A Modified Nonlinear Schrodinger Equation for Interactions Between Waves and Shear Currents

A nonlinear Schrodinger equation for the propagation of two-dimensional surface gravity waves on linear shear currents in finite water depth is derived. In the derivation, linear shear currents are assumed to be a linear combination of depth-uniform currents and constant vorticity. Therefore, the equation includes the combined effects of depthuniform currents and constant vorticity. Furthermore, the influence of linear shear currents on the Peregrine breather is also studied. It is demonstrated that depth-uniform opposing currents can reduce the breather extension in finite water depth, but following currents has the adverse impact, indicating that a wave packets with freak waves formed on following currents contains more hazardous waves in finite water depth. However, the corresponding and coexisting vorticity can counteract the influence of currents. If the water depth is deep enough, both depth-uniform currents and vorticity have negligible effect on the characteristics of Peregrine breather.