Davey–Stewartson equation

From HandWiki

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

References

External links