Sur les ondes de Stokes et une conjecture de Levi-Civita

In his work [6] on Stokes waves (stationary periodic gravity waves), Levi-Civita conjectured that, for any given propagation speed c > 0, the wavelengths are not larger than 2πc²/g, where g > 0 is the acceleration due to gravity (see also [3]). We state a result on the existence of Stokes waves with arbitrarily large wavelengths, that shows no such upper-bound on the wavelength exists, and therefore that Levi-Civita's conjecture is false (see [2] for a complete proof). These long waves arise by way of sub-harmonic bifurcations. This vindicates numerical results of Saffman [9] and offers a rigorous complement to the analysis of Baesens and MacKay [1].