# Davey–Stewartson equation

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In fluid dynamics, the Davey–Stewartson equation (DSE) was introduced in a paper by A. Davey and Keith Stewartson (1974) to describe the evolution of a three-dimensional wave-packet on water of finite depth.

## Contents

It is a system of partial differential equations for a complex (wave-amplitude) field ${\displaystyle u\,}$ and a real (mean-flow) field ${\displaystyle \phi \,}$:

${\displaystyle iu_{t}+c_{0}u_{xx}+u_{yy}=c_{1}|u|^{2}u+c_{2}u\phi _{x},\,}$
${\displaystyle \phi _{xx}+c_{3}\phi _{yy}=(|u|^{2})_{x}.\,}$

The DSE is an example of a soliton equation in 2+1 dimensions. The corresponding Lax representation for it is given in Boiti, Martina & Pempinelli (1995).

In 1+1 dimensions the DSE reduces to the nonlinear Schrödinger equation

${\displaystyle iu_{t}+u_{xx}+2k|u|^{2}u=0.\,}$

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 modulated nonlinear surface gravity waves, propagating over a horizontal sea bed.

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## References

• Boiti, M.; Martina, L.; Pempinelli, F. (December 1995), "Multidimensional localized solitons", Chaos, Solitons & Fractals, 5 (12): 2377–2417, arXiv:, Bibcode:1995CSF.....5.2377B, doi:10.1016/0960-0779(94)E0106-Y, ISSN   0960-0779
• Davey, A.; Stewartson, K. (1974), "On three dimensional packets of surface waves", Proc. R. Soc. A, 338 (1613): 101–110, Bibcode:1974RSPSA.338..101D, doi:10.1098/rspa.1974.0076
• Sattinger, David H.; Tracy, C. A.; Venakides, S., eds. (1991), , Contemporary Mathematics, 122, Providence, RI: American Mathematical Society, ISBN   0-8218-5129-2, MR   1135850