卡东穆塞夫-彼得韦亚斯维利方程有解析解[2]
行波解
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)}
Kadomtsev-Petviahivili equation 3D plot
Kadomtsev-Petviashivili pde animation
Sech 函数亮孤立子解 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
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
tanh 函数解
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；
雅可比椭圆函数解 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椭圆函数解[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}}
Kadomtsev Petviashivili pde elliptic function solution 3d plot
