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 Ginzburg–Landau equation Nonlinear systems Ishimori equation
References Boiti, M.; Martina, L.; Pempinelli, F. (December 1995), "Multidimensional localized solitons", Chaos, Solitons & Fractals, 5 (12): 2377–2417, arXiv:patt-sol/9311002, Bibcode:1995CSF.....5.2377B, doi:10.1016/0960-0779(94)E0106-Y, ISSN 0960-0779, S2CID 1232249 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, S2CID 121348168 Sattinger, David H.; Tracy, C. A.; Venakides, S., eds. (1991), Inverse Scattering and Applications, Contemporary Mathematics, vol. 122, Providence, RI: American Mathematical Society, ISBN 0-8218-5129-2, MR 1135850
External links Davey-Stewartson_system at the dispersive equations wiki.