Equivariant critical point theory and bifurcation of $3d$ gravity-capillary Stokes waves

Abstract

We establish novel existence results of $3d$ gravity-capillary periodic traveling waves. In particular we prove the bifurcation of multiple, geometrically distinct truly $3d$ Stokes waves having the same momentum of any non-resonant $2d$ Stokes wave. This unexpected clustering phenomenon of Stokes waves, observed in physical fluids, is a fundamental consequence of the Hamiltonian nature of the water waves equations, their symmetry groups, and novel topological arguments. We employ a variational Lyapunov-Schmidt reduction combined with equivariant Morse-Conley theory for a functional defined on a joined topological space invariant under a $2$-torus action. Although the reduction is a priori singular near the hyperplanes of $2d$-waves, we circumvent this difficulty by exhaustive use of the symmetry groups. This approach yields a complete bifurcation picture of $3d $ gravity-capillary Stokes waves.