Existence of pure capillary solitary waves in constant vorticity flows

TL;DR

This paper proves the existence of pure capillary solitary waves in 2D flows with constant vorticity, showing vorticity enables solitary waves where they do not exist in irrotational flows.

Contribution

It demonstrates that constant vorticity allows for pure capillary solitary waves, filling a gap left by previous nonexistence results for irrotational flows.

Findings

Existence of solitary waves in vorticity flows established

A Hamiltonian and normal-form analysis was used

A KdV-type equation describes the wave profile

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.