# Davey-Stewartson system

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

i u_t + c_0 u_xx + u_yy = c_1 |u|^2 u + c_2 u phi_x phi_xx + c_3 phi_yy = partial_x ( |u|^2 )

The field phi depends elliptically on u when c_3 is positive and thus one usually only specifies the initial data for u, not phi. This equation is a modification of the cubic nonlinear Schrodinger equation and is completely integrable in the cases (c_0, c_1, c_2, c_3) = (-1,1,-2,1) (DS-I) and (1,-1,2,-1) (DS-II). When c_3 > 0 the situation becomes a nonlinear Schrodinger equation with non-local nonlinearity, and can be treated by Strichartz estimates [GhSau1990]; for 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.