Misplaced Pages

Davey–Stewartson equation

Article snapshot taken from[REDACTED] with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

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 {\displaystyle A\,} and a real (mean-flow) field B {\displaystyle B} :

i ∂ A ∂ t + c 0 ∂ 2 A ∂ x 2 + ∂ A ∂ y 2 = c 1 | A | 2 A + c 2 A ∂ B ∂ x , {\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}},}
∂ B ∂ x 2 + c 3 ∂ 2 B ∂ y 2 = ∂ | A | 2 ∂ x . {\displaystyle {\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 & Pempinelli (1995).

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

i ∂ A ∂ t + ∂ 2 A ∂ x 2 + 2 k | A | 2 A = 0. {\displaystyle 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

References

External links


[REDACTED]

This article about theoretical physics is a stub. You can help Misplaced Pages by expanding it.

Categories:
Davey–Stewartson equation Add topic