Exact solutions of a three-dimensional nonlinear Schrödinger equation applied to gravity waves

Abstract

The three-dimensional evolution of packets of gravity waves is studied using a nonlinear Schrödinger equation (the Davey–Stewartson equation). It is shown that permanent wave groups of the elliptic en and dn functions and their common limiting solitary sech forms exist and propagate along directions making an angle less than ψ c = tan −1 (1/√2) = 35° with the underlying wave field, whilst, along directions making an angle greater than ψ c , there exist permanent wave groups of elliptic sn and negative solitary tanh form. Furthermore, exact general solutions are given showing wave groups travelling along the two characteristic directions at ψ c or − ψ c . These latter solutions tend to form regions of large wave slope and are used to discuss the waves produced by a ship, in particular the nonlinear evolution of the leading edge of the pattern.