Dispersion relations for steady periodic water waves of fixed mean-depth with two rotational layers

Calin Iulian Martin; Adrián Rodríguez-Sanjurjo

doi:10.3934/dcds.2019209

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2019, Volume 39, Issue 9: 5149-5169. Doi: 10.3934/dcds.2019209

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Dispersion relations for steady periodic water waves of fixed mean-depth with two rotational layers

Calin Iulian Martin^1, , and
Adrián Rodríguez-Sanjurjo^2,

1.
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria
2.
School of Mathematical Sciences, University College Cork, Western Road, T12 XF62, Cork, Ireland

^* Corresponding author: Calin Iulian Martin
^* Corresponding author: Calin Iulian Martin

Received: June 30, 2018

Revised: March 31, 2019

Published: May 2019

C. I. Martin acknowledges the support of the Science Foundation Ireland (SFI)–research grant 13/CDA/2117 and the support of the Austrian Science Fund (FWF)–research grant P 30878-N32. A. Rodríguez-Sanjurjo acknowledges the support of the Science Foundation Ireland (SFI)–research grant 13/CDA/2117

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Abstract

The aim of this paper is to obtain the dispersion relation for small-amplitude periodic travelling water waves propagating over a flat bed with a specified mean depth under the presence of a discontinuous piecewise constant vorticity. An analysis of the dispersion relation for a model with two rotational layers each having a non-zero constant vorticity is presented. Moreover, we present a stability result for the bifurcation inducing laminar flow solutions.

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Mathematics Subject Classification: Primary: 35Q31, 35Q35; Secondary: 74J15, 74J30, 76B15, 76E05.

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Figure 1. Sketch of the graph of the polynomial providing the dispersion relation for two negative vorticities under the condition (44). The root x01 does not satisfy the non-stagnation condition while the root x02 verifies it if and only if (47) is assumed

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