科特韦赫-德弗里斯方程 （英語：Korteweg-De Vries equation ），一般简称KdV方程 ，是1895年由荷兰数学家科特韦赫 和德弗里斯 共同发现的一种偏微分方程 。关于实自变量x 和t 的函数 φ所满足的KdV方程形式如下：
∂
t
ϕ
−
6
ϕ
∂
x
ϕ
+
∂
x
3
ϕ
=
0
{\displaystyle \partial _{t}\phi -6\phi \partial _{x}\phi +\partial _{x}^{3}\phi =0}
KdV方程的解为簇集的孤立子 （又称孤子 ，孤波 ）。
KdV 方程有多种孤波解[ 1] [ 2]
ϕ
(
x
,
t
)
=
1
2
c
s
e
c
h
2
[
c
2
(
x
−
c
t
−
a
)
]
{\displaystyle \phi (x,t)={\frac {1}{2}}\,c\,\mathrm {sech} ^{2}\left[{{\sqrt {c}} \over 2}(x-c\,t-a)\right]}
ϕ
(
x
,
t
)
=
k
t
a
n
h
[
k
(
x
+
2
t
k
2
+
c
)
]
{\displaystyle \phi (x,t)=k\,\mathrm {tanh} [k(x+2tk^{2}+c)]}
ϕ
(
x
,
t
)
=
a
+
b
t
a
n
h
(
1
+
c
x
+
d
t
)
2
{\displaystyle \phi (x,t)=a+b\,\mathrm {tanh} (1+cx+dt)^{2}}
利用Maple tanh 法可得 孤立子解：[ 3]
u
(
x
,
t
)
=
(
1
/
6
)
∗
(
4
∗
C
2
3
−
C
3
)
/
C
2
−
2
∗
C
2
2
∗
c
s
c
(
C
1
+
C
2
∗
x
+
C
3
∗
t
)
2
{\displaystyle {u(x,t)=(1/6)*(4*_{C}2^{3}-_{C}3)/_{C}2-2*_{C}2^{2}*csc(_{C}1+_{C}2*x+_{C}3*t)^{2}}}
u
(
x
,
t
)
=
(
1
/
6
)
∗
(
4
∗
C
2
3
−
C
3
)
/
C
2
−
2
∗
C
2
2
∗
s
e
c
(
C
1
+
C
2
∗
x
+
C
3
∗
t
)
2
{\displaystyle {u(x,t)=(1/6)*(4*_{C}2^{3}-_{C}3)/_{C}2-2*_{C}2^{2}*sec(_{C}1+_{C}2*x+_{C}3*t)^{2}}}
u
(
x
,
t
)
=
−
(
1
/
6
)
∗
(
4
∗
C
2
3
+
C
3
)
/
C
2
−
2
∗
C
2
2
∗
c
s
c
h
(
C
1
+
C
2
∗
x
+
C
3
∗
t
)
2
{\displaystyle u(x,t)=-(1/6)*(4*_{C}2^{3}+_{C}3)/_{C}2-2*_{C}2^{2}*csch(_{C}1+_{C}2*x+_{C}3*t)^{2}}
u
(
x
,
t
)
=
−
(
1
/
6
)
∗
(
4
∗
C
2
3
+
C
3
)
/
C
2
+
2
∗
C
2
2
∗
s
e
c
h
(
C
1
+
C
2
∗
x
+
C
3
∗
t
)
2
{\displaystyle {u(x,t)=-(1/6)*(4*_{C}2^{3}+_{C}3)/_{C}2+2*_{C}2^{2}*sech(_{C}1+_{C}2*x+_{C}3*t)^{2}}}
u
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x
,
t
)
=
(
1
/
6
)
∗
(
8
∗
C
2
3
−
C
3
)
/
C
2
−
2
∗
C
2
2
∗
c
o
t
h
(
C
1
+
C
2
∗
x
+
C
3
∗
t
)
2
{\displaystyle {u(x,t)=(1/6)*(8*_{C}2^{3}-_{C}3)/_{C}2-2*_{C}2^{2}*coth(_{C}1+_{C}2*x+_{C}3*t)^{2}}}
u
(
x
,
t
)
=
(
1
/
6
)
∗
(
8
∗
C
2
3
−
C
3
)
/
C
2
−
2
∗
C
2
2
∗
t
a
n
h
(
C
1
+
C
2
∗
x
+
C
3
∗
t
)
2
{\displaystyle {u(x,t)=(1/6)*(8*_{C}2^{3}-_{C}3)/_{C}2-2*_{C}2^{2}*tanh(_{C}1+_{C}2*x+_{C}3*t)^{2}}}
u
(
x
,
t
)
=
−
(
1
/
6
)
∗
(
8
∗
C
2
3
+
C
3
)
/
C
2
−
2
∗
C
2
2
∗
c
o
t
(
C
1
+
C
2
∗
x
+
C
3
∗
t
)
2
{\displaystyle {u(x,t)=-(1/6)*(8*_{C}2^{3}+_{C}3)/_{C}2-2*_{C}2^{2}*cot(_{C}1+_{C}2*x+_{C}3*t)^{2}}}
u
(
x
,
t
)
=
−
(
1
/
6
)
∗
(
8
∗
C
2
3
+
C
3
)
/
C
2
−
2
∗
C
2
2
∗
t
a
n
(
C
1
+
C
2
∗
x
+
C
3
∗
t
)
2
{\displaystyle {u(x,t)=-(1/6)*(8*_{C}2^{3}+_{C}3)/_{C}2-2*_{C}2^{2}*tan(_{C}1+_{C}2*x+_{C}3*t)^{2}}}
u
(
x
,
t
)
=
(
1
/
6
)
∗
(
−
8
∗
C
3
3
+
4
∗
C
3
3
∗
C
1
2
−
C
4
)
/
C
3
+
2
∗
C
3
2
∗
J
a
c
o
b
i
D
N
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2
+
C
3
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x
+
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4
∗
t
,
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1
