The sub-harmonic bifurcation of Stokes waves

Steady periodic water waves on the free surface of an infinitely deep irrot ational how under gravity without surface tension (Stokes waves) can be des cribed in terms of solutions of a quasi-linear equation which involves the Hilbert transform and which is the Euler-Lagrange equation of a simple func tional. The unknowns are a 2 pi-periodic function w which gives the wave pr ofile and the Froude number, a dimensionless parameter reflecting the wavel ength when the wave speed is fixed (and vice versa). Although this equation is exact, it is quadratic (with no higher order term s) and the global structure of its solution set can be studied using elemen ts of the theory of rear analytic varieties and variational techniques. In this paper it is shown that there bifurcates from the first eigenvalue o f the linearised problem a uniquely defined are-wise connected set of solut ions with prescribed minimal period which, although it is not necessarily m aximal as a connected set of solutions and may possibly self-intersect, has a local real analytic parametrisation and contains a wave of greatest heig ht in its closure (suitably defined). Moreover it contains infinitely many points which are either turning points or points where solutions with the p rescribed minimal period bifurcate. (The numerical evidence is that only th e former occurs, and this remains an open question.) It is also shown that there are infinitely many values of the Froude number at which Stokes waves, having a minimal wavelength that is an arbitrarily large integer multiple of the basic wavelength, bifurcate from the primary branch. These are the sub-harmonic bifurcations in the paper's title. (In 1 925 Levi-Civita speculated that the minimal wavelength of a Stokes wave pro pagating with speed c did not exceed 2 pi c(2)/g. This is disproved by our result on sub-harmonic bifurcation, since it shows that there are Stokes wa ves with bounded propagation speeds but arbitrarily large minimal wavelengt hs.) Although the work of BENJAMIN & FEIR and others [9, 10] has shown Stokes wa ves on deep water to be unstable, they retain a central place in theoretica l hydrodynamics. The mathematical tools used to study them here rue real an alytic-function theory, spectral theory of periodic linear pseudo-different ial operators and Morse theory, all combined with the deep influence of a p aper by PLOTNIKOV [36].