这张在法国大西洋 岸雷岛 （RHE）鲸鱼灯塔拍摄的照片，显示浅海上田字形的椭圆余弦波列。这种浅水中的孤波 可以由卡东穆夫-彼得韦亚斯维利方程模拟。 卡东穆塞夫-彼得韦亚斯维利方程 (Kadomtsev-Petviashvili equation)，简称KP方程，是1970年苏联物理学家波里斯·卡东穆塞夫 和弗拉基米尔-彼得韦亚斯维利创立以模拟非线性波动的非线性偏微分方程[ 1]
∂
x
(
∂
t
u
+
u
∂
x
u
+
ϵ
2
∂
x
x
x
u
)
+
λ
∂
y
y
u
=
0
{\displaystyle \displaystyle \partial _{x}(\partial _{t}u+u\partial _{x}u+\epsilon ^{2}\partial _{xxx}u)+\lambda \partial _{yy}u=0}
其中
λ
=
±
1
{\displaystyle \lambda =\pm 1}
卡东穆塞夫-彼得韦亚斯维利方程有解析解[ 2]
u
(
x
,
y
,
t
)
=
C
5
+
12.
∗
C
2
∗
tanh
(
C
1
+
C
2
∗
x
+
C
3
∗
y
−
(
.50000000000000000000
∗
(
8.
∗
C
2
4
+
C
3
2
)
)
∗
t
/
C
2
)
{\displaystyle u(x,y,t)=C5+12.*_{C}2*\tanh(_{C}1+_{C}2*x+_{C}3*y-(.50000000000000000000*(8.*_{C}2^{4}+_{C}3^{2}))*t/_{C}2)}
代人参数： C5 = 1, _C1 = 0, _C2 = 1, _C3 = 3
得：
u
=
1
+
12.
∗
t
a
n
h
(
x
+
3
∗
y
−
8.5000000000000000000
∗
t
)
{\displaystyle u=1+12.*tanh(x+3*y-8.5000000000000000000*t)}
利用sech函数展开法可得卡东穆塞夫-彼得韦亚斯维利方程的sech函数解和tanh函数解[ 3]
u
:=
a
∗
s
e
c
h
(
a
∗
x
+
b
∗
y
+
c
∗
z
−
(
a
4
+
3
∗
b
2
+
3
∗
c
2
)
/
a
)
∗
t
{\displaystyle u:=a*sech(a*x+b*y+c*z-(a^{4}+3*b^{2}+3*c^{2})/a)*t}
参数：a = -2 .. 2, b = -2 .. 2, c = 0
u
:=
2
∗
a
2
∗
t
a
n
h
(
a
∗
x
+
b
∗
y
+
(
8
∗
a
4
−
3
∗
b
2
)
/
a
)
2
∗
t
{\displaystyle u:=2*a^{2}*tanh(a*x+b*y+(8*a^{4}-3*b^{2})/a)^{2}*t}
[ 4]
参数：a = 2, b = -2；
通过朗斯基行列式 展开法可得卡东塞穆夫-彼得韦亚斯维利方程多个雅可比橢圓函數解[ 5]
u
4
:=
(
−
4
∗
m
2
∗
k
[
1
]
2
∗
g
)
(
1
−
m
2
∗
s
n
(
ξ
[
1
]
,
k
)
∗
s
i
n
(
ξ
[
2
]
)
+
d
n
(
ξ
[
1
]
,
k
)
∗
c
o
s
(
ξ
[
2
]
)
∗
c
n
(
ξ
[
1
]
,
k
)
)
2
)
{\displaystyle u4:={\frac {(-4*m^{2}*k[1]^{2}*g)}{({\sqrt {1-m^{2}}}*sn(\xi [1],k)*sin(\xi [2])+dn(\xi [1],k)*cos(\xi [2])*cn(\xi [1],k))^{2})}}}
其中：
g
=
(
m
2
−
1
)
∗
s
n
(
ξ
[
1
]
,
k
)
2
+
(
2
−
2
∗
m
2
)
∗
s
n
(
ξ
[
1
]
,
k
)
4
+
c
o
s
(
ξ
[
2
]
)
2
;
−
2
∗
s
n
(
ξ
[
1
]
,
k
)
2
∗
c
o
s
(
ξ
[
2
]
)
2
+
m
2
∗
s
n
(
ξ
[
1
]
,
k
)
4
∗
c
o
s
(
ξ
[
2
]
)
2
{\displaystyle g=(m^{2}-1)*sn(\xi [1],k)^{2}+(2-2*m^{2})*sn(\xi [1],k)^{4}+cos(\xi [2])^{2};-2*sn(\xi [1],k)^{2}*cos(\xi [2])^{2}+m^{2}*sn(\xi [1],k)^{4}*cos(\xi [2])^{2}}
ξ
[
1
]
=
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
(
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
+
γ
[
1
]
{\displaystyle \xi [1]=k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1]}
ξ
[
2
]
=
1
−
m
2
∗
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
(
−
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
)
−
γ
[
2
]
{\displaystyle \xi [2]={\sqrt {1-m^{2}}}*(k[1]*x+\lambda [1]*y+(-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2]}
代入后得：
f
4
:=
−
4
∗
m
2
∗
k
[
1
]
2
∗
(
(
m
2
−
1
)
∗
J
a
c
o
b
i
S
N
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
(
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
{\displaystyle f4:=-4*m^{2}*k[1]^{2}*((m^{2}-1)*JacobiSN(k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}}
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
+
γ
[
1
]
,
k
)
2
+
(
2
−
2
∗
m
2
)
∗
J
a
c
o
b
i
S
N
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