)
2
{\displaystyle {u(x,t)=(1/6)*(-8*_{C}3^{3}+4*_{C}3^{3}*_{C}1^{2}-_{C}4)/_{C}3+2*_{C}3^{2}*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}}}
u
(
x
,
t
)
=
(
1
/
6
)
∗
(
−
8
∗
C
3
3
+
4
∗
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3
3
∗
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4
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/
C
3
+
(
2
∗
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2
−
2
∗
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2
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2
)
∗
J
a
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i
N
D
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2
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C
3
∗
x
+
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4
∗
t
,
C
1
)
2
{\displaystyle {u(x,t)=(1/6)*(-8*_{C}3^{3}+4*_{C}3^{3}*_{C}1^{2}-_{C}4)/_{C}3+(2*_{C}3^{2}-2*_{C}3^{2}*_{C}1^{2})*JacobiND(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}}}
u
(
x
,
t
)
=
(
1
/
6
)
∗
(
4
∗
C
3
3
∗
C
1
2
+
4
∗
C
3
3
−
C
4
)
/
C
3
−
2
∗
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3
2
∗
J
a
c
o
b
i
N
S
(
C
2
+
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3
∗
x
+
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4
∗
t
,
C
1
)
2
{\displaystyle {u(x,t)=(1/6)*(4*_{C}3^{3}*_{C}1^{2}+4*_{C}3^{3}-_{C}4)/_{C}3-2*_{C}3^{2}*JacobiNS(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}}}
u
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x
,
t
)
=
(
1
/
6
)
∗
(
4
∗
C
3
3
∗
C
1
2
+
4
∗
C
3
3
−
C
4
)
/
C
3
−
2
∗
C
3
2
∗
C
1
2
∗
J
a
c
o
b
i
S
N
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+
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3
∗
x
+
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4
∗
t
,
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1
)
2
{\displaystyle {u(x,t)=(1/6)*(4*_{C}3^{3}*_{C}1^{2}+4*_{C}3^{3}-_{C}4)/_{C}3-2*_{C}3^{2}*_{C}1^{2}*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}}}
u
(
x
,
t
)
=
−
(
1
/
6
)
∗
(
8
∗
C
3
3
∗
C
1
2
−
4
∗
C
3
3
+
C
4
)
/
C
3
+
(
−
2
∗
C
3
2
+
2
∗
C
3
2
∗
C
1
2
)
∗
J
a
c
o
b
i
N
C
(
C
2
+
C
3
∗
x
+
C
4
∗
t
,
C
1
)
2
{\displaystyle {u(x,t)=-(1/6)*(8*_{C}3^{3}*_{C}1^{2}-4*_{C}3^{3}+_{C}4)/_{C}3+(-2*_{C}3^{2}+2*_{C}3^{2}*_{C}1^{2})*JacobiNC(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}}}
u
(
x
,
t
)
=
−
(
1
/
6
)
∗
(
8
∗
C
3
3
∗
C
1
2
−
4
∗
C
3
3
+
C
4
)
/
C
3
+
2
∗
C
3
2
∗
C
1
2
∗
J
a
c
o
b
i
C
N
(
C
2
+
C
3
∗
x
+
C
4
∗
t
,
C
1
)
2
{\displaystyle {u(x,t)=-(1/6)*(8*_{C}3^{3}*_{C}1^{2}-4*_{C}3^{3}+_{C}4)/_{C}3+2*_{C}3^{2}*_{C}1^{2}*JacobiCN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}}}
9.81207
−
7.70406
∗
I
+
5.44331
∗
a
r
c
t
a
n
h
(
10.4881
/
(
−
110.
