库普-库珀施密特方程 (Kaup-Kupershmidt Equation)是一个非线性偏微分方程：[1]
∂
4
u
(
x
,
t
)
∂
x
4
+
∂
u
(
x
,
t
)
∂
x
+
45
(
∂
u
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x
,
t
)
∂
x
∗
u
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x
,
t
)
2
−
(
75
/
2
)
∗
∂
2
u
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x
,
t
)
∂
x
2
∗
∂
u
(
x
,
t
)
∂
x
−
15
∗
u
(
x
,
t
)
∗
∂
3
u
(
x
,
t
)
∂
x
3
{\displaystyle {\frac {\partial ^{4}u(x,t)}{\partial x^{4}}}+{\frac {\partial u(x,t)}{\partial x}}+45({\frac {\partial u(x,t)}{\partial x}}*u(x,t)^{2}-(75/2)*{\frac {\partial ^{2}u(x,t)}{\partial x^{2}}}*{\frac {\partial u(x,t)}{\partial x}}-15*u(x,t)*{\frac {\partial ^{3}u(x,t)}{\partial x^{3}}}}
利用Maple 软件包TWSolution，随所选定展开函数不同，可得多种行波解[2]
tanh 展开
g
[
2
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:=
u
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x
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=
−
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/
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r
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s
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)
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2
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∗
t
a
n
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r
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I
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s
q
r
t
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2
)
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∗
x
+
C
3
∗
t
)
2
{\displaystyle g[2]:={u(x,t)=-(2/3)*(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}*tanh(_{C}1+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
3
]
:=
u
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x
,
t
)
=
−
(
2
/
3
)
∗
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
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2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
+
(
−
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r
t
(
2
)
+
(
1
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∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
t
a
n
h
(
C
1
+
(
−
(
1
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)
∗
s
q
r
t
(
2
)
+
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1
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2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[3]:={u(x,t)=-(2/3)*(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}+(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}*tanh(_{C}1+(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
4
]
:=
u
(
x
,
t
)
=
−
(
2
/
3
)
∗
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
+
(
(
1
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2
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∗
s
q
r
t
(
2
)
−
(
1
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2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
t
a
n
h
(
C
1
+
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[4]:={u(x,t)=-(2/3)*((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}+((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}*tanh(_{C}1+((1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
5
]
:=
u
(
x
,
t
)
=
−
(
2
/
3
)
∗
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
+
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
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2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
t
a
n
h
(
C
1
+
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[5]:={u(x,t)=-(2/3)*((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}+((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}*tanh(_{C}1+((1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
6
]
:=
u
(
x
,
t
)
=
−
(
4
/
3
)
∗
(
−
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
+
2
∗