{\displaystyle -3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)^{2}+(2-2*m^{2})*JacobiSN(k[1]*x+\lambda [1]*y}
+
(
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
+
γ
[
1
]
,
k
)
4
+
{\displaystyle +(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)^{4}+}
c
o
s
(
(
1
−
m
2
)
∗
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
(
−
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
)
{\displaystyle cos({\sqrt {(1-m^{2})}}*(k[1]*x+\lambda [1]*y+(-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)}
−
γ
[
2
]
)
2
)
/
(
s
q
r
t
(
1
−
m
2
)
∗
J
a
c
o
b
i
S
N
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
(
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
{\displaystyle -\gamma [2])^{2})/(sqrt(1-m^{2})*JacobiSN(k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-}
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
+
γ
[
1
]
,
k
)
∗
s
i
n
(
(
1
−
m
2
)
∗
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
{\displaystyle 3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)*sin({\sqrt {(}}1-m^{2})*(k[1]*x+\lambda [1]*y+}
(
−
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
)
−
γ
[
2
]
)
+
J
a
c
o
b
i
D
N
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
{\displaystyle (-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2])+JacobiDN(k[1]*x+\lambda [1]*y+}
(
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
+
γ
[
1
]
,
k
)
∗
c
o
s
(
1
−
m
2
∗
(
k
[
1
]
∗
x
+
λ
[
1
]
∗
y
+
{\displaystyle (4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)*cos({\sqrt {1-m^{2}}}*(k[1]*x+\lambda [1]*y+}
(
−
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
)
−
γ
[
2
]
)
∗
J
a
c
o
b
i
C
N
(
k
[
1
]
∗
x
+
{\displaystyle (-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2])*JacobiCN(k[1]*x+}
λ
[
1
]
∗
y
+
(
4
∗
m
2
∗
k
[
1
]
3
+
16
∗
k
[
1
]
3
−
3
∗
σ
2
∗
λ
[
1
]
2
/
k
[
1
]
)
∗
t
+
γ
[
1
]
,
k
)
)
2
{\displaystyle \lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k))^{2}}
^ Kodomtsev,B.B and Petviashivili V.I. On the stability of solitary waves in weakly dispersive media Dokl. Akad Nauk SSSR 192 753-6(1970) Soviet Phys. Dok 15,539-41(1970)
^ Erk Infeld & George Rowlands, Nonlinear Waves,Solitons and Chaos p224-233 Cambridge University Press,2000
^ AHMET BEKIR and ÖZKAN GÜNER Bright and dark soliton solutions of the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation and generalized Benjamin equation,journal of Physics, August 2013 Vol. 81, No. 2, pp. 203–214
^ AHMET BEKIR and ÖZKAN GÜNER
^ 吕大昭等 Novel Interaction Solutions to Kadomtsev–Petviashvili Equation，Commun. Theor. Phys. (Beijing, China) 54 (2010) pp. 484–488
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王东明著 《消去法及其应用》 科学出版社 2002
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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
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George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759
