Difference between revisions of "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:  
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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>
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<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>
 
<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 modification of the 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).   When <math>c_3 > 0</math> the situation becomes a nonlinear Schrodinger equation with non-local nonlinearity, 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].
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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]].
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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
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$$u=\exp\left(i\frac{x^2-y^2}{1-4t}\right)\frac{1-4t}{(1-4t)^2+x^2+y^2}.$$
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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]].
 
The Davey-Stewartson system is a special case of the [[Zakharov-Schulman system]].
  
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[[Category:Integrability]]
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[[Category:Schrodinger]]
 
[[Category:Equations]]
 
[[Category:Equations]]

Latest revision as of 13:41, 29 May 2012

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

The field phi depends elliptically on u when is positive and thus one usually only specifies the initial data for , not . This equation is a two-dimensional modification of the one-dimensional cubic nonlinear Schrodinger equation and is completely integrable in the cases (DS-I) and (DS-II). The equation has a focusing nonlinearity when $ c_0 > 0 $. When 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 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 norm, but blows up in the $L^{\infty}$ and 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.