Davey–Stewartson equation

Davey–Stewartson equation

In fluid dynamics, the Davey–Stewartson equation (DSE) was introduced in a paper by Davey & Stewartson (1974) 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 u\, and a real (mean-flow) field \phi\,:

i u_t + c_0 u_{xx} + u_{yy} = c_1 |u|^2 u + c_2 u \phi_x,\,
\phi_{xx} + c_3 \phi_{yy} = ( |u|^2 )_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 u_t + u_{xx} + 2k |u|^2 u =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

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