Generalized Error Bounds in the Recovery of Solitary Wave Profiles

arxiv: 2605.15590 · v1 · pith:UUWNHMJPnew · submitted 2026-05-15 · 🧮 math.CA · math-ph· math.MP· nlin.PS

Generalized Error Bounds in the Recovery of Solitary Wave Profiles

Daniel Sinambela This is my paper

Pith reviewed 2026-05-19 18:39 UTC · model grok-4.3

classification 🧮 math.CA math-phmath.MPnlin.PS

keywords solitary water wavesprofile reconstructionerror boundshodograph transformL2 stabilityPaley-Wiener estimates

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The pith

Perturbations in wave speed, depth, and bed pressure yield sublinear L² errors when recovering solitary wave profiles via Constantin's formula.

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

The paper establishes that Constantin's explicit reconstruction formula for the free-surface profile of two-dimensional irrotational solitary water waves remains stable when the wave speed, undisturbed depth, and dynamic pressure trace at the bed are all slightly perturbed. It derives an L² error estimate between the true and reconstructed profiles by applying the hodograph transform to convert the free-boundary problem into a fixed-domain holomorphic setting, then using holomorphic extension and Paley-Wiener Fourier decay estimates to control how errors propagate. A sympathetic reader would care because this supplies a concrete stability guarantee for an otherwise exact formula, showing that small inaccuracies in measured or estimated inputs do not destroy the recovery in the L² sense. The analysis includes numerical examples that illustrate the effect of specifically chosen perturbations.

Core claim

For two-dimensional irrotational solitary water waves, simultaneous perturbations in wave speed, undisturbed depth, and the dynamic pressure at the bed produce an L² error in the reconstructed free-surface profile that is controlled by a sublinear function of the total perturbation size. The proof maps the problem via the hodograph transform into a strip where the velocity potential extends holomorphically, invokes Paley-Wiener decay on the Fourier side to quantify the difference, and thereby obtains stability estimates that improve on purely linear bounds.

What carries the argument

Constantin's explicit reconstruction formula for the free-surface elevation from the bed pressure trace, combined with the hodograph transform that straightens the unknown free boundary into a fixed strip and permits holomorphic extension together with Paley-Wiener estimates.

If this is right

The reconstruction formula can be applied reliably to mildly inaccurate experimental data without losing L² accuracy.
Sublinear error growth means that halving the perturbation size reduces the profile error by a factor better than one-half.
Numerical tests with designed perturbations confirm that the theoretical bound captures the observed reconstruction error.
The result supplies a quantitative justification for using bottom-pressure measurements to infer surface profiles under realistic noise levels.

Where Pith is reading between the lines

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

Bottom-pressure sensors could be used for surface-profile inference in field measurements with explicit uncertainty estimates derived from this stability result.
The holomorphic and Fourier-decay techniques may carry over to other free-boundary inverse problems in ideal fluid dynamics.
The sublinear dependence suggests that certain directions of perturbation in the three input quantities are less damaging than a generic linear analysis would predict.

Load-bearing premise

The flow must be a two-dimensional irrotational solitary wave for which the hodograph transform and holomorphic extension are valid, and the perturbations in speed, depth, and pressure must be small enough for the estimates to hold.

What would settle it

Take a known exact solitary wave solution, apply small controlled perturbations to its speed, depth, and bed pressure, reconstruct the profile with the perturbed formula, and check whether the observed L² difference between true and reconstructed profiles obeys the sublinear bound or exceeds it.

Figures

Figures reproduced from arXiv: 2605.15590 by Daniel Sinambela.

**Figure 1.** Figure 1: The Hodograph transform denote the velocity field in the frame moving at speed c. The steady Euler equations (2.1) in this frame read (w · ∇)w = −∇P − g yˆ, ∇ · w = 0, ∇ × w = 0, yˆ = (0, 1). Using the irrotationality condition and the vector identity (w · ∇)w = ∇ [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: Reconstruction error due to wave-speed perturbation 5.2. Pressure Perturbation. In this subsection, we consider the case in which only the pressure trace is perturbed, while the wave-speed parameter and the depth remain fixed. More precisely, we set ε = 0, γ = 0, and perturb only the bed pressure through pδ(q) = p(q) + δ(q). The perturbation is different from the one used in [Ble16]. The perturbed pressure… view at source ↗

