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

1. Problem Formulation

The paper addresses the existence and multiplicity of truly 3-dimensional (3D) periodic traveling waves (Stokes waves) for the gravity-capillary water waves equations in a 3D domain with a flat bottom.

• Physical Setting: An incompressible, inviscid, irrotational fluid under gravity ($g$) and surface tension ($\kappa$). The fluid occupies a domain $D_\eta$ defined over a 2D torus $T^2_\Gamma = \mathbb{R}^2/\Gamma$, where $\Gamma$ is an arbitrary lattice.
• Mathematical Model: The system is governed by the Zakharov-Craig-Sulem equations, which form a Hamiltonian system with a symplectic structure. The equations are reversible ($\mathbb{Z}_2$ symmetry) and invariant under spatial translations ($T^2_\Gamma$ symmetry).
• The Core Question: While 2D Stokes waves (constant in one direction) are well-understood, the existence of genuinely 3D waves (non-constant in all directions) bifurcating from a flat state at a resonant speed $c^*$ is a complex problem. Specifically, the authors investigate the scenario where the momentum of the wave is collinear with a resonant wave vector. In this case, 2D waves always exist; the challenge is to prove the existence of additional, geometrically distinct 3D waves with the same momentum.

2. Methodology

The authors employ a sophisticated combination of variational methods, Lyapunov-Schmidt reduction, and equivariant topology.

A. Variational Lyapunov-Schmidt Reduction

1. Linearization: The linearized operator $L_{c^*}$ at the flat state has a finite-dimensional kernel $V = \ker(L_{c^*})$ spanned by resonant plane waves.
2. Decomposition: The phase space is decomposed into $V$ (resonant modes) and its symplectic orthogonal complement $W$.
3. Reduction: Using the Implicit Function Theorem, the infinite-dimensional problem is reduced to finding critical points of a reduced functional $\Phi(c, v)$ on the finite-dimensional kernel $V$.
4. Speed Construction ($c(v)$): A major technical hurdle is that the speed $c$ cannot be fixed a priori when $v$ approaches the subspace of 2D waves ($V_{2d}$). The authors construct a speed function $c(v)$ that solves the bifurcation equations. Crucially, they handle the singularity near $V_{2d}$ by exploiting the $T^2_\Gamma$ symmetry, defining $c(v)$ continuously across the 2D subspaces, unlike previous works which required $v$ to stay away from 2D waves.

B. Straightening of Momentum

The reduced momentum map $I(v)$ is not purely quadratic near the origin due to higher-order terms. The authors prove a "Straightening" theorem (Theorem 6.2), establishing an equivariant homeomorphism $\zeta$ that transforms the level sets of the reduced momentum $I(v)$ into the level sets of a purely quadratic momentum $\mathcal{I}(v)$. This allows the topological analysis to be performed on a simpler, explicit geometric object.

C. Equivariant Critical Point Theory (Part I)

The paper develops a rigorous framework for Equivariant Morse-Conley theory on singular metric spaces.

• Space Structure: The level sets of the momentum, denoted $M_a$, are shown to be topological joins of spheres: $M_a \cong (S^{2d_- -1} \times S^{2d_+ -1}) \star S^1$. Here, $S^1$ represents the 2D wave orbit, and the product of spheres represents the 3D wave components.
• Singularity Handling: The space $M_a$ is not a manifold (it has singularities where 2D and 3D waves coexist). The authors use equivariant cohomology and the annihilator of local cohomology to derive a lower bound for the number of critical orbits.
• Key Theorem (3.14): They prove that for a continuous invariant function on a join space $M \star S^n$ (where $S^n$ is a stationary orbit), the number of critical values distinct from the stationary level is bounded below by the equivariant cup-length of $M$ plus one.

3. Key Contributions and Results

Theorem 1.1: Multiplicity in the Collinear Case

This is the paper's primary novelty.

• Scenario: Consider a resonant speed $c^*$ with at least 3 resonant wave vectors ($\#V \ge 3$). Let the momentum $a$ be collinear with exactly one resonant vector $j_0$.
• Result: In addition to the unique 2D Stokes wave orbit (along $j_0$), there exist at least $N = \#V - 2$ geometrically distinct truly 3D Stokes waves with the same momentum $a$.
• Significance: This proves a "clustering phenomenon" where multiple 3D waves bifurcate alongside the 2D wave, a phenomenon observed in physical fluids but previously unproven rigorously in this specific resonant/collinear setting.

Theorem 1.2: Non-Collinear Case (Improvement)

• Result: If the momentum $a$ is not collinear with any resonant vector, there exist at least $\#V - 1$ geometrically distinct 3D waves.
• Improvement: This improves upon the seminal work of Craig and Nicholls [15] by providing a uniform bound on the amplitude and increasing the count of distinct orbits by a factor of 2 (accounting for the $\mathbb{Z}_2$ reflection symmetry).

Theorem 1.3: Non-Existence and Classification

• Result: The paper provides a complete bifurcation picture.
  • If the momentum $a$ lies outside a specific convex cone $C$ generated by the resonant vectors, no small-amplitude Stokes waves exist.
  • If $a$ is on the boundary of $C$, any solution must be 2D.
  • If $a$ is in the interior of $C$ and non-collinear, any solution is truly 3D.

4. Technical Innovations

1. Handling Singular Reductions: The construction of the speed function $c(v)$ is continuous and analytic everywhere, including the singular hyperplanes of 2D waves. This is achieved by a careful analysis of the symmetry groups and the behavior of the reduced functional near these subspaces.
2. Topology of Join Spaces: The identification of the momentum level sets as a topological join ($X \star S^1$) is crucial. It allows the application of equivariant cohomology to count critical points even when the space is not a manifold.
3. Equivariant Cup-Length: The authors compute the equivariant cup-length for the product of spheres involved in the join, deriving the lower bound $\#V - 2$ for the number of 3D waves.

5. Significance

• Mathematical Physics: The paper resolves a long-standing open problem regarding the multiplicity of 3D water waves in the presence of surface tension (gravity-capillary case). It confirms that the Hamiltonian structure and symmetries of the water wave equations force the existence of multiple 3D solutions even when 2D solutions exist.
• Methodological Impact: The development of Equivariant Morse-Conley theory for singular spaces (specifically joins with spheres) provides a powerful new tool for bifurcation theory in Hamiltonian PDEs. This approach can likely be applied to other problems involving symmetry-breaking and resonant bifurcations in infinite-dimensional systems.
• Completeness: The work provides a "complete bifurcation picture," classifying solutions based on their momentum relative to the resonant cone, distinguishing clearly between 2D and 3D regimes.

In summary, the paper combines deep topological arguments with precise analytical estimates to prove the existence of a rich family of 3D water waves, demonstrating that the interplay between symmetry, resonance, and topology leads to unexpected multiplicity in fluid dynamics.