# Davey-Stewartson system

The Davey-Stewartson system [DavSte1974] in 2 spatial dimensions involves a complex field u and a real field phi:

${\displaystyle i\partial _{t}u+c_{0}\partial _{x}^{2}u+\partial _{y}^{2}u=c_{1}|u|^{2}u+c_{2}u\partial _{x}phi}$

${\displaystyle \partial _{x}^{2}phi+c_{3}\partial _{y}^{2}phi=\partial _{x}(|u|^{2})}$

The field phi depends elliptically on u when ${\displaystyle c_{3}}$ is positive and thus one usually only specifies the initial data for ${\displaystyle u}$, not ${\displaystyle \phi }$. This equation is a modification of the cubic nonlinear Schrodinger equation and is completely integrable in the cases ${\displaystyle (c_{0},c_{1},c_{2},c_{3})=(-1,1,-2,1)}$ (DS-I) and ${\displaystyle (1,-1,2,-1)}$ (DS-II). When ${\displaystyle c_{3}>0}$ the situation becomes a nonlinear Schrodinger equation with non-local nonlinearity, and can be treated by Strichartz estimates [GhSau1990]; for ${\displaystyle c_{3}<0}$ the situation is quite different but one can still use derivative NLS techniques (i.e. gauge transforms and energy methods) to obtain some local existence results [LiPo1993]. Further results are in [HaSau1995].

The Davey-Stewartson system is a special case of the Zakharov-Schulman system.