雙重sinh-Gordon方程 （Double sinh-Gordon equation）是一個非線性偏微分方程 。[ 1] [ 2] [ 3] [ 4] [ 5]
u
x
t
=
a
s
i
n
h
(
u
)
+
b
s
i
n
h
(
2
u
)
{\displaystyle u_{xt}=asinh(u)+bsinh(2u)}
v
=
C
5
∗
J
a
c
o
b
i
C
N
(
C
2
+
C
3
∗
x
−
(
a
∗
C
5
2
−
2
∗
b
∗
C
5
2
−
2
∗
b
−
a
)
∗
t
/
(
C
3
∗
(
C
5
2
−
1
)
)
,
(
(
−
2
∗
a
∗
C
5
2
+
a
∗
C
5
4
+
a
+
2
∗
b
−
2
∗
b
∗
C
5
4
)
∗
(
a
∗
C
5
2
−
2
∗
b
∗
C
5
2
−
a
)
)
∗
C
5
/
(
−
2
∗
a
∗
C
5
2
+
a
∗
C
5
4
+
a
+
2
∗
b
−
2
∗
b
∗
C
5
4
)
)
{\displaystyle {v=_{C}5*JacobiCN(_{C}2+_{C}3*x-(a*_{C}5^{2}-2*b*_{C}5^{2}-2*b-a)*t/(_{C}3*(_{C}5^{2}-1)),{\sqrt {(}}(-2*a*_{C}5^{2}+a*_{C}5^{4}+a+2*b-2*b*_{C}5^{4})*(a*_{C}5^{2}-2*b*_{C}5^{2}-a))*_{C}5/(-2*a*_{C}5^{2}+a*_{C}5^{4}+a+2*b-2*b*_{C}5^{4}))}}
v
=
C
5
∗
J
a
c
o
b
i
D
N
(
C
2
+
C
3
∗
x
−
C
5
2
∗
(
a
∗
C
5
2
−
2
∗
b
∗
C
5
2
−
a
)
∗
t
/
(
C
3
∗
(
−
2
∗
C
5
2
+
1
+
C
5
4
)
)
,
(
(
−
2
∗
a
∗
C
5
2
+
a
∗
C
5
4
+
a
+
2
∗
b
−
2
∗
b
∗
C
5
4
)
∗
(
a
∗
C
5
2
−
2
∗
b
∗
C
5
2
−
a
)
)
/
(
(
a
∗
C
5
2
−
2
∗
b
∗
C
5
2
−
a
)
∗
C
5
)
)
{\displaystyle {v=_{C}5*JacobiDN(_{C}2+_{C}3*x-_{C}5^{2}*(a*_{C}5^{2}-2*b*_{C}5^{2}-a)*t/(_{C}3*(-2*_{C}5^{2}+1+_{C}5^{4})),{\sqrt {(}}(-2*a*_{C}5^{2}+a*_{C}5^{4}+a+2*b-2*b*_{C}5^{4})*(a*_{C}5^{2}-2*b*_{C}5^{2}-a))/((a*_{C}5^{2}-2*b*_{C}5^{2}-a)*_{C}5))}}
v
=
C
5
∗
J
a
c
o
b
i
N
C
(
C
2
+
C
3
∗
x
+
(
a
∗
C
5
2
−
2
∗
b
∗
C
5
2
−
2
∗
b
−
a
)
∗
t
/
(
C
3
∗
(
C
5
2
−
1
)
)
,
(
−
(
−
2
∗
a
∗
C
5
2
+
a
∗
C
5
4
+
a
+
2
∗
b
−
2
∗
b
∗
C
5
4
)
∗
(
a
∗
C
5
2
−
2
∗
b
−
a
)
)
/
(
−
2
∗
a
∗
C
5
2
+
a
∗
C
5
4
+
a
+
2
∗
b
−
2
∗
b
∗
C
5
4
)
)
{\displaystyle {v=_{C}5*JacobiNC(_{C}2+_{C}3*x+(a*_{C}5^{2}-2*b*_{C}5^{2}-2*b-a)*t/(_{C}3*(_{C}5^{2}-1)),{\sqrt {(}}-(-2*a*_{C}5^{2}+a*_{C}5^{4}+a+2*b-2*b*_{C}5^{4})*(a*_{C}5^{2}-2*b-a))/(-2*a*_{C}5^{2}+a*_{C}5^{4}+a+2*b-2*b*_{C}5^{4}))}}
v
=
C
5
∗
J
a
c
o
b
i
N
D
(
C
2
+
C
3
∗
x
−
(
a
∗
C
5
2
−
2
∗
b
−
a
)
∗
t
/
(
C
3
∗
(
−
2
∗
C
5
2
+
1
+
C
5
4
)
)
,
(
−
(
−
2
∗
a
∗
C
5
2
+
a
∗
C
5
4
+
a
+
2
∗
b
−
2
∗
b
∗
C
5
4
)
∗
(
a
∗
C
5
2
−
2
∗
b
−
a
)
)
/
(
a
∗
C
5
2
−
2
∗
b
−
a
)
)
