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 $A$ and a real (mean-flow) field $B$:

$i\frac{\partial A}{\partial t}+c_0\frac{{\partial }^{2}A}{\partial x^2}+\frac{\partial A}{\partial y^2}=c_1|A{|}^{2}A+c_{2}A\frac{\partial B}{\partial x},$

$\frac{\partial B}{\partial x^2}+c_3\frac{{\partial }^{2}B}{\partial y^2}=\frac{\partial |A{|}^2}{\partial x}.$

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

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

$i\frac{\partial A}{\partial t}+\frac{{\partial }^{2}A}{\partial x^2}+2k|A{|}^{2}A=0.$

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.

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.