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Global bifurcation for water waves with capillary effects and constant vorticity

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Abstract

We study periodic capillary and capillary-gravity waves traveling over a water layer of constant vorticity and finite depth. Inverting the curvature operator, we formulate the mathematical model as an operator equation for a compact perturbation of the identity. By means of global bifurcation theory, we then construct global continua consisting of solutions of the water wave problem which may feature stagnation points. A characterization of these continua is also included.

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Notes

  1. For example, the sequence \((\eta _p)_p\) defined by \(\eta _p(x):=\cos (px)/p\) for \(x\in {\mathbb R}\) and \(p\ge 1\) is uniformly bounded in \(C^1(\mathbb S)\) and \(\inf _\beta \sup _p\Vert \eta _p'\Vert _\beta =\infty \) for all \(\beta \in (0,1)\). Though \(\eta _p\) is a real-analytic map for all \(p\ge 1\), the sequence \((\eta _p)_p\) does not have a convergence subsequence in \(C^1(\mathbb S)\).

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Authors and Affiliations

  1. Institut für Mathematik, Universität Wien, Nordbergstraße 15, 1090 , Vienna, Austria

    Bogdan-Vasile Matioc

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  1. Bogdan-Vasile Matioc
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Correspondence to Bogdan-Vasile Matioc.

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Communicated by J. Escher.

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Matioc, BV. Global bifurcation for water waves with capillary effects and constant vorticity. Monatsh Math 174, 459–475 (2014). https://doi.org/10.1007/s00605-013-0583-1

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