{\displaystyle {v=_{C}5*JacobiND(_{C}2+_{C}3*x-(a*_{C}5^{2}-2*b-a)*t/(_{C}3*(-2*_{C}5^{2}+1+_{C}5^{4})),{\sqrt {(}}-(-2*a*_{C}5^{2}+a*_{C}5^{4}+a+2*b-2*b*_{C}5^{4})*(a*_{C}5^{2}-2*b-a))/(a*_{C}5^{2}-2*b-a))}}
v
=
(
a
∗
(
2
∗
b
+
a
)
)
∗
c
s
c
(
C
1
+
C
2
∗
x
−
(
2
∗
b
+
a
)
∗
t
/
C
2
)
/
a
{\displaystyle {v={\sqrt {(}}a*(2*b+a))*csc(_{C}1+_{C}2*x-(2*b+a)*t/_{C}2)/a}}
v
=
(
a
∗
(
2
∗
b
+
a
)
)
∗
c
s
c
(
C
2
+
C
3
∗
x
−
(
2
∗
b
+
a
)
∗
t
/
C
3
)
/
a
{\displaystyle {v={\sqrt {(}}a*(2*b+a))*csc(_{C}2+_{C}3*x-(2*b+a)*t/_{C}3)/a}}
v
=
(
a
∗
(
2
∗
b
+
a
)
)
∗
s
e
c
(
C
1
+
C
2
∗
x
−
(
2
∗
b
+
a
)
∗
t
/
C
2
)
/
a
{\displaystyle {v={\sqrt {(}}a*(2*b+a))*sec(_{C}1+_{C}2*x-(2*b+a)*t/_{C}2)/a}}
v
=
(
a
∗
(
2
∗
b
+
a
)
)
∗
s
e
c
h
(
C
1
+
C
2
∗
x
+
(
2
∗
b
+
a
)
∗
t
/
C
2
)
/
a
{\displaystyle {v={\sqrt {(}}a*(2*b+a))*sech(_{C}1+_{C}2*x+(2*b+a)*t/_{C}2)/a}}
v
=
(
−
a
∗
(
2
∗
b
+
a
)
)
∗
c
s
c
h
(
C
1
+
C
2
∗
x
+
(
2
∗
b
+
a
)
∗
t
/
C
2
)
/
a
{\displaystyle {v={\sqrt {(}}-a*(2*b+a))*csch(_{C}1+_{C}2*x+(2*b+a)*t/_{C}2)/a}}
v
=
(
(
a
−
2
∗
b
)
∗
a
)
∗
c
o
s
h
(
C
2
+
C
3
∗
x
−
(
a
−
2
∗
b
)
∗
t
/
C
3
)
/
(
a
−
2
∗
b
)
{\displaystyle {v={\sqrt {(}}(a-2*b)*a)*cosh(_{C}2+_{C}3*x-(a-2*b)*t/_{C}3)/(a-2*b)}}
v
=
(
(
a
−
2
∗
b
)
∗
(
2
∗
b
+
a
)
)
∗
t
a
n
h
(
C
1
+
C
2
∗
x
+
(
1
/
8
)
∗
(
a
2
−
4
∗
b
2
)
∗
t
/
(
C
2
∗
b
)
)
/
(
a
−
2
∗
b
)
{\displaystyle {v={\sqrt {(}}(a-2*b)*(2*b+a))*tanh(_{C}1+_{C}2*x+(1/8)*(a^{2}-4*b^{2})*t/(_{C}2*b))/(a-2*b)}}
其中
v
=
t
a
n
h
(
(
1
/
2
)
∗
u
)
{\displaystyle v=tanh((1/2)*u)}
u
(
x
,
t
)
=
2
a
r
c
t
a
n
h
(
1.5
∗
J
a
c
o
b
i
C
N
(
1.2
+
1.3
∗
x
+
3.2307692307692307692
∗
t
,
1.0555973258234951998
)
)
{\displaystyle u(x,t)=2arctanh(1.5*JacobiCN(1.2+1.3*x+3.2307692307692307692*t,1.0555973258234951998))}
u
(
x
,
t
)
=
2
a
r
c
t
a
n
h
(
1.5
∗
J
a
c
o
b
i
D
N
(
1.2
+
1.3
∗
x
+
3.6000000000000000000
∗
t
,
.94733093343134184593
)
)
{\displaystyle u(x,t)=2arctanh(1.5*JacobiDN(1.2+1.3*x+3.6000000000000000000*t,.94733093343134184593))}
u
(
x
,
t
)
=
2
∗
a
r
c
t
a
n
h
(
1.5
∗
J
a
c
o
b
i
N
C
(
−
1.2
−
1.3
∗
x
+
3.2307692307692307692
∗
t
,
.33806170189140663100
∗
I
)
)
