# 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 two-dimensional modification of the one-dimensional 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). The equation has a focusing nonlinearity when $c_0 > 0$. When ${\displaystyle c_{3}>0}$ 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 ${\displaystyle c_{3}<0}$ 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.

In the integrable focusing case, Sun1994 and Sun1995 has obtained global existence and uniqueness results for small initial data by using the inverse scattering transform. A particularly interesting solution of the focusing Davey-Stewartson II equation was found by Oza1992 and is

$$u=\exp\left(i\frac{x^2-y^2}{1-4t}\right)\frac{1-4t}{(1-4t)^2+x^2+y^2}.$$

This solution preserves the ${\displaystyle L^{2}}$ norm, but blows up in the $L^{\infty}$ and ${\displaystyle H^{1}}$ norms. There has been some numerical investigation of such behavior by BesMauSti2004, McCFokPel2005 and KleMuiRoi2011 using a numerical scheme devised by WhiWei1994. It is unclear whether such behavior is generic.

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