Davey-Stewartson system

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

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

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

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.