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

TL;DR

This paper proves the existence and bifurcation of complex three-dimensional gravity-capillary Stokes waves, revealing a clustering phenomenon linked to the Hamiltonian structure and symmetries of water wave equations.

Contribution

It introduces a novel bifurcation analysis for 3D gravity-capillary waves using equivariant Morse-Conley theory and variational reduction, uncovering new wave clustering phenomena.

Findings

Existence of multiple 3D Stokes waves with the same momentum.

Bifurcation of geometrically distinct 3D waves from 2D waves.

Identification of a clustering phenomenon in 3D wave solutions.

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.