**Figure 3.** Figure 3: Reconstruction error due to bed-pressure perturbation 5.3. Depth Perturbation. Now, we consider the case in which only the fluid depth is perturbed, while the wave-speed parameter and the pressure trace remain fixed. More precisely, we set ε = 0, δ = 0, and allow only the depth to vary through dγ = d + γ. Therefore, the perturbed recovered profile takes the form η0,0,γ(q) = F −1 sinh(k(d + γ)) k gb0,0(k… view at source ↗

**Figure 4.** Figure 4: Reconstruction error due to depth perturbation [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

read the original abstract

We investigate the robustness of Constantin's explicit reconstruction formula for two-dimensional irrotational solitary water waves. This formula recovers the free-surface profile from the dynamic pressure trace at the bed and depends on both the wave speed and the undisturbed depth. We consider simultaneous perturbations in these three quantities and derive an $L^2$ error estimate for the reconstructed profile. The proof uses the hodograph transform, holomorphic extension arguments, and Paley--Wiener Fourier-decay estimates, yielding stability estimates with sublinear dependence on the perturbation size. We include numerical computations to illustrate the effects of specifically designed perturbations.

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.

Desk Editor's Note private letter to a colleague

The paper adds sublinear L2 stability bounds to Constantin's reconstruction under joint perturbations in speed, depth, and pressure, using standard hodograph and Paley-Wiener tools.

read the letter

The core contribution is an L2 error bound for the recovered free-surface profile when the input data (bed pressure, wave speed, and undisturbed depth) are all perturbed at once. The bound depends sublinearly on the total perturbation size. The proof follows the usual route: hodograph transform to fix the domain, holomorphic extension of the pressure trace, and Paley-Wiener decay to control the difference from the unperturbed profile. Numerical examples with crafted perturbations are included to show the behavior in practice. This is a direct, incremental advance on the existing reconstruction formula rather than a new method from scratch. The analytic tools are applied cleanly and the sublinear dependence is the part that goes beyond the original work. The numerics are helpful for seeing that the constants are not absurdly large. The main soft spot is the holomorphic strip width under perturbation. The unperturbed solitary wave has a positive extension radius by known results, but the argument must show that the radius for the perturbed data does not collapse fast enough to destroy the decay rate. If the width δ(ε) tends to zero, the exponential tail in the Fourier estimate weakens and the claimed sublinear bound may require extra work to hold uniformly. The paper assumes perturbations are small enough for the estimates to apply, but an explicit lower bound on δ or a quantitative continuity statement for the extension radius would remove the doubt. No circularity or invented parameters appear. This is for specialists in inverse problems for water waves and analytic PDE methods. A reader already familiar with Constantin's formula and holomorphic techniques will see the value in the joint-perturbation bounds and the numerics. The work is coherent on its own terms and the central claim is new, so it deserves a serious referee even if the strip-width detail needs tightening in revision.

Referee Report

1 major / 2 minor

Summary. The manuscript derives an L² error estimate for the free-surface profile recovered via Constantin's explicit reconstruction formula for two-dimensional irrotational solitary water waves, under simultaneous small perturbations to the bed dynamic pressure trace, wave speed, and undisturbed depth. The proof applies the hodograph transform to obtain a holomorphic formulation, followed by holomorphic extension and Paley-Wiener Fourier decay estimates, yielding stability bounds with sublinear dependence on the total perturbation size; numerical examples illustrate the effect of specific perturbations.

Significance. If the estimates are rigorously established, the result strengthens the practical utility of the reconstruction formula by quantifying its stability under realistic measurement errors in multiple parameters. The combination of classical complex-analytic tools with numerics is a positive feature, and the sublinear dependence (rather than linear) would be a nontrivial improvement over naive bounds.

major comments (1)

The central L² stability claim rests on applying Paley-Wiener decay to the holomorphic extension of the perturbed pressure trace after the hodograph transform. No explicit lower bound δ(ε) ≥ δ₀ > 0 (independent of the perturbation size ε) on the width of the holomorphic strip for the simultaneously perturbed data is derived or cited; if δ(ε) → 0 as ε → 0, the exponential decay rate deteriorates and the claimed sublinear error bound may fail to hold uniformly. The manuscript should supply this uniform control or show how the hodograph map prevents shrinkage for the joint perturbations in speed, depth, and pressure.

minor comments (2)

The numerical section would benefit from a clearer statement of the exact perturbation forms (e.g., functional expressions or parameter values) to facilitate reproducibility.
Notation for the perturbed quantities (speed, depth, pressure) should be introduced consistently in the statement of the main theorem to avoid ambiguity when comparing to the unperturbed case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the uniform control of the holomorphic strip width. We address this point below and will incorporate the requested clarification in the revised manuscript.

read point-by-point responses

Referee: The central L² stability claim rests on applying Paley-Wiener decay to the holomorphic extension of the perturbed pressure trace after the hodograph transform. No explicit lower bound δ(ε) ≥ δ₀ > 0 (independent of the perturbation size ε) on the width of the holomorphic strip for the simultaneously perturbed data is derived or cited; if δ(ε) → 0 as ε → 0, the exponential decay rate deteriorates and the claimed sublinear error bound may fail to hold uniformly. The manuscript should supply this uniform control or show how the hodograph map prevents shrinkage for the joint perturbations in speed, depth, and pressure.

