The sub-harmonic bifurcation of Stokes waves

Steady periodic water waves on the free surface of an infinitely deep irrotational how under gravity without surface tension (Stokes waves) can be described in terms of solutions of a quasi-linear equation which involves the Hilbert transform and which is the Euler-Lagrange equation of a simple functional. The unknowns are a 2 pi-periodic function w which gives the wave profile and the Froude number, a dimensionless parameter reflecting the wavelength when the wave speed is fixed (and vice versa). Although this equation is exact, it is quadratic (with no higher order terms) and the global structure of its solution set can be studied using elements of the theory of rear analytic varieties and variational techniques. In this paper it is shown that there bifurcates from the first eigenvalue of the linearised problem a uniquely defined are-wise connected set of solutions with prescribed minimal period which, although it is not necessarily maximal as a connected set of solutions and may possibly self-intersect, has a local real analytic parametrisation and contains a wave of greatest height in its closure (suitably defined). Moreover it contains infinitely many points which are either turning points or points where solutions with the prescribed minimal period bifurcate. (The numerical evidence is that only the 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 1925 Levi-Civita speculated that the minimal wavelength of a Stokes wave propagating 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 waves with bounded propagation speeds but arbitrarily large minimal wavelengths.) Although the work of BENJAMIN & FEIR and others [9, 10] has shown Stokes waves on deep water to be unstable, they retain a central place in theoretical hydrodynamics. The mathematical tools used to study them here rue real analytic-function theory, spectral theory of periodic linear pseudo-differential operators and Morse theory, all combined with the deep influence of a paper by PLOTNIKOV [36].