特里忒蔡卡方程 (Tritzeica equation)是一个最早由罗马尼亚数学家George Tritzeica在1907年在微分几何领域研究的非线性偏微分方程[ 1] [ 2]
罗马尼亚数学家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})}
作变换
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
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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，再用反代换
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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