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\,}
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
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 ISBN 0-8218-5129-2 MR 1135850
External links