(
−
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
∗
t
a
n
h
(
C
1
+
(
−
(
1
/
44
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
44
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[6]:={u(x,t)=-(4/3)*(-(1/22)*sqrt(2)*11^{(}3/4)-(1/22*I)*sqrt(2)*11^{(}3/4))^{2}+2*(-(1/22)*sqrt(2)*11^{(}3/4)-(1/22*I)*sqrt(2)*11^{(}3/4))^{2}*tanh(_{C}1+(-(1/44)*sqrt(2)*11^{(}3/4)-(1/44*I)*sqrt(2)*11^{(}3/4))*x+_{C}3*t)^{2}}}
g
[
7
]
:=
u
(
x
,
t
)
=
−
(
4
/
3
)
∗
(
−
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
+
2
∗
(
−
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
∗
t
a
n
h
(
C
1
+
(
−
(
1
/
44
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
44
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[7]:={u(x,t)=-(4/3)*(-(1/22)*sqrt(2)*11^{(}3/4)+(1/22*I)*sqrt(2)*11^{(}3/4))^{2}+2*(-(1/22)*sqrt(2)*11^{(}3/4)+(1/22*I)*sqrt(2)*11^{(}3/4))^{2}*tanh(_{C}1+(-(1/44)*sqrt(2)*11^{(}3/4)+(1/44*I)*sqrt(2)*11^{(}3/4))*x+_{C}3*t)^{2}}}
g
[
8
]
:=
u
(
x
,
t
)
=
−
(
4
/
3
)
∗
(
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
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22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
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4
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2
+
2
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(
1
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22
)
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q
r
t
(
2
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11
(
3
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4
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(
1
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s
q
r
t
(
2
)
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11
(
3
/
4
)
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2
∗
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a
n
h
(
C
1
+
(
(
1
/
44
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
44
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[8]:={u(x,t)=-(4/3)*((1/22)*sqrt(2)*11^{(}3/4)-(1/22*I)*sqrt(2)*11^{(}3/4))^{2}+2*((1/22)*sqrt(2)*11^{(}3/4)-(1/22*I)*sqrt(2)*11^{(}3/4))^{2}*tanh(_{C}1+((1/44)*sqrt(2)*11^{(}3/4)-(1/44*I)*sqrt(2)*11^{(}3/4))*x+_{C}3*t)^{2}}}
g
[
9
]
:=
u
(
x
,
t
)
=
−
(
4
/
3
)
∗
(
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
+
2
∗
(
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
∗
t
a
n
h
(
C
1
+
(
(
1
/
44
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
44
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[9]:={u(x,t)=-(4/3)*((1/22)*sqrt(2)*11^{(}3/4)+(1/22*I)*sqrt(2)*11^{(}3/4))^{2}+2*((1/22)*sqrt(2)*11^{(}3/4)+(1/22*I)*sqrt(2)*11^{(}3/4))^{2}*tanh(_{C}1+((1/44)*sqrt(2)*11^{(}3/4)+(1/44*I)*sqrt(2)*11^{(}3/4))*x+_{C}3*t)^{2}}}
Kaup Kupershmidt eq tanh method animation2
Kaup Kupershmidt eq tanh method animation7
Kaup Kupershmidt eq tanh method animation8
JacobiSN 展开
g
[
2
]
:=
u
(
x
,
t
)
=
−
(
1
/
2
)
∗
C
3
2
−
(
1
/
6
)
∗
s
q
r
t
(
−
3
∗
C
3
4
−
4
)
+
(
(
1
/
2
)
∗
C
3
2
+
(
1
/
2
)
∗
s
q
r
t
(
−
3
∗
C
3
4
−
4
)
)
∗
J
a
c
o
b
i
S
N
(
C
2
+
C
3
∗
x
+
C
4
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t
,
(
1
/
2
)
∗
s
q
r
t
(
2
∗
C
3
2
+
2
∗
s
q
r
t
(
−
3
∗
C
3
4
−
4
)
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/
C
3
)
2
{\displaystyle g[2]:={u(x,t)=-(1/2)*_{C}3^{2}-(1/6)*sqrt(-3*_{C}3^{4}-4)+((1/2)*_{C}3^{2}+(1/2)*sqrt(-3*_{C}3^{4}-4))*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*sqrt(2*_{C}3^{2}+2*sqrt(-3*_{C}3^{4}-4))/_{C}3)^{2}}}
g
[
3
]
:=
u
(
x
,
t
)
=
−
(
1
/
2
)
∗
C
3
2
+
(
1
/
6
)
∗
s
q
r
t
(
−
3
∗
C
3
4
−
4
)
+
(
(
1
/
2
)
∗
C
3
2
−
(
1
/
2
)
∗
s
q
r
t
(
−
3
∗
C
3
4
−
4
)
)
∗
J
a
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b
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N
(
C
2
+
C
3
∗
x
+
C
4
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t
