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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.

Nonlinear systems
Ishimori equation

References

Boiti, M.; Martina, L.; Pempinelli, F. (1995), "Multidimensional localized solitons", Chaos Solitons Fractals 5 (12): 2377–2417, Bibcode 1995CSF.....5.2377B, doi:10.1016/0960-0779(94)E0106-Y, MR 1368226
Davey, A.; Stewartson, K. (1974), "On three dimensional packets of surface waves", Proc. R. Soc. A 338 (1613): 101–110, Bibcode 1974RSPSA.338..101D, doi:10.1098/rspa.1974.0076
Sattinger, David H.; Tracy, C. A.; Venakides, S., eds. (1991), Inverse Scattering and Applications, Contemporary Mathematics, 122, Providence, RI: American Mathematical Society, ISBN 0-8218-5129-2, MR 1135850