Davey–Stewartson equation
In fluid dynamics, the Davey–Stewartson equation (DSE) was introduced in a paper by A. Davey and Keith Stewartson to describe the evolution of a three-dimensional wave-packet on water of finite depth. It is a system of partial differential equations for a complex (wave-amplitude) field [math]\displaystyle{ A\, }[/math] and a real (mean-flow) field [math]\displaystyle{ B }[/math]:
- [math]\displaystyle{ i \frac{\partial A}{\partial t} + c_0 \frac{\partial^2 A}{\partial x^2} + \frac{\partial A}{\partial y^2} = c_1 |A|^2 A + c_2 A\frac{\partial B}{\partial x}, }[/math]
- [math]\displaystyle{ \frac{\partial B}{\partial x^2} + c_3 \frac{\partial^2 B}{\partial y^2} = \frac{\partial |A|^2}{\partial x}. }[/math]
The DSE is an example of a soliton equation in 2+1 dimensions. The corresponding Lax representation for it is given in (Boiti Martina).
In 1+1 dimensions the DSE reduces to the nonlinear Schrödinger equation
- [math]\displaystyle{ i \frac{\partial A}{\partial t} + \frac{\partial^2 A}{\partial x^2} + 2k |A|^2 A =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 modulated nonlinear surface gravity waves, propagating over a horizontal sea bed.
See also
- Ginzburg–Landau equation
- Nonlinear systems
- Ishimori equation
References
- Boiti, M.; Martina, L.; Pempinelli, F. (December 1995), "Multidimensional localized solitons", Chaos, Solitons & Fractals 5 (12): 2377–2417, doi:10.1016/0960-0779(94)E0106-Y, ISSN 0960-0779, Bibcode: 1995CSF.....5.2377B
- Davey, A.; Stewartson, K. (1974), "On three dimensional packets of surface waves", Proc. R. Soc. A 338 (1613): 101–110, doi:10.1098/rspa.1974.0076, Bibcode: 1974RSPSA.338..101D
- Sattinger, David H.; Tracy, C. A.; Venakides, S., eds. (1991), Inverse Scattering and Applications, Contemporary Mathematics, 122, Providence, RI: American Mathematical Society, ISBN 0-8218-5129-2, https://archive.org/details/inversescatterin0000amsi
External links
- Davey-Stewartson_system at the dispersive equations wiki.
 |  |
