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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
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