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

References

External links