Two dimensional solitary water waves with constant vorticity, Part I: the deep gravity case

TL;DR

This paper proves the existence of smooth, solitary water waves with constant vorticity in deep water, resembling Benjamin-Ono solitons, near critical velocity, using a fixed point approach.

Contribution

It introduces a new existence result for solitary waves with constant vorticity in deep water, connecting them to Benjamin-Ono solitons.

Findings

Existence of solitary waves near critical velocity.

Wave profiles resemble rescaled Benjamin-Ono solitons.

Solitary waves are smooth with asymptotic expansions.

Abstract

We consider the two dimensional pure gravity water waves with nonzero constant vorticity in infinite depth, working in the holomorphic coordinates introduced by Hunter, Ifrim, and Tataru. We show that close to the critical velocity corresponding to zero frequency, a solitary wave exists. We use a fixed point argument to construct the solitary wave whose profile resembles a rescaled Benjamin-Ono soliton. The solitary wave is smooth and has an asymptotic expansion in terms of powers of the Benjamin-Ono soliton.