On the symmetry of spatially periodic two-dimensional water waves

Florian  Kogelbauer

doi:10.3934/dcds.2016107

Article Contents

2016, Volume 36, Issue 12: 7057-7061. Doi: 10.3934/dcds.2016107

This issue Previous Article On hyperbolicity in the renormalization of near-critical area-preserving maps Next Article Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$

On the symmetry of spatially periodic two-dimensional water waves

Florian Kogelbauer^1,

1.
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, Vienna, A-1090

Received: November 30, 2015

Revised: December 31, 2015

Published: May 2017

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Abstract

We show that a spatially periodic solution to the irrotational two-dimensional gravity water wave problem, with the property that the horizontal velocity component at the flat bed is symmetric, while the acceleration at the flat bed is anti-symmetric with respect to a common axis of symmetry, necessarily constitutes a traveling wave. The proof makes use complex variables and structural properties of the governing equations for nonlinear water waves.

Keywords:

Irrotational water waves,
traveling wave.

Mathematics Subject Classification: Primary: 35Q35; Secondary: 35B50.

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