,
(
1
/
2
)
∗
s
q
r
t
(
2
∗
C
3
2
−
2
∗
s
q
r
t
(
−
3
∗
C
3
4
−
4
)
)
/
C
3
)
2
{\displaystyle g[3]:={u(x,t)=-(1/2)*_{C}3^{2}+(1/6)*sqrt(-3*_{C}3^{4}-4)+((1/2)*_{C}3^{2}-(1/2)*sqrt(-3*_{C}3^{4}-4))*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,(1/2)*sqrt(2*_{C}3^{2}-2*sqrt(-3*_{C}3^{4}-4))/_{C}3)^{2}}}
g
[
4
]
:=
u
(
x
,
t
)
=
−
4
∗
C
3
2
−
(
2
/
33
)
∗
s
q
r
t
(
−
1452
∗
C
3
4
−
11
)
+
(
4
∗
C
3
2
+
(
2
/
11
)
∗
s
q
r
t
(
−
1452
∗
C
3
4
−
11
)
)
∗
J
a
c
o
b
i
S
N
(
C
2
+
C
3
∗
x
+
C
4
∗
t
,
(
1
/
22
)
∗
s
q
r
t
(
242
∗
C
3
2
+
11
∗
s
q
r
t
(
−
1452
∗
C
3
4
−
11
)
)
/
C
3
)
2
{\displaystyle g[4]:={u(x,t)=-4*_{C}3^{2}-(2/33)*sqrt(-1452*_{C}3^{4}-11)+(4*_{C}3^{2}+(2/11)*sqrt(-1452*_{C}3^{4}-11))*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,(1/22)*sqrt(242*_{C}3^{2}+11*sqrt(-1452*_{C}3^{4}-11))/_{C}3)^{2}}}
g
[
5
]
:=
u
(
x
,
t
)
=
−
4
∗
C
3
2
+
(
2
/
33
)
∗
s
q
r
t
(
−
1452
∗
C
3
4
−
11
)
+
(
4
∗
C
3
2
−
(
2
/
11
)
∗
s
q
r
t
(
−
1452
∗
C
3
4
−
11
)
)
∗
J
a
c
o
b
i
S
N
(
C
2
+
C
3
∗
x
+
C
4
∗
t
,
(
1
/
22
)
∗
s
q
r
t
(
242
∗
C
3
2
−
11
∗
s
q
r
t
(
−
1452
∗
C
3
4
−
11
)
)
/
C
3
)
2
{\displaystyle g[5]:={u(x,t)=-4*_{C}3^{2}+(2/33)*sqrt(-1452*_{C}3^{4}-11)+(4*_{C}3^{2}-(2/11)*sqrt(-1452*_{C}3^{4}-11))*JacobiSN(_{C}2+_{C}3*x+_{C}4*t,(1/22)*sqrt(242*_{C}3^{2}-11*sqrt(-1452*_{C}3^{4}-11))/_{C}3)^{2}}}
Kaup Kupershmidt JacobiSN method animation2
Kaup Kupershmidt JacobiSN method animation3
Kaup Kupershmidt JacobiSN method animation4
sech 展开
g
[
2
]
:=
u
(
x
,
t
)
=
(
1
/
3
)
∗
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
−
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
s
e
c
h
(
C
1
+
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[2]:={u(x,t)=(1/3)*(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}-(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}*sech(_{C}1+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
3
]
:=
u
(
x
,
t
)
=
(
1
/
3
)
∗
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
−
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
s
e
c
h
(
C
1
+
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[3]:={u(x,t)=(1/3)*(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}-(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}*sech(_{C}1+(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
4
]
:=
u
(
x
,
t
)
=
(
1
/
3
)
∗
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
−
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
s
e
c
h
(
C
1
+
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[4]:={u(x,t)=(1/3)*((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}-((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}*sech(_{C}1+((1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
5
]
:=
u
(
x
,
t
)
=
(
1
/
3
)
∗
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
−
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
s
e
c
h
(
C
1
+
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[5]:={u(x,t)=(1/3)*((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}-((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}*sech(_{C}1+((1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
6
]
:=
u
(
x
,
t
)
=
(
2
/
3
)
∗
(
−
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
−
2
∗
(
−
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
∗
s
e
c
h
(
C
1
+
(
−
(
1
/
44
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
44
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[6]:={u(x,t)=(2/3)*(-(1/22)*sqrt(2)*11^{(}3/4)-(1/22*I)*sqrt(2)*11^{(}3/4))^{2}-2*(-(1/22)*sqrt(2)*11^{(}3/4)-(1/22*I)*sqrt(2)*11^{(}3/4))^{2}*sech(_{C}1+(-(1/44)*sqrt(2)*11^{(}3/4)-(1/44*I)*sqrt(2)*11^{(}3/4))*x+_{C}3*t)^{2}}}
g
[
7
]
:=
u
(
x
,
t
)
=
(
2
/
3
)
∗
(
−
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
−
2
∗
(
−
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
∗
s
e
c
