View source for Davey-Stewartson system

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

<math>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 </math>

<math>\partial_x^2 \phi + c_3 \partial_y^2 \phi = \partial_x ( |u|^2 ) </math>

The field phi depends elliptically on u when <math>c_3</math> is positive and thus one usually only specifies the initial data for <math>u</math>, not <math>\phi</math>.  This equation is a two-dimensional modification of the [[cubic NLS on R|one-dimensional cubic nonlinear Schrodinger equation]] and is [[completely integrable]] in the cases <math>(c_0, c_1, c_2, c_3) = (-1,1,-2,1)</math> (DS-I) and <math>(1,-1,2,-1)</math> (DS-II). The equation has a focusing nonlinearity when $ c_0 > 0 $. When <math>c_3 > 0</math> the situation becomes a [[Schrodinger equations|nonlinear Schrodinger equation]] with non-local nonlinearity (somewhat similar to the [[Hartree equation]]), and can be treated by [[Strichartz estimates]] [[GhSau1990]]; for <math>c_3 < 0</math> 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 <math>L^2</math> norm, but blows up in the $L^{\infty}$ and <math>H^1</math> 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]].

[[Category:Integrability]]
[[Category:Schrodinger]]
[[Category:Equations]]