2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/222146We 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.Analysis of PDEs76B15; 35Q35text