Authors: We agree that an explicit uniform lower bound on the width of the holomorphic strip is essential to ensure the Paley-Wiener decay rate remains uniform and the sublinear L² bound holds. In the revised version we will add a new lemma establishing that, for all sufficiently small joint perturbations ε in (speed, depth, pressure), the transformed data after the hodograph map admit holomorphic extension to a strip of width at least δ₀/2, where δ₀ > 0 is the width for the unperturbed solitary wave. The argument relies on the continuous dependence of the hodograph transform on the three parameters in the C¹ topology together with the explicit form of Constantin’s reconstruction formula; small perturbations cannot push the image of the fluid domain arbitrarily close to the boundary of the strip. With this uniform δ₀ the exponential Fourier decay is controlled independently of ε, and the sublinear error estimate follows as stated. We will also include a brief numerical check confirming that the effective strip width remains bounded away from zero in the computed examples. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies classical analytic tools to external reconstruction formula

full rationale

The central result is an L² stability estimate derived from the hodograph transform, holomorphic extension, and Paley-Wiener decay applied to simultaneous perturbations of speed, depth, and pressure in Constantin's reconstruction formula. No equation reduces by construction to a fitted parameter or self-referential definition. No load-bearing uniqueness theorem or ansatz is imported via self-citation. The proof chain is independent of the target bound and rests on standard function-theoretic estimates whose validity is external to the present paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of Constantin's reconstruction formula (taken from prior literature) and on standard properties of the hodograph transform and holomorphic functions in the complex plane for irrotational flows. No free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (1)

domain assumption The flow is two-dimensional, irrotational, and the wave is a solitary traveling wave of permanent form.
This is the setting in which Constantin's formula and the hodograph transform are known to apply.

pith-pipeline@v0.9.0 · 5622 in / 1263 out tokens · 53778 ms · 2026-05-19T18:39:01.659637+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The proof uses the hodograph transform, holomorphic extension arguments, and Paley–Wiener Fourier-decay estimates, yielding stability estimates with sublinear dependence on the perturbation size.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 3.4 (Holomorphic extension of the bed pressure) … P* holomorphic in the hodograph strip S* :={q+ip : q∈R, −2d < p < 0}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

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Wave Motion , FJOURNAL =

Vasan, Vishal and Oliveras, Katie and Henderson, Diane and Deconinck, Bernard , TITLE =. Wave Motion , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.wavemoti.2017.08.003 , URL =

work page doi:10.1016/j.wavemoti.2017.08.003 2017

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Clamond, Didier , TITLE =. J. Fluid Mech. , FJOURNAL =. 2013 , PAGES =. doi:10.1017/jfm.2013.253 , URL =

work page doi:10.1017/jfm.2013.253 2013

[3] [3]

and Thomas, G

Henry, D. and Thomas, G. P. , TITLE =. Philos. Trans. Roy. Soc. A , FJOURNAL =. 2018 , NUMBER =. doi:10.1098/rsta.2017.0102 , URL =

work page doi:10.1098/rsta.2017.0102 2018

[4] [4]

Clamond, Didier and Henry, David , TITLE =. J. Fluid Mech. , FJOURNAL =. 2020 , PAGES =. doi:10.1017/jfm.2020.729 , URL =

work page doi:10.1017/jfm.2020.729 2020

[5] [5]

Journal of Fluid Mechanics , volume=

General procedure for free-surface recovery from bottom pressure measurements: application to rotational overhanging waves , author=. Journal of Fluid Mechanics , volume=. 2023 , publisher=

work page 2023

[6] [6]

Journal of Physical Oceanography , volume=

Relation between orbital velocities, pressure, and surface elevation in nonlinear nearshore water waves , author=. Journal of Physical Oceanography , volume=

work page

[7] [7]

Journal of Fluid Mechanics , volume=

Steady water waves with arbitrary surface pressure: their recovery from bottom-pressure measurements , author=. Journal of Fluid Mechanics , volume=. 2024 , publisher=

work page 2024

[8] [8]

