The Davey-Stewartson system [DavSte1974] in 2 spatial dimensions involves a complex field u and a real field phi:
The field phi depends elliptically on u when is positive and thus one usually only specifies the initial data for , not . This equation is a modification of the cubic nonlinear Schrodinger equation and is completely integrable in the cases (DS-I) and (DS-II). When the situation becomes a nonlinear Schrodinger equation with non-local nonlinearity, and can be treated by Strichartz estimates [GhSau1990]; for 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.