Kadomtsev-Petviahivili equation 3D plot
Kadomtsev-Petviashivili pde animation
Kadomtsev Petviashivili pde sech solution 3d plot
卡东穆塞夫-彼得韦亚斯维利方程sech函数亮孤立子
KP方程暗tanh函数暗孤立子
![{\displaystyle u4:={\frac {(-4*m^{2}*k[1]^{2}*g)}{({\sqrt {1-m^{2}}}*sn(\xi [1],k)*sin(\xi [2])+dn(\xi [1],k)*cos(\xi [2])*cn(\xi [1],k))^{2})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c2132eb6e06bb3124acd0f9e8206516877edcf6)
![{\displaystyle g=(m^{2}-1)*sn(\xi [1],k)^{2}+(2-2*m^{2})*sn(\xi [1],k)^{4}+cos(\xi [2])^{2};-2*sn(\xi [1],k)^{2}*cos(\xi [2])^{2}+m^{2}*sn(\xi [1],k)^{4}*cos(\xi [2])^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5a5d40a7f0fd0d1ffe96ed53ce8eef12c05d813)
![{\displaystyle \xi [1]=k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81d3c21a85c439a737d9ccf8cd011e93a136735c)
![{\displaystyle \xi [2]={\sqrt {1-m^{2}}}*(k[1]*x+\lambda [1]*y+(-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c79f15cc0b827b45d523263d173c7118650f8e1f)
![{\displaystyle f4:=-4*m^{2}*k[1]^{2}*((m^{2}-1)*JacobiSN(k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c6de7c73f0f5f3341ab9863ae5b3298c5772c52)
![{\displaystyle -3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)^{2}+(2-2*m^{2})*JacobiSN(k[1]*x+\lambda [1]*y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f6f3aa22e2d753e441baac5d4c55ce7573758fb)
![{\displaystyle +(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)^{4}+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f96ac6d59d29ff916eb7d343fcbe2600316aa6a)
![{\displaystyle cos({\sqrt {(1-m^{2})}}*(k[1]*x+\lambda [1]*y+(-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/120441e5c5bc8dcc1401c69b7a49bd54dcc5ad4e)
![{\displaystyle -\gamma [2])^{2})/(sqrt(1-m^{2})*JacobiSN(k[1]*x+\lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b81dc5fd1e7e556e1c50361d9d0a85f4d05fc00)
![{\displaystyle 3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)*sin({\sqrt {(}}1-m^{2})*(k[1]*x+\lambda [1]*y+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59db65c30c4cce75a956c249210e63fa475aaf66)
![{\displaystyle (-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2])+JacobiDN(k[1]*x+\lambda [1]*y+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7efad1ca0b0f4c4f7255c122a3f0adc70c1f2cd8)
![{\displaystyle (4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k)*cos({\sqrt {1-m^{2}}}*(k[1]*x+\lambda [1]*y+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/675e7fe0bec8443c0362cc4ec7e31ce9d8d1da09)
![{\displaystyle (-4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t)-\gamma [2])*JacobiCN(k[1]*x+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f42ef1652973680628cafe8b641fe6a1b82ed86)
![{\displaystyle \lambda [1]*y+(4*m^{2}*k[1]^{3}+16*k[1]^{3}-3*\sigma ^{2}*\lambda [1]^{2}/k[1])*t+\gamma [1],k))^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5db7390bfeb8e410275d9195ddbd90e07d176a23)
Kadomtsev Petviashivili pde elliptic function solution 3d plot