Davey–Stewartson equation

Template:Short description 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 Template:Harvtxt.

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

• Template:Citation
• Template:Citation
• Template:Citation

External links

• Davey-Stewartson_system at the dispersive equations wiki.