∗
c
s
c
(
1.40000
+
1.50000
∗
x
+
1.60000
∗
t
)
2
+
110.
)
)
{\displaystyle 9.81207-7.70406*I+5.44331*arctanh(10.4881/{\sqrt {(}}-110.*csc(1.40000+1.50000*x+1.60000*t)^{2}+110.))}
9.81207
−
7.70406
∗
I
−
5.44331
∗
a
r
c
t
a
n
(
10.4881
/
(
−
110.
∗
c
s
c
h
(
1.40000
+
1.50000
∗
x
+
1.60000
∗
t
)
2
−
110.
)
)
{\displaystyle 9.81207-7.70406*I-5.44331*arctan(10.4881/{\sqrt {(}}-110.*csch(1.40000+1.50000*x+1.60000*t)^{2}-110.))}
9.81207
−
7.70406
∗
I
+
5.44331
∗
a
r
c
t
a
n
(
10.4881
/
(
−
110.
∗
c
s
c
h
(
1.40000
+
1.50000
∗
x
+
1.60000
∗
t
)
2
−
110.
)
)
{\displaystyle 9.81207-7.70406*I+5.44331*arctan(10.4881/{\sqrt {(}}-110.*csch(1.40000+1.50000*x+1.60000*t)^{2}-110.))}
KdV方程行波图 KdV方程行波图 KdV方程行波图 KdV方程行波图
KdV方程行波图 KdV方程行波图 KdV方程行波图 KdV方程行波图
KdV方程行波图 KdV方程行波图 KdV方程行波图 KdV方程行波图
KdV方程行波图 KdV方程行波图 KdV方程行波图 KdV方程行波图
KdV方程行波图 KdV方程行波图 KdV方程行波图 KdV方程行波图
KdV方程在物理学的许多领域都有应用，例如等离子体磁流波、离子声波、非谐振晶格振动、低温非线性晶格声子波包的热激发、液体气体混合物的压力表等。
KdV方程也可以用逆散射 技术求解。
Korteweg, D. J. and de Vries, F. "On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves." Philosophical Magazine, 39 , 422--443, 1895.
P. G. Drazin. Solitons . Cambridge University Press, 1983.
^ 阎振亚著 《复杂非线性波动构造性理论及其应用》 29页 科学出版社 2007
^ Graham W.Griffiths William E.Schiesser Traveling Wave Analysis of Partial Differential Equations p422-430
^ Graham W.Griffiths William E.Schiesser Traveling Wave Analysis of Partial Differential Equations p391-404
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*阎振亚著 《复杂非线性波的构造性理论及其应用》 科学出版社 2007年
李志斌编著 《非线性数学物理方程的行波解》 科学出版社
王东明著 《消去法及其应用》 科学出版社 2002
*何青 王丽芬编著 《Maple 教程》 科学出版社 2010 ISBN 9787030177445
Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
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Dongming Wang, Elimination Practice,Imperial College Press 2004
David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
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