Generalizing an idea of Arnold, we discuss the hydrodynamic stability a la Lyapunov of solitary water-waves which are rotational solutions of the Euler equation, travelling with constant phase speed F. In the reference frame moving with the wave profile, the solitary wave is described by a solution of [formula omitted]. We show that if [formula omitted], as in other applications of Arnold's idea, and if [formula omitted] at the air sea surface, a rather realistic request, the system is stable for (a) small vertical or horizontal space scale perturbations; (b) perturbations with a very long vertical space scale and very small horizontal space scale or with a very long horizontal space scale and very small vertical space scale. Finally we show that the system is also stable for irrotational perturbations. © 1986, Taylor & Francis Group, LLC. All rights reserved.

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