  We present a numerical study of essentially nonlinear dynamics of surface
gravity waves on deep water with constant vorticity using governing equations
in conformal coordinates. The dispersion relation of surface gravity waves on
shear flow is known to have two branches, one of which is weakly dispersive for
long waves. Weakly nonlinear evolution of the waves of this branch can be
described by the Benjamin-Ono equation, which is integrable and has soliton and
multi-soliton solutions. Currently, the extent to which the properties of such
solitary waves obtained within the weakly nonlinear model are preserved in the
exact Euler equations is unknown. We investigate the behaviour of this class of
solitary waves without the restrictive assumption of weak nonlinearity by using
the exact Euler equations. The evolution of localized initial perturbations
leading to the formation of single or multiple solitary waves is modeled, and
the properties of finite-amplitude solitary waves are discussed. We show that
within the framework of the exact equations, two-soliton collisions are almost
elastic, but in contrast to solutions of the Benjamin-Ono equation the waves
receive a phase shift as a result of the interaction.
