the small parameters: ε for the nonlinearity amplitude, δ for the dispersion (long-wave) parameter, γ for the weak transversality, and μ for the slow bottom variation. We adopt the ordering ε = O(δ²) = O(γ²) = O(μ), consistent with the standard KP scaling. The Jacobian J is assumed to vary on the slow scale, so that its derivatives are O(μ) and thus higher-order in the expansion. We will provide the multiple-scale ansatz and show the truncation at the appropriate order, ensuring the consistency holds within this regime. revision: yes

  2. Referee: The statement that the new equation 'provides a consistent extension of several well known weakly nonlinear dispersive wave models' is not supported by explicit reductions. The manuscript should demonstrate, by setting the Jacobian to a constant or to specific slowly varying profiles, that the standard KP, KdV, or other referenced models are recovered at leading order, including any necessary rescalings of variables and coefficients.

    Authors: We acknowledge that explicit demonstrations of the limiting cases would enhance the manuscript. In the revision, we will add a section on consistency checks and limiting cases. First, when the Jacobian is constant (flat bottom, μ=0), after rescaling the variables to standard form, the equation reduces to the classical KP equation. Second, in the one-dimensional limit (γ=0), it reduces to the KdV equation. We will also consider a specific slowly varying Jacobian, such as a linear function representing a constant slope, and derive the corresponding variable-coefficient model, showing it matches known extensions in the literature. These reductions will include the necessary rescalings and coefficient adjustments. revision: yes

Circularity Check

0 steps flagged

No circularity: standard asymptotic reduction from Euler equations

full rationale

The paper derives the KP-type model via asymptotic expansions applied to the Euler equations expressed in conformal coordinates. The slow-variation assumption on the Jacobian is stated explicitly as an input to the truncation and is not derived from or fitted to the target equation itself. No self-citations are invoked to justify uniqueness or to smuggle in an ansatz, and no parameter is fitted to a subset of data then relabeled as a prediction. The resulting equation is presented as a consistent extension of prior models, but the derivation chain remains independent of those models and does not reduce to them by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard assumption that the Jacobian of the conformal map varies slowly enough for the asymptotic reduction to be valid; no free parameters or new entities are introduced beyond this domain assumption.

axioms (1)
  • domain assumption The Jacobian of the conformal map is a slowly varying function
    Invoked to define effective depth and justify the asymptotic analysis for topography.

pith-pipeline@v0.9.0 · 5444 in / 1184 out tokens · 45772 ms · 2026-05-14T23:51:06.112205+00:00 · methodology

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Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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    Grimshaw, R. (1981). Evolution equations for long nonlinear internal waves in stratified shear flows.Studies in Applied Mathematics, 65:159–188

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    Nachbin, A. (2003). A terrain-following boussinesq system.SIAM Journal on Applied Mathematics, 63(3):905–922

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    Nachbin, A. (2023). Water wave models using conformal coordinates.Physica D: Nonlinear Phenomena, 445:133646

    work page 2023

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    Poletto, J. V. P., Andrade, D., Flamarion, M. V., and Ribeiro-Jr, R. (2025). Full euler equations for waves generated by vertical seabed displacements. SIAM Journal on Applied Mathematics, 85(4):1344–1360

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    Ruban, V. P. (2025). Nonlinear dynamics of waves over a nonuniformly periodic bottom.JETP Letters, 122:411–416

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    Viotti, C., Dutykh, D., Dudley, J. M., and Dias, F. (2013). Emergence of coherent wave groups in deep-water random sea.Phys. Rev. E, 87:063001. 13

    work page 2013