Coastal Engineering , volume=

Field data-based evaluation of methods for recovering surface wave elevation from pressure measurements , author=. Coastal Engineering , volume=. 2019 , publisher=

work page 2019

[9] [9]

2011 , publisher=

Fourier analysis: an introduction , author=. 2011 , publisher=

work page 2011

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and Lannes, D

Bonneton, P. and Lannes, D. , TITLE =. J. Fluid Mech. , FJOURNAL =. 2017 , PAGES =. doi:10.1017/jfm.2017.666 , URL =

work page doi:10.1017/jfm.2017.666 2017

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2016 , type =

Bleile, Jessica , title =. 2016 , type =

work page 2016

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and Constantin, A

Clamond, D. and Constantin, A. , TITLE =. J. Fluid Mech. , FJOURNAL =. 2013 , PAGES =. doi:10.1017/jfm.2012.490 , URL =

work page doi:10.1017/jfm.2012.490 2013

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, TITLE =

Constantin, A. , TITLE =. J. Fluid Mech. , FJOURNAL =. 2014 , PAGES =. doi:10.1017/jfm.2014.81 , URL =

work page doi:10.1017/jfm.2014.81 2014

[14] [14]

Ocean Engineering , volume=

On the recovery of surface wave by pressure transfer function , author=. Ocean Engineering , volume=. 2005 , publisher=

work page 2005

[15] [15]

Chen, Robin Ming and Walsh, Samuel , TITLE =. J. Differential Equations , FJOURNAL =. 2018 , NUMBER =. doi:10.1016/j.jde.2017.09.002 , URL =

work page doi:10.1016/j.jde.2017.09.002 2018

[16] [16]

Chen, Robin Ming and Hur, Vera Mikyoung and Walsh, Samuel , TITLE =. J. Math. Fluid Mech. , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s00021-016-0265-6 , URL =

work page doi:10.1007/s00021-016-0265-6 2017

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Coastal Engineering 1968 , pages=

Wave measurements by a pressure type wave gauge , author=. Coastal Engineering 1968 , pages=

work page 1968

[18] [18]

Coastal Engineering , volume=

Measuring waves with pressure transducers , author=. Coastal Engineering , volume=. 1987 , publisher=

work page 1987

[19] [19]

2011 , PAGES =

Constantin, Adrian , TITLE =. 2011 , PAGES =. doi:10.1137/1.9781611971873 , URL =

work page doi:10.1137/1.9781611971873 2011

[20] [20]

, TITLE =

Constantin, A. , TITLE =. J. Fluid Mech. , FJOURNAL =. 2012 , PAGES =. doi:10.1017/jfm.2012.114 , URL =

work page doi:10.1017/jfm.2012.114 2012

[21] [21]

Craig, Walter and Sternberg, Peter , TITLE =. Arch. Rational Mech. Anal. , FJOURNAL =. 1992 , NUMBER =. doi:10.1007/BF00375690 , URL =

work page doi:10.1007/bf00375690 1992

[22] [22]

and Oliveras, K

Deconinck, B. and Oliveras, K. L. and Vasan, V. , TITLE =. J. Nonlinear Math. Phys. , FJOURNAL =. 2012 , PAGES =. doi:10.1142/S1402925112400141 , URL =

work page doi:10.1142/s1402925112400141 2012

[23] [23]

Escher, Joachim and Schlurmann, Torsten , TITLE =. J. Nonlinear Math. Phys. , FJOURNAL =. 2008 , PAGES =. doi:10.2991/jnmp.2008.15.s2.4 , URL =

work page doi:10.2991/jnmp.2008.15.s2.4 2008

[24] [24]

Oliveras, K. L. and Vasan, V. and Deconinck, B. and Henderson, D. , TITLE =. SIAM J. Appl. Math. , FJOURNAL =. 2012 , NUMBER =. doi:10.1137/110853285 , URL =

work page doi:10.1137/110853285 2012

[25] [25]

Report on Waves: Made to the Meetings of the British Association in 1842-43 , author=

work page

[26] [26]

Constantin, Adrian , TITLE =. IMA J. Appl. Math. , FJOURNAL =. 2012 , NUMBER =. doi:10.1093/imamat/hxs033 , URL =

work page doi:10.1093/imamat/hxs033 2012

[27] [27]

Constantin, Adrian and Strauss, Walter , TITLE =. Comm. Pure Appl. Math. , FJOURNAL =. 2010 , NUMBER =. doi:10.1002/cpa.20299 , URL =

work page doi:10.1002/cpa.20299 2010