庫普-庫珀施密特方程 (Kaup-Kupershmidt Equation)是一個非線性偏微分方程：[ 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
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2
−
(
75
/
2
)
∗
∂
2
u
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x
,
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∂
x
2
∗
∂
u
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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 軟體包TWSolution，隨所選定展開函數不同，可得多種行波解[ 2]
tanh 展開
g
[
2
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:=
u
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/
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∗
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a
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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
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s
q
r
t
(
2
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+
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I
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s
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r
t
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)
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2
+
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s
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r
t
(
2
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+
(
1
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∗
I
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∗
s
q
r
t
(
2
)
)
2
∗
t
a
n
h
(
C
1
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s
q
r
t
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(
1
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∗
I
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∗
s
q
r
t
(
2
)
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∗
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
/
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[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
/
22
∗
I
)
∗
s
q
r
t
(
2
)
∗
11
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3
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4
)
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2
+
2
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(
(
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22
)
∗
s
q
r
t
(
2
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∗
11
(
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4
)
−
(
1
/
22
∗
I
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s
q
r
t
(
2
)
∗
11
(
3
/
4
)
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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[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 展開
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
∗
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[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
c
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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
)
)
/
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 展開
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 展開
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
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{\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}}}
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{\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 展開
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{\displaystyle {u(x,t)=_{C}4}}
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{\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}}}
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{\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
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{\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
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{\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
^ Qinghua Feng New Analytical Solutions For (3+1) Dimensional Kaup-Kupershmidt Equation,2012 International Conference on Computer Technology and Science (ICCTS 2012) IPCSIT vol. 47 (2012)
^ Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Page 27
*谷超豪 《孤立子 理論中的達布變換 及其幾何應用》 上海科學技術出版社
*閻振亞著 《複雜非線性波的構造性理論及其應用》 科學出版社 2007年
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
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 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e60c07a2632dff5ab7504a4713b3c8085a72b9e7)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35332e2310deb20eeb83339fd237f9f5186a0ae4)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7440d2af4f00e0ee1438a764026abc8b82e4cd28)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d278e178e362ab67f0b7077035811f980a329bd)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/214953f703fbd2084529daff45bb615d64021501)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5df9be550db647065e2607294b93985e8aafb6d8)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74d5b8f4408a87c4da9c0fdf5cbb17669b70e896)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/175bf762e367302d5f2474a110f04b6feea83791)
Kaup Kupershmidt eq tanh method animation2
Kaup Kupershmidt eq tanh method animation7
Kaup Kupershmidt eq tanh method animation8
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d429e2c55b327bade8fb8564591ccc846608954)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6126290c3e3f3dad310b90db0e392d7973c3651e)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb9b1f036a118ec0a552c3a246600e3e93dbbba)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc63179d73846cf92227d530ceff1f8a2154906)
Kaup Kupershmidt JacobiSN method animation2
Kaup Kupershmidt JacobiSN method animation3
Kaup Kupershmidt JacobiSN method animation4
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a3fea3b11e19da0b0b0cd2edf42a5f60541c03)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4cd43320209fe8339164e858aaf75025a6984af)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7112bc2abcc2bb92114bd726a7f50f1c8eea7b72)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f548bc1ea9ef1a8a25556569eac2fe42d1d2da03)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8914bddf838516f6a43c009e4a7b8532749ac87)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee17ff87801ead50057de216526769870eefec57)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb2421aac7cc10bbe153c79fef36b0a5d903b52)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8b47615e55f85800f2acdeba00317fc0dc1849)
Kaup Kupershmidt sech method animation2
Kaup Kupershmidt sech method animation4
Kaup Kupershmidt sech method animation8
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90ef317aa108e05bc8a0dfd5f987d4a97a350d3f)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/794fb788dc46e8c3f90247a5567c42f8f165fc0f)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eea9aa85a7569072cf228d3f04d6b920faeeeac5)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a01d1b2c7b84353279fb4edfc464f739cfc890)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/124c2352398582aba5fc1287969b91434cc555ee)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dceedf56820b0ec3a332151cb2aa3246ad1c4f22)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fbb8e89a6b1f199a921cdb7ada0a32b98508b9a)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df501726ab96d8f1299abb5d08589f8ded3308b0)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef38e9e2a5c44a698f2c4dc22f4c2d9747d295c)
Kaup Kupershmidt sec method animation3
Kaup Kupershmidt sec method animation5
Kaup Kupershmidt sec method animation10
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c10d12b5581559d10676d4d7463949c9270b2608)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/680cc6ba5446377c62c33eb2e554104c599cf6e0)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b86780c613f3968780ce94b1f051036f29ff91)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/518f542260e3eccaf23e362112d9b96fdaa91975)
Kaup Kupershmidt csch method animation2
Kaup Kupershmidt csch method animation3
Kaup Kupershmidt csch method animation5
