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In [[fluid dynamics]], the '''Davey–Stewartson equation (DSE)''' was introduced in a paper by {{harvtxt|Davey|Stewartson|1974}} to describe the evolution of a three-dimensional [[wave packet|wave-packet]] on water of finite depth.
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It is a system of partial differential equations for a complex ([[amplitude|wave-amplitude]]) [[field (physics)|field]] <math>u\,</math> and a real ([[mean flow|mean-flow]]) field <math>\phi\,</math>:
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:<math>i u_t + c_0 u_{xx} + u_{yy} = c_1 |u|^2 u + c_2 u \phi_x,\,</math>
 
:<math>\phi_{xx} + c_3 \phi_{yy} = ( |u|^2 )_x.\,</math>
 
The DSE is an example of a [[soliton]] equation in 2+1 dimensions. The corresponding Lax representation for it is given in {{harvtxt|Boiti|Martina|Pempinelli|1995}}.
 
In 1+1 dimensions the DSE reduces to the [[nonlinear Schrödinger equation]]
 
:<math>i u_t + u_{xx} + 2k |u|^2 u =0.\,</math>
 
Itself, the DSE is the particular reduction of the [[Zakharov–Schulman system]].  On the other hand, the equivalent counterpart of the DSE is the [[Ishimori equation]].  
 
The DSE is the result of a [[multiple-scale analysis]] of [[modulation|modulated]] nonlinear [[surface gravity wave]]s, propagating over a horizontal sea bed.
 
==See also==
 
* [[Nonlinear systems]]
* [[Ishimori equation]]
 
==References==
*{{citation|unused_data=Solitons in science and engineering: theory and applications.|mr=1368226
|last=Boiti|first= M.|last2= Martina|first2= L.|last3= Pempinelli|first3= F.
|title=Multidimensional localized solitons
|journal=Chaos Solitons Fractals |volume=5 |year=1995|issue=12|pages= 2377–2417
|doi=10.1016/0960-0779(94)E0106-Y |bibcode = 1995CSF.....5.2377B }}
*{{citation|last=Davey|first= A.|authorlink2=Keith Stewartson|last2=Stewartson |first2=K.|title=On three dimensional packets of surface waves|journal= Proc. R. Soc. A |volume=338 |year=1974|issue=1613|pages= 101–110
|doi=10.1098/rspa.1974.0076|bibcode = 1974RSPSA.338..101D }}
*{{citation|editor1-first=David H.|editor1-last= Sattinger|editor2-first= C. A. |editor2-last=Tracy |editor3-first=S. |editor3-last= Venakides |title=Inverse Scattering and Applications |year= 1991
|mr=1135850 |series= Contemporary Mathematics |volume= 122 |publisher= American Mathematical Society |place= Providence, RI |isbn=0-8218-5129-2 }}
 
==External links==
*[http://tosio.math.toronto.edu/wiki/index.php/Davey-Stewartson_system Davey-Stewartson_system] at the dispersive equations wiki.
 
{{DEFAULTSORT:Davey-Stewartson equation}}
[[Category:Partial differential equations]]
[[Category:Equations of fluid dynamics]]

Latest revision as of 03:56, 29 December 2014

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