Davey-Stewartson system

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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 two-dimensional modification of the one-dimensional 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 (somewhat similar to the Hartree equation), 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.