34 - Modeling the Davey-Stewartson (DS) Equations

Abstract

This chapter discusses a model of the Davey–Stewartson (DS) equations. The DS model is exactly integrable in shallow water and almost integrable in deep water. The model has easily identifiable coherent structures and waves––including solitons, unstable rogue-wave type modes, and Stokes waves––and the velocity field contains vortices. These waveforms are packets and are the “nonlinear Fourier components” in the theory of the inverse scattering transform (IST) and its presumed “slowly varying” or “adiabatic” extensions. The model can easily be modified to include current, winds, and bathymetry. Because the model is based upon the IST, the nonlinear Fourier analysis procedures for analyzing data, and a hyperfast numerical model, which is two or three orders of magnitude faster than typical fast Fourier transform (FFT)-type numerical integrations of the equation, is developed. The identifiable coherent structures are nonlinear wave packets whose maximum waves are often referred to as “rogue waves.” In 2 + 1 dimensions, the Euler equations can be reduced to the DS equations in the fields Φ(x, y, t) and ψ(x, y, t), where Φ(x, y, t) is the (normalized, long wave part of the) velocity potential and ψ(x, y, t) is the complex envelope of a narrow-banded wave train, which may be modulated in both the x and y directions. As a result a directional, narrow-banded sea state can be accounted for by the DS equations. The normalized form of the DS equations is discussed.