Two-dimensional solitary water waves with constant vorticity, Part II: the deep capillary case

We consider the two-dimensional capillary water waves with nonzero constant vorticity in infinite depth. We first derive the Babenko equation that describes the profile of the solitary wave. When the velocity $c$ is close to a critical velocity and a sign condition involving the physical parameters is met, the Babenko equation can be reduced to the stationary focusing cubic nonlinear Schr\"odinger equation plus perturbative error. We show the existence of a critical value of a dimensionless physical parameter below which at least two families of velocities satisfy the focusing condition and above which only one does. This gives the existence of small-amplitude solitary wave solutions for the water wave system with constant vorticity.