# Davey–Stewartson equation

In fluid dynamics, the Davey–Stewartson equation (DSE) was introduced in a paper by Davey & Stewartson (1974) to describe the evolution of a three-dimensional wave-packet on water of finite depth.

It is a system of partial differential equations for a complex (wave-amplitude) field ${\displaystyle u\,}$ and a real (mean-flow) field ${\displaystyle \phi \,}$:

${\displaystyle iu_{t}+c_{0}u_{xx}+u_{yy}=c_{1}|u|^{2}u+c_{2}u\phi _{x},\,}$
${\displaystyle \phi _{xx}+c_{3}\phi _{yy}=(|u|^{2})_{x}.\,}$

The DSE is an example of a soliton equation in 2+1 dimensions. The corresponding Lax representation for it is given in Boiti, Martina & Pempinelli (1995).

In 1+1 dimensions the DSE reduces to the nonlinear Schrödinger equation

${\displaystyle iu_{t}+u_{xx}+2k|u|^{2}u=0.\,}$

Itself, the DSE is the particular reduction of the Zakharov–Schulman system. On the other hand, the equivalent counterpart of the DSE is the Ishimori equation.

The DSE is the result of a multiple-scale analysis of modulated nonlinear surface gravity waves, propagating over a horizontal sea bed.