  This paper is a study of the water wave problem in a two-dimensional domain
of infinite depth in the presence of nonzero constant vorticity. A goal is to
describe the effects of uniform shear flow on the modulation of weakly
nonlinear quasi-monochromatic surface gravity waves. Starting from the
Hamiltonian formulation of this problem and using techniques from Hamiltonian
transformation theory, we derive a Hamiltonian Dysthe equation for the time
evolution of the wave envelope. Consistent with previous studies, we observe
that the uniform shear flow tends to enhance or weaken the modulational
instability of Stokes waves depending on its direction and strength. Our method
also provides a non-perturbative procedure to reconstruct the surface elevation
from the wave envelope, based on the Birkhoff normal form transformation to
eliminate all non-resonant triads. This model is tested against direct
numerical simulations of the full Euler equations and against a related Dysthe
equation recently derived by Curtis, Carter and Kalisch (J. Fluid Mech. 855,
2018) in the context of constant vorticity. Very good agreement is found for a
range of values of the vorticity.
