特里忒蔡卡方程 (Tritzeica equation)是一個最早由羅馬尼亞數學家George Tritzeica在1907年在微分幾何領域研究的非線性偏微分方程[1] [2]
羅馬尼亞數學家George Tritzeica
u
x
y
=
e
x
p
(
u
x
,
y
)
−
e
x
p
(
−
2
∗
u
x
,
y
)
{\displaystyle u_{xy}=exp(u_{x,y})-exp(-2*u_{x,y})}
作變換
w
(
x
,
y
)
=
e
x
p
(
u
(
x
,
y
)
)
{\displaystyle w(x,y)=exp(u(x,y))}
得
w
(
x
,
y
)
y
,
x
∗
w
(
x
,
y
)
−
w
(
x
,
y
)
x
∗
w
(
x
,
y
)
y
−
w
(
x
,
y
)
3
+
1
=
0
{\displaystyle w(x,y)_{y,x}*w(x,y)-w(x,y)_{x}*w(x,y)_{y}-w(x,y)^{3}+1=0}
求得行波解，再用反代換
u
(
x
,
y
)
=
l
n
(
w
(
x
,
y
)
)
{\displaystyle u(x,y)=ln(w(x,y))}
u
(
x
,
y
)
=
l
n
(
−
1
/
2
−
(
1
/
2
∗
I
)
∗
(
3
)
+
(
3
/
4
+
(
3
/
4
∗
I
)
∗
(
3
)
)
∗
c
s
c
(
C
1
+
C
2
∗
x
+
(
3
/
4
)
∗
(
1
/
2
+
(
1
/
2
∗
I
)
∗
(
3
)
)
∗
y
/
C
2
)
2
)
{\displaystyle u(x,y)=ln(-1/2-(1/2*I)*{\sqrt {(}}3)+(3/4+(3/4*I)*{\sqrt {(}}3))*csc(_{C}1+_{C}2*x+(3/4)*(1/2+(1/2*I)*{\sqrt {(}}3))*y/_{C}2)^{2})}
u
(
x
,
y
)
=
l
n
(
−
1
/
2
−
(
1
/
2
∗
I
)
∗
(
3
)
+
(
3
/
4
+
(
3
/
4
∗
I
)
∗
(
3
)
)
∗
s
e
c
(
C
1
+
C
2
∗
x
+
(
3
/
4
)
∗
(
1
/
2
+
(
1
/
2
∗
I
)
∗
(
3
)
)
∗
y
/
C
2
)
2
)
{\displaystyle u(x,y)=ln(-1/2-(1/2*I)*{\sqrt {(}}3)+(3/4+(3/4*I)*{\sqrt {(}}3))*sec(_{C}1+_{C}2*x+(3/4)*(1/2+(1/2*I)*{\sqrt {(}}3))*y/_{C}2)^{2})}
u
(
x
,
y
)
=
l
n
(
−
1
/
2
+
(
1
/
2
∗
I
)
∗
(
3
)
+
(
3
/
4
−
(
3
/
4
∗
I
)
∗
(
3
)
)
∗
c
s
c
(
C
1
+
C
2
∗
x
+
(
3
/
4
)
∗
(
1
/
2
−
(
1
/
2
∗
I
∗
(
3
)
)
∗
y
/
C
2
)
2
)
{\displaystyle u(x,y)=ln(-1/2+(1/2*I)*{\sqrt {(}}3)+(3/4-(3/4*I)*{\sqrt {(}}3))*csc(_{C}1+_{C}2*x+(3/4)*(1/2-(1/2*I*{\sqrt {(}}3))*y/_{C}2)^{2})}
u
(
x
,
y
)
=
l
n
(
1
/
4
−
(
1
/
4
∗
I
)
∗
(
3
)
+
(
−
3
/
4
+
(
3
/
4
∗
I
)
∗
(
3
)
)
∗
c
o
t
h
(
C
1
+
C
2
∗
x
+
(
3
/
4
)
∗
(
−
1
/
2
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
3
)
)
∗
y
/
C
2
)
2
)
{\displaystyle u(x,y)=ln(1/4-(1/4*I)*{\sqrt {(}}3)+(-3/4+(3/4*I)*{\sqrt {(}}3))*coth(_{C}1+_{C}2*x+(3/4)*(-1/2+(1/2*I)*sqrt(3))*y/_{C}2)^{2})}
u
(
x
,
y
)
=
l
n
(
1
/
4
−
(
1
/
4
∗
I
)
∗
(
3
)
+
(
−
3
/
4
+
(
3
/
4
∗
I
)
∗
(
3
)
)
∗
t
a
n
h
(
C
1
+
C
2
∗
x
+
(
3
/
4
)
∗
(
−
1
/
2
+
(
1
/
2
∗
I
)
∗
s
q
r
t
(
3
)
)
∗
y
/
C
2
)
2
)
{\displaystyle u(x,y)=ln(1/4-(1/4*I)*{\sqrt {(}}3)+(-3/4+(3/4*I)*{\sqrt {(}}3))*tanh(_{C}1+_{C}2*x+(3/4)*(-1/2+(1/2*I)*sqrt(3))*y/_{C}2)^{2})}
