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
(
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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C
2
+
C
3
∗
x
+
C
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}*JacobiDN(_{C}2+_{C}3*x+_{C}4*t,_{C}1)^{2}}}
u
(
x
,
t
)
=
(
1
/
6
)
∗
(
−
8
∗
C
3
3
+
4
∗
C
3
3
∗
C
1
2
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C
4
)
/
C
3
+
(
2
∗
C
3
2
−
2
∗
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3
2
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C
1
2
)
∗
J
a
c
o
b
i
N
D
(
C
2
+
C
3
∗
x
+
C
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
∗
C
3
2
∗
J
a
c
o
b
i
N
S
(
C
2
+
C
3
∗
x
+
C
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
(
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
(
C
2
+
C
3
∗
x
+
C
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}*_{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
*谷超豪 《孤立子 理论中的达布变换 及其几何应用》 上海科学技术出版社
*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
Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
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
George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759
![{\displaystyle \phi (x,t)={\frac {1}{2}}\,c\,\mathrm {sech} ^{2}\left[{{\sqrt {c}} \over 2}(x-c\,t-a)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/129751a6e06e076ca786e38d49260ed60586f0df)
![{\displaystyle \phi (x,t)=k\,\mathrm {tanh} [k(x+2tk^{2}+c)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4407f6394ecdc0534654bd093872ff8c387e8f8)
钟形孤波解
扭形孤波解
暗孤波解
