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Long wave approximation for water waves under a Coriolis forcing and the Ostrovsky equation

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Benjamin Melinand (IMB)

Analysis of PDEs · Submitted 29 Mar 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Therefore, this work provides the first mathematical justification of the Ostrovsky equation.

how it is stated · asserted 1.00 read the record →

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arXiv ID1603.08782 (version 3)
Pith IDYZDFWIXH, Pith's permanent record of this paper and its review
CitationBenjamin Melinand (IMB). "Long wave approximation for water waves under a Coriolis forcing and the Ostrovsky equation." arXiv:1603.08782 (2016). https://pith.science/paper/YZDFWIXH
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,711 input tokens.

What the abstract itself states, as probabilities: claims first 0.88 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.44 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

This paper is devoted to the study of the long wave approximation for water waves under the influence of the gravity and a Coriolis forcing. We start by deriving a generalization of the Boussinesq equations in 1D (in space) and we rigorously justify them as an asymptotic model of the water waves equations. These new Boussinesq equations are not the classical Boussinesq equations. A new term due to the vorticity and the Coriolis forcing appears that can not be neglected. Then, we study the Boussinesq regime and we derive and fully justify different asymptotic models when the bottom is flat : a linear equation linked to the Klein-Gordon equation admitting the so-called Poincar{\'e} waves; the Ostrovsky equation, which is a generalization of the KdV equation in presence of a Coriolis forcing, when the rotation is weak; and finally the KdV equation when the rotation is very weak. Therefore, this work provides the first mathematical justification of the Ostrovsky equation. Finally, we derive a generalization of the Green-Naghdi equations in 1D in space for small topography variations and we show that this model is consistent with the water waves equations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 This paper is devoted to the study of the long wave approximation for water waves under the influence of the gravity and a Coriolis forcing. background 0.57 0.52 0.04 asserted 0.92
s1 We start by deriving a generalization of the Boussinesq equations in 1D (in space) and we rigorously justify them as an asymptotic model of the water waves equations. contribution 0.54 0.96 0.06 asserted 1.00
s2 These new Boussinesq equations are not the classical Boussinesq equations. result 0.78 0.61 0.04 asserted 1.00
s3 A new term due to the vorticity and the Coriolis forcing appears that can not be neglected. result 0.78 0.77 0.06 asserted 1.00
s4 Then, we study the Boussinesq regime and we derive and fully justify different asymptotic models when the bottom is flat : a linear equation linked to the Klein-Gordon equation admitting the so-called Poincar{\'e} waves; the Ostrovsky equation, which is a generalization of the KdV equation in presence of a Coriolis forcing, when the rotation is weak; and finally the KdV equation when the rotation is very weak. contribution 0.67 0.95 0.14 asserted 1.00
s5 Therefore, this work provides the first mathematical justification of the Ostrovsky equation. result 0.43 0.91 0.75 asserted 1.00
s6 Finally, we derive a generalization of the Green-Naghdi equations in 1D in space for small topography variations and we show that this model is consistent with the water waves equations. contribution 0.91 0.96 0.07 asserted 0.92

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

This paper is devoted to the study of the long wave approximation for water waves under the influence of the gravity and a Coriolis forcing. We start by deriving a generalization of the Boussinesq equations in 1D (in space) and we rigorously justify them as an asymptotic model of the water waves equations. These new Boussinesq equations are not the classical Boussinesq equations. A new term due to the vorticity and the Coriolis forcing appears that can not be neglected. Then, we study the Boussinesq regime and we derive and fully justify different asymptotic models when the bottom is flat : a linear equation linked to the Klein-Gordon equation admitting the so-called Poincar{\'e} waves; the Ostrovsky equation, which is a generalization of the KdV equation in presence of a Coriolis forcing, when the rotation is weak; and finally the KdV equation when the rotation is very weak. Therefore, this work provides the first mathematical justification of the Ostrovsky equation. Finally, we derive a generalization of the Green-Naghdi equations in 1D in space for small topography variations and we show that this model is consistent with the water waves equations.

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