Existence of pure capillary solitary waves in constant vorticity flows

Abstract: 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.