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Whitham modulation theory for the Ostrovsky equation

Whitfield, AJ; Johnson, ER; (2017) Whitham modulation theory for the Ostrovsky equation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 473 (2197) , Article 20160709. 10.1098/rspa.2016.0709. Green open access

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Abstract

This paper derives the Whitham modulation equations for the Ostrovsky equation. The equations are then used to analyse localised cnoidal wavepacket solutions of the Ostrovsky equation in the weak rotation limit. The analysis is split into two main parameter regimes: the Ostrovsky equation with normal dispersion relevant to typical oceanic parameters and the Ostrovsky equation with anomalous dispersion relevant to strongly sheared oceanic flows and other physical systems. For anomalous dispersion a new steady, symmetric cnoidal wavepacket solution is presented. The new wavepacket can be represented as a solution of the modulation equations and an analytical solution for the outer solution of the wavepacket is given. For normal dispersion the modulation equations are used to describe the unsteady finite-amplitude wavepacket solutions produced from the rotation-induced decay of a Korteweg-de Vries solitary wave. Again, an analytical solution for the outer solution can be given. The centre of the wavepacket closely approximates a train of solitary waves with the results suggesting that the unsteady wavepacket is a localised, modulated cnoidal wavetrain. The formation of wavepackets from solitary wave initial conditions is considered, contrasting the rapid formation of the packets in anomalous dispersion with the slower formation of unsteady packets under normal dispersion.

Type: Article
Title: Whitham modulation theory for the Ostrovsky equation
Open access status: An open access version is available from UCL Discovery
DOI: 10.1098/rspa.2016.0709
Publisher version: http://dx.doi.org/10.1098/rspa.2016.0709
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: wavepackets, Ostrovsky equation, modulation equations
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/1532007
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