On analyticity of travelling water waves

Abstract

In this paper we establish the existence and analyticity of periodic solutions of a classical free-boundary model of the evolution of three-dimensional, capillary–gravity waves on the surface of an ideal fluid. The result is achieved through the application of bifurcation theory to a boundary perturbation formulation of the problem, and it yields analyticity jointly with respect to the perturbation parameter and the spatial variables. The travelling waves we find can be interpreted as resulting from the (nonlinear) interaction of two two-dimensional wavetrains, giving rise to a periodic travelling pattern. Our analyticity theorem extends the most sophisticated results known to date in the absence of resonance; ‘short crested waves’, which result from the interaction of two wavetrains with unit amplitude ratio are realized as a special case. Our method of proof also sheds light on the convergence and conditioning properties of classical boundary perturbation methods for the numerical approximation of travelling surface waves. Indeed, we demonstrate that the rather unstable numerical behaviour of these approaches can be attributed to the strong but subtle cancellations in the formulas underlying their classical implementations. These observations motivate the derivation and use of an alternative, stable, formulation which, in addition to providing our method of proof, suggests new stabilized implementations of boundary perturbation algorithms.