h
(
C
1
+
(
−
(
1
/
44
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
44
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[7]:={u(x,t)=(2/3)*(-(1/22)*sqrt(2)*11^{(}3/4)+(1/22*I)*sqrt(2)*11^{(}3/4))^{2}-2*(-(1/22)*sqrt(2)*11^{(}3/4)+(1/22*I)*sqrt(2)*11^{(}3/4))^{2}*sech(_{C}1+(-(1/44)*sqrt(2)*11^{(}3/4)+(1/44*I)*sqrt(2)*11^{(}3/4))*x+_{C}3*t)^{2}}}
g
[
8
]
:=
u
(
x
,
t
)
=
(
2
/
3
)
∗
(
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
−
2
∗
(
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
∗
s
e
c
h
(
C
1
+
(
(
1
/
44
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
44
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[8]:={u(x,t)=(2/3)*((1/22)*sqrt(2)*11^{(}3/4)-(1/22*I)*sqrt(2)*11^{(}3/4))^{2}-2*((1/22)*sqrt(2)*11^{(}3/4)-(1/22*I)*sqrt(2)*11^{(}3/4))^{2}*sech(_{C}1+((1/44)*sqrt(2)*11^{(}3/4)-(1/44*I)*sqrt(2)*11^{(}3/4))*x+_{C}3*t)^{2}}}
g
[
9
]
:=
u
(
x
,
t
)
=
(
2
/
3
)
∗
(
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
−
2
∗
(
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
∗
s
e
c
h
(
C
1
+
(
(
1
/
44
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
44
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[9]:={u(x,t)=(2/3)*((1/22)*sqrt(2)*11^{(}3/4)+(1/22*I)*sqrt(2)*11^{(}3/4))^{2}-2*((1/22)*sqrt(2)*11^{(}3/4)+(1/22*I)*sqrt(2)*11^{(}3/4))^{2}*sech(_{C}1+((1/44)*sqrt(2)*11^{(}3/4)+(1/44*I)*sqrt(2)*11^{(}3/4))*x+_{C}3*t)^{2}}}
Kaup Kupershmidt sech method animation2
Kaup Kupershmidt sech method animation4
Kaup Kupershmidt sech method animation8
sec、coth 展开
g
[
2
]
:=
u
(
x
,
t
)
=
−
(
1
/
3
)
∗
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
+
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
s
e
c
(
C
1
+
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[2]:={u(x,t)=-(1/3)*(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}*sec(_{C}1+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
3
]
:=
u
(
x
,
t
)
=
−
(
1
/
3
)
∗
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
+
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
s
e
c
(
C
1
+
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[3]:={u(x,t)=-(1/3)*(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}+(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}*sec(_{C}1+(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
4
]
:=
u
(
x
,
t
)
=
−
(
1
/
3
)
∗
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
+
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
s
e
c
(
C
1
+
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[4]:={u(x,t)=-(1/3)*((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}+((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}*sec(_{C}1+((1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
5
]
:=
u
(
x
,
t
)
=
−
(
1
/
3
)
∗
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
+
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
s
e
c
(
C
1
+
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[5]:={u(x,t)=-(1/3)*((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}+((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}*sec(_{C}1+((1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
6
]
:=
u
(
x
,
t
)
=
−
(
2
/
3
)
∗
(
−
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
+
2
∗
(
−
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
∗
s
e
c
(
C
1
+
(
−
(
1
/
44
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
44
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[6]:={u(x,t)=-(2/3)*(-(1/22)*sqrt(2)*11^{(}3/4)-(1/22*I)*sqrt(2)*11^{(}3/4))^{2}+2*(-(1/22)*sqrt(2)*11^{(}3/4)-(1/22*I)*sqrt(2)*11^{(}3/4))^{2}*sec(_{C}1+(-(1/44)*sqrt(2)*11^{(}3/4)-(1/44*I)*sqrt(2)*11^{(}3/4))*x+_{C}3*t)^{2}}}
g
[
7
]
:=
u
(
x
,
t
)
=
−
(
2
/
3
)
∗
(
−
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
+
2
∗
(
−
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
∗
s
e
c
(
C
1
+
(
−
