# American Institute of Mathematical Sciences

August  2014, 34(8): 3125-3133. doi: 10.3934/dcds.2014.34.3125

## A characterization of the symmetric steady water waves in terms of the underlying flow

 1 University of Vienna, Nordbergstraße 15, 1090, Vienna, Austria

Received  April 2013 Revised  September 2013 Published  January 2014

In this paper we present a characterization of the symmetric rotational periodic gravity water waves of finite depth and without stagnation points in terms of the underlying flow. Namely, we show that such a wave is symmetric and has a single crest and trough per period if and only if there exists a vertical line within the fluid domain such that all the fluid particles located on that line minimize there simultaneously their distance to the fluid bed as they move about. Our analysis uses the moving plane method, sharp elliptic maximum principles, and the principle of analytic continuation.
Citation: Bogdan-Vasile Matioc. A characterization of the symmetric steady water waves in terms of the underlying flow. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3125-3133. doi: 10.3934/dcds.2014.34.3125
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