Existence of pure capillary solitary waves in constant vorticity flows

We prove the existence of pure capillary solitary waves for the 2D finite-depth Euler equations with nonzero constant vorticity. In the irrotational case, nonexistence of solitary waves was established by Ifrim–Pineau–Tataru–Taylor, so our theorem isolates constant vorticity as a mechanism that enables solitary waves in the pure-capillary regime. The proof uses a spatial-dynamics Hamiltonian formulation of the travelling-wave equations and a nonlinear change of variables that flattens the free surface while putting the symplectic form into Darboux coordinates. Near a distinguished curve in the vorticity–capillarity parameter space, the linearization has a two-dimensional center subspace; a parameter-dependent center-manifold reduction yields a canonical planar Hamiltonian system. A cubic normal-form expansion and long-wave scaling produce a KdV-type profile equation with a reversible homoclinic orbit, which persists under the full dynamics and generates the solitary-wave solutions.

💡 Research Summary

The paper establishes the existence of pure‑capillary solitary waves in a two‑dimensional, finite‑depth, incompressible, inviscid fluid when a non‑zero constant vorticity is present. In the irrotational case, recent works (Ifrim–Pineau–Tataru–Taylor, 2022) proved that no such solitary waves exist, so the present result isolates constant vorticity as the decisive mechanism that enables solitary structures in the pure‑capillary regime.

The authors begin by formulating the free‑surface Euler equations with surface tension σ and constant vorticity ω. By writing the velocity as a shear flow (−ωy,0) plus an irrotational perturbation ∇φ, and assuming the free surface is a graph y=η(t,x), they obtain a coupled boundary‑value problem for φ, its harmonic conjugate ψ, and η. Introducing the traveling‑wave coordinate ξ=x−ct and normalizing depth and speed to one, the system reduces to a set of quasi‑linear equations (3.4) in ξ.

A crucial step is the spatial‑dynamics and Hamiltonian reformulation. A nonlinear change of variables flattens the free surface and simultaneously puts the symplectic form into Darboux coordinates, yielding an infinite‑dimensional reversible Hamiltonian system (M_{ω,σ},Ω_{ω,σ},H_{ω,σ}) on a Hilbert space of functions (ϕ,θ) together with scalar variables (z,η). The Hamiltonian functional encodes kinetic energy, the effect of vorticity, and the surface‑tension contribution.

Linearizing around the trivial flat‑surface state, the authors identify a distinguished curve in the (ω,σ) parameter plane given by ωd/c=1+ε with ε≪1. Along this curve the linear operator possesses a two‑dimensional center subspace (purely imaginary eigenvalues) while the remaining spectrum is hyperbolic. Using Mielke’s center‑manifold theorem, they verify the required resolvent bounds and reduce the dynamics to a finite‑dimensional canonical Hamiltonian system on the center manifold.

Next, a cubic normal‑form expansion is performed on the reduced Hamiltonian. After a long‑wave scaling ξ=ε^{-1/2}X, the leading‑order equations become the stationary Korteweg–de Vries (KdV) profile equation
 Q’’ = Q + (1/2)Q².
This equation admits the explicit homoclinic solution Q(X)=−3 sech²(X/2), which corresponds to a reversible heteroclinic orbit in the planar Hamiltonian phase plane.

The homoclinic orbit persists under the full (untruncated) dynamics by the invariant‑manifold theory, producing a solitary‑wave solution of the original Euler system. In physical variables the surface elevation satisfies
 η(ξ)=d ε Q(ε^{1/2}ξ/d)+O(ε²),
with the wave speed c constrained by σ/(c²d)>1/3. The solution is C²‑smooth and decays exponentially as |ξ|→∞.

The paper’s significance lies in three aspects. First, it demonstrates that constant vorticity supplies an effective dispersive mechanism that balances the pure‑capillary nonlinearity, allowing solitary waves even when gravity is absent. Second, it showcases a powerful analytical framework—spatial dynamics, Hamiltonian structure, center‑manifold reduction, and normal‑form theory—applied to a free‑surface fluid problem with a nontrivial symplectic form. Third, it provides a rigorous counterpart to earlier non‑existence results, confirming that the presence of vorticity is essential for solitary waves in the pure‑capillary finite‑depth setting.

The authors also compare their approach with recent infinite‑depth constructions based on Babenko‑type integral equations, noting that the finite‑depth geometry leads to a different nonlocal operator and dispersion relation. Future directions suggested include the stability analysis of the constructed solitary waves, the interaction of multiple solitary pulses, and extensions to variable vorticity or three‑dimensional configurations.