{\displaystyle u(x,t)=2*arctanh(1.5*JacobiNC(-1.2-1.3*x+3.2307692307692307692*t,.33806170189140663100*I))}
u
(
x
,
t
)
=
2
∗
a
r
c
t
a
n
h
(
1.5
∗
J
a
c
o
b
i
N
D
(
1.2
+
1.3
∗
x
+
.36923076923076923077
∗
t
,
2.9580398915498080213
∗
I
)
)
{\displaystyle u(x,t)=2*arctanh(1.5*JacobiND(1.2+1.3*x+.36923076923076923077*t,2.9580398915498080213*I))}
u
(
x
,
t
)
=
−
2
∗
a
r
c
t
a
n
h
(
(
3
)
∗
c
s
c
(
15.1
−
1.2
∗
x
+
2.5000000000000000000
∗
t
)
)
{\displaystyle u(x,t)=-2*arctanh({\sqrt {(}}3)*csc(15.1-1.2*x+2.5000000000000000000*t))}
u
(
x
,
t
)
=
−
2
∗
a
r
c
t
a
n
h
(
(
3
)
∗
c
s
c
(
−
1.2
−
1.3
∗
x
+
2.3076923076923076923
∗
t
)
)
{\displaystyle u(x,t)=-2*arctanh({\sqrt {(}}3)*csc(-1.2-1.3*x+2.3076923076923076923*t))}
u
(
x
,
t
)
=
2
∗
a
r
c
t
a
n
h
(
(
3
)
∗
s
e
c
(
15.1
−
1.2
∗
x
+
2.5000000000000000000
∗
t
)
)
{\displaystyle u(x,t)=2*arctanh({\sqrt {(}}3)*sec(15.1-1.2*x+2.5000000000000000000*t))}
u
(
x
,
t
)
=
2
∗
a
r
c
t
a
n
h
(
(
3
)
∗
s
e
c
h
(
−
15.1
+
1.2
∗
x
+
2.5000000000000000000
∗
t
)
)
{\displaystyle u(x,t)=2*arctanh({\sqrt {(}}3)*sech(-15.1+1.2*x+2.5000000000000000000*t))}
u
(
x
,
t
)
=
2
∗
a
r
c
t
a
n
h
(
(
3
)
∗
s
e
c
h
(
1.2
+
1.3
∗
x
+
2.3076923076923076923
∗
t
)
)
{\displaystyle u(x,t)=2*arctanh({\sqrt {(}}3)*sech(1.2+1.3*x+2.3076923076923076923*t))}
u
(
x
,
t
)
=
2
∗
a
r
c
t
a
n
h
(
(
−
3
)
∗
c
s
c
h
(
−
15.1
+
1.2
∗
x
+
2.5000000000000000000
∗
t
)
)
{\displaystyle u(x,t)=2*arctanh({\sqrt {(}}-3)*csch(-15.1+1.2*x+2.5000000000000000000*t))}
{\displaystyle }
{\displaystyle }
{\displaystyle }
^ Andrei D. Polyanin, Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS SECOND EDITION CRC Press, A Chapman & Hall Book ISBN 9781420087239
^ Zeitschrift Für Naturforschung: A journal of physical sciences 2004 p933-937
^ A. M. WAZWAZ Exact solutions to the double sinh-gordon equation by the tanh method and a variable separated ODE method,Computers & Mathematics with Applications,Volume 50, Issues 10–12, November–December 2005, Pages 1685–1696
^ Issues in Logic, Operations, and Computational Mathematics and Geometry 2013 p484
^ Mathematical Reviews - Page 3708 2007
