A Nonlinear Schrodinger Equation for Gravity-Capillary Waves on Deep Water with Constant Vorticity

The surface gravity-capillary waves on deep water with constant vorticity in the region bounded by the free surface and the infinitely deep plane bottom are considered. A nonlinear Schrodinger equation is derived from a system of exact nonlinear integro-differential equations in conformal variables written in the implicit form taking into account surface tension. In deriving the nonlinear Schrodinger equation, the role of the mean flow is taken into account. The nonlinear Schrodinger equation is investigated for modulation instability. A soliton solution of the nonlinear Schrodinger equation that represents a soliton of the "ninth wave" type is obtained.