這張在法國大西洋 岸雷島 （RHE）鯨魚燈塔拍攝的照片，顯示淺海上田字形的橢圓餘弦波列。這種淺水中的孤波 可以由卡東穆夫-彼得韋亞斯維利方程模擬。 卡東穆塞夫-彼得韋亞斯維利方程 (Kadomtsev-Petviashvili equation)，簡稱KP方程，是1970年蘇聯物理學家波里斯·卡東穆塞夫 和弗拉基米爾-彼得韋亞斯維利創立以模擬非線性波動的非線性偏微分方程[ 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}
卡東穆塞夫-彼得韋亞斯維利方程有解析解[ 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)}
代人參數： 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函數展開法可得卡東穆塞夫-彼得韋亞斯維利方程的sech函數解和tanh函數解[ 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}
參數：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]
參數：a = 2, b = -2；
通過朗斯基行列式 展開法可得卡東塞穆夫-彼得韋亞斯維利方程多個雅可比橢圓函數解[ 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]}
代入後得：
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
^ 呂大昭等 Novel Interaction Solutions to Kadomtsev–Petviashvili Equation，Commun. Theor. Phys. (Beijing, China) 54 (2010) pp. 484–488
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李志斌編著 《非線性數學物理方程的行波解》 科學出版社
王東明著 《消去法及其應用》 科學出版社 2002
*何青 王麗芬編著 《Maple 教程》 科學出版社 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
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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
卡東穆塞夫-彼得韋亞斯維利方程sech函數亮孤立子
KP方程暗tanh函數暗孤立子
![{\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