(
1
/
44
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
44
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[7]:={u(x,t)=-(2/3)*(-(1/22)*sqrt(2)*11^{(}3/4)+(1/22*I)*sqrt(2)*11^{(}3/4))^{2}+2*(-(1/22)*sqrt(2)*11^{(}3/4)+(1/22*I)*sqrt(2)*11^{(}3/4))^{2}*sec(_{C}1+(-(1/44)*sqrt(2)*11^{(}3/4)+(1/44*I)*sqrt(2)*11^{(}3/4))*x+_{C}3*t)^{2}}}
g
[
8
]
:=
u
(
x
,
t
)
=
−
(
2
/
3
)
∗
(
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
+
2
∗
(
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
∗
s
e
c
(
C
1
+
(
(
1
/
44
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
−
(
1
/
44
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[8]:={u(x,t)=-(2/3)*((1/22)*sqrt(2)*11^{(}3/4)-(1/22*I)*sqrt(2)*11^{(}3/4))^{2}+2*((1/22)*sqrt(2)*11^{(}3/4)-(1/22*I)*sqrt(2)*11^{(}3/4))^{2}*sec(_{C}1+((1/44)*sqrt(2)*11^{(}3/4)-(1/44*I)*sqrt(2)*11^{(}3/4))*x+_{C}3*t)^{2}}}
g
[
9
]
:=
u
(
x
,
t
)
=
−
(
2
/
3
)
∗
(
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
+
2
∗
(
(
1
/
22
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
2
∗
s
e
c
(
C
1
+
(
(
1
/
44
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
+
(
1
/
44
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
(
3
/
4
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[9]:={u(x,t)=-(2/3)*((1/22)*sqrt(2)*11^{(}3/4)+(1/22*I)*sqrt(2)*11^{(}3/4))^{2}+2*((1/22)*sqrt(2)*11^{(}3/4)+(1/22*I)*sqrt(2)*11^{(}3/4))^{2}*sec(_{C}1+((1/44)*sqrt(2)*11^{(}3/4)+(1/44*I)*sqrt(2)*11^{(}3/4))*x+_{C}3*t)^{2}}}
g
[
10
]
:=
u
(
x
,
t
)
=
−
(
2
/
3
)
∗
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
+
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
c
o
t
h
(
C
1
+
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[10]:={u(x,t)=-(2/3)*(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}*coth(_{C}1+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
Kaup Kupershmidt sec method animation3
Kaup Kupershmidt sec method animation5
Kaup Kupershmidt sec method animation10
csch 展开
u
(
x
,
t
)
=
C
4
{\displaystyle {u(x,t)=_{C}4}}
g
[
2
]
:=
u
(
x
,
t
)
=
(
1
/
3
)
∗
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
+
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
c
s
c
h
(
C
1
+
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[2]:={u(x,t)=(1/3)*(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}*csch(_{C}1+(-(1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
3
]
:=
u
(
x
,
t
)
=
(
1
/
3
)
∗
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
+
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
c
s
c
h
(
C
1
+
(
−
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[3]:={u(x,t)=(1/3)*(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}+(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}*csch(_{C}1+(-(1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
4
]
:=
u
(
x
,
t
)
=
(
1
/
3
)
∗
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
+
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
c
s
c
h
(
C
1
+
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
−
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[4]:={u(x,t)=(1/3)*((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}+((1/2)*sqrt(2)-(1/2*I)*sqrt(2))^{2}*csch(_{C}1+((1/2)*sqrt(2)-(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
g
[
5
]
:=
u
(
x
,
t
)
=
(
1
/
3
)
∗
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
+
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
2
∗
c
s
c
h
(
C
1
+
(
(
1
/
2
)
∗
s
q
r
t
(
2
)
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
2
)
)
∗
x
+
C
3
∗
t
)
2
{\displaystyle g[5]:={u(x,t)=(1/3)*((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}+((1/2)*sqrt(2)+(1/2*I)*sqrt(2))^{2}*csch(_{C}1+((1/2)*sqrt(2)+(1/2*I)*sqrt(2))*x+_{C}3*t)^{2}}}
Kaup Kupershmidt csch method animation2
Kaup Kupershmidt csch method animation3
Kaup Kupershmidt csch method animation5