u
(
x
,
y
)
=
l
n
(
1
/
4
−
(
1
/
4
∗
I
)
∗
(
3
)
+
(
3
/
4
−
(
3
/
4
∗
I
)
∗
(
3
)
)
∗
c
o
t
(
C
1
+
C
2
∗
x
+
(
3
/
4
)
∗
(
1
/
2
−
(
1
/
2
∗
I
)
∗
(
3
)
)
∗
y
/
C
2
)
2
)
{\displaystyle u(x,y)=ln(1/4-(1/4*I)*{\sqrt {(}}3)+(3/4-(3/4*I)*{\sqrt {(}}3))*cot(_{C}1+_{C}2*x+(3/4)*(1/2-(1/2*I)*{\sqrt {(}}3))*y/_{C}2)^{2})}
u
(
x
,
y
)
=
l
n
(
1
/
4
−
(
1
/
4
∗
I
)
∗
(
3
)
+
(
3
/
4
−
(
3
/
4
∗
I
)
∗
(
3
)
)
∗
t
a
n
(
C
1
+
C
2
∗
x
+
(
3
/
4
)
∗
(
1
/
2
−
(
1
/
2
∗
I
)
∗
(
3
)
)
∗
y
/
C
2
)
2
)
{\displaystyle u(x,y)=ln(1/4-(1/4*I)*{\sqrt {(}}3)+(3/4-(3/4*I)*{\sqrt {(}}3))*tan(_{C}1+_{C}2*x+(3/4)*(1/2-(1/2*I)*{\sqrt {(}}3))*y/_{C}2)^{2})}
w
(
x
,
y
)
=
(
8
/
3
)
∗
C
4
2
−
(
1
/
3
)
∗
R
o
o
t
O
f
(
64
∗
C
4
6
+
27
−
24
∗
C
4
4
∗
Z
−
6
∗
C
4
2
∗
Z
2
+
Z
3
)
−
4
∗
C
4
2
∗
J
a
c
o
b
i
D
N
(
C
2
+
(
1
/
2
)
∗
C
4
∗
x
+
C
4
∗
t
,
(
1
/
2
)
∗
R
o
o
t
O
f
(
−
R
o
o
t
O
f
(
64
∗
C
4
6
+
27
−
24
∗
C
4
4
∗
Z
−
6
∗
C
4
2
∗
Z
2
+
Z
3
)
+
Z
2
)
/
C
4
)
2
{\displaystyle w(x,y)=(8/3)*_{C}4^{2}-(1/3)*RootOf(64*_{C}4^{6}+27-24*_{C}4^{4}*_{Z}-6*_{C}4^{2}*_{Z}^{2}+_{Z}^{3})-4*_{C}4^{2}*JacobiDN(_{C}2+(1/2)*_{C}4*x+_{C}4*t,(1/2)*RootOf(-RootOf(64*_{C}4^{6}+27-24*_{C}4^{4}*_{Z}-6*_{C}4^{2}*_{Z}^{2}+_{Z}^{3})+_{Z}^{2})/_{C}4)^{2}}
Tritzeica equation traveling wave plot Tritzeica equation traveling wave plot Tritzeica equation traveling wave plot Tritzeica equation traveling wave plot
Tritzeica equation traveling wave plot Tritzeica equation traveling wave plot Tritzeica equation traveling wave plot Tritzeica equation traveling wave plot
Tritzeica equation traveling wave plot Tritzeica equation traveling wave plot Tritzeica equation traveling wave plot Tritzeica equation traveling wave plot
Tritzeica equation traveling wave plot Tritzeica equation traveling wave plot Tritzeica equation traveling wave plot Tritzeica equation traveling wave plot
^ G. Tzitz´eica, 「Geometric infinitesimale-sur une nouvelle classes
de surfaces,」Comptes Rendus de l』Acad´emie des Sciences, vol. 144,pp. 1257–1259, 1907.
^ Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p540-542 CRC PRESS
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