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Dispersion Equation for Water Waves with Vorticity and Stokes Waves on Flows with Counter-Currents

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Abstract

The two-dimensional free-boundary problem of steady periodic waves with vorticity is considered for water of finite depth. We investigate how flows with small-amplitude Stokes waves on the free surface bifurcate from a horizontal parallel shear flow in which counter-currents may be present. Two bifurcation mechanisms are described: one for waves with fixed Bernoulli’s constant, and the other for waves with fixed wavelength. In both cases the corresponding dispersion equations serve for defining wavelengths from which Stokes waves bifurcate. Necessary and sufficient conditions for the existence of roots of these equations are obtained. Two particular vorticity distributions are considered in order to illustrate the general results.

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Authors and Affiliations

  1. Department of Mathematics, Linköping University, 581 83, Linköping, Sweden

    Vladimir Kozlov

  2. Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, V.O., Bol’shoy pr. 61, 199178, St Petersburg, Russian Federation

    Nikolay Kuznetsov

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  1. Vladimir Kozlov
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  2. Nikolay Kuznetsov
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Correspondence to Nikolay Kuznetsov.

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Communicated by P. Rabinowitz

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Kozlov, V., Kuznetsov, N. Dispersion Equation for Water Waves with Vorticity and Stokes Waves on Flows with Counter-Currents. Arch Rational Mech Anal 214, 971–1018 (2014). https://doi.org/10.1007/s00205-014-0787-0

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