Abstract
We show that horizontally symmetric water waves are traveling waves. The result is valid for the Euler equations, and is based on a general principle that applies to a large class of nonlinear partial differential equations, including some of the most famous model equations for water waves. A detailed analysis is given for weak solutions of the Camassa–Holm equation. In addition, we establish the existence of nonsymmetric linear rotational waves for the Euler equations.
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