Davey-Stewartson system: Difference between revisions

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The '''Davey-Stewartson system''' [DavSte1974] in 2 spatial dimensions involves a complex field u and a real field phi:  
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>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>
\partial_x^2 phi + c_3 \partial_y^2 phi = partial_x ( |u|^2 )
 
<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].
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].

Revision as of 03:53, 28 July 2006

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 modification of the cubic nonlinear Schrodinger equation and is completely integrable in the cases (DS-I) and (DS-II). When the situation becomes a nonlinear Schrodinger equation with non-local nonlinearity, 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].

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