# American Institute of Mathematical Sciences

December  2016, 36(12): 7057-7061. doi: 10.3934/dcds.2016107

## On the symmetry of spatially periodic two-dimensional water waves

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

Received  December 2015 Revised  January 2016 Published  October 2016

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.
Citation: Florian Kogelbauer. On the symmetry of spatially periodic two-dimensional water waves. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7057-7061. doi: 10.3934/dcds.2016107
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