5.4. FIRST INTEGRALS AND LINEARIZATION 121
and the two remaining equations in (5.232) as
W
x
C pW
u
C
pq
u
W
q
D 0; (5.234a)
W
y
C qW
u
C
pq
u
W
p
D 0: (5.234b)
ese are easily solved giving (5.230). If we choose
W D x
u
p
; (5.235)
then from (5.233), (5.231), and (5.203) we have
X D q; Y D y
u
q
; U D x; P D
u
pq
; Q D
q
p
; (5.236)
and under this contact transformation
U
XX
D
2uu
x
u
y
u
xy
C u
2
u
xx
u
yy
u
2
xy
u
2
x
u
2
y
u
3
x
u
2
y
u
yy
D 0: (5.237)
We note that through (5.236), we are able to explicitly solve for x; y; and u, giving
x D U; y D Y
XP
Q
; u D
X
2
P
Q
; (5.238)
and one can show that under this transformation, (5.229) becomes U
XX
D 0.
5.4.3 ELLIPTIC MA EQUATIONS
We now extend our results to elliptic Monge–Ampere equations. We again consider
Ar C Bs C C t C D
rt s
2
D E; (5.239)
and suppose that it admits two general first integrals, say
F
.
˛
1
˙ iˇ
1
; ˛
2
˙ iˇ
2
/
D 0; (5.240)
where F is an arbitrary function of ˛
i
˙ iˇ
i
; i D 1; 2; and ˛
i
; ˇ
i
are real functions of x; y; u; p,
and q. We will show it is possible to transform (5.239) to U
XX
C U
Y Y
D 0.
We define
X D ˛
1
; Y D ˇ
1
; P D ˇ
2
; Q D ˛
2
(5.241)
and we’ll assume that U exists such that the contact conditions are satisfied. Consider
U
XX
C U
Y Y
D 0:
122 5. FIRST INTEGRALS
is can be written as
@P
@X
C
@Q
@Y
D
@.P; Y /
@.X; Y /
C
@.X; Q/
@.X; Y /
D
@.P; Y /
@.x; y/
=
@.X; Y /
@.x; y/
C
@.X; Q/
@.x; y/
=
@.X; Y /
@.x; y/
D 0;
(5.242)
giving
@.P; Y /
@.x; y/
C
@.X; Q/
@.x; y/
D 0;
or
P
x
Y
y
P
y
Y
x
C X
x
Q
y
X
y
Q
x
D 0: (5.243)
We now bring in (5.241), so (5.243) becomes
ˇ
2
x
C pˇ
2
u
C rˇ
2
p
C sˇ
2
q
ˇ
1
y
C qˇ
1
u
C sˇ
1
p
C tˇ
1
q
ˇ
1
x
C pˇ
1
u
C rˇ
1
p
C sˇ
1
q
ˇ
2
y
C qˇ
2
u
C sˇ
2
p
C tˇ
2
q
C
˛
1
x
C p˛
1
u
C r˛
1
p
C s˛
1
q
˛
2
y
C q˛
2
u
C s˛
2
p
C t˛
2
q
˛
2
x
C p˛
2
u
C r˛
2
p
C s˛
2
q
˛
1
y
C q˛
1
u
C s˛
1
p
C t˛
1
q
D 0:
(5.244)
As F in (5.240) are first integrals, they satisfy the following (assuming that D ¤ 0):
˛
1
x
iˇ
1
x
C p
.
˛
1
u
iˇ
1
u
/
C
D
˛
1
p
iˇ
1
p
C
.
˛
1
iˇ
1
/
˛
1
q
iˇ
1
q
D 0; (5.245a)
˛
1
y
iˇ
1
y
C q
.
˛
1
u
iˇ
1
u
/
C
.
˛
1
C iˇ
1
/
˛
1
p
iˇ
1
p
A
D
˛
1
q
iˇ
1
q
D 0; (5.245b)
˛
2
x
iˇ
2
x
C p
.
˛
2
u
iˇ
2
u
/
C
D
˛
2
p
iˇ
2
p
C
.
˛
2
iˇ
2
/
˛
2
q
iˇ
2
q
D 0; (5.245c)
˛
2
y
iˇ
2
y
C q
.
˛
2
u
iˇ
2
u
/
C
.
˛
2
C iˇ
2
/
˛
2
p
iˇ
2
p
A
D
˛
2
q
iˇ
2
q
D 0; (5.245d)
where
D
2
2
BD C AC CDE D 0; D a ˙ i b: (5.246)
We isolate real and imaginary parts of (5.245) and solve for ˛
i
x
; ˛
i
y
; ˇ
i
x
; ˇ
i
y
and eliminate
these in (5.244). is leads to
A
D
r C 2a s C
C
D
t C rt s
2
D a
2
C b
2
AC
D
2
: (5.247)
From (5.246), we have that
B 2aD D 0; (5.248a)
a
2
b
2
D
2
aBD C AC C DE D 0; (5.248b)
5.4. FIRST INTEGRALS AND LINEARIZATION 123
from which we deduce
a
2
C b
2
D
2
D AC C DE. Eliminating a and b in (5.247) gives
Ar C Bs C C t C D
rt s
2
D E;
which is (5.239). e next two examples illustrate these results.
Example 5.15 Consider
.1 C u
2
y
/u
xx
2u
x
u
y
u
xy
C u
2
x
u
yy
D 0: (5.249)
As was shown in Example 5.3, this PDE admits the first integrals
F
u ˙ iy;
q i
p
D 0: (5.250)
Here we try
X D u; Y D y; P D
1
p
; Q D
q
p
: (5.251)
Unfortunately, the contact conditions are not satisfied, but with the slight adjustment,
X D u; Y D y; P D
1
p
; Q D
q
p
; (5.252)
they lead to
U
D
x
to which we add to (5.252) and under
X D u; Y D y; U D x; P D
1
p
; Q D
q
p
: (5.253)
or
x D U; y D Y; u D X; p D
1
U
X
; q D
U
Y
U
X
(5.254)
the PDE (5.249) is transformed to U
XX
C U
Y Y
D 0.
Example 5.16 Consider
u
xx
C u
yy
C 2u
xx
u
yy
2u
2
xy
D 0: (5.255)
As was shown in Example 5.8, this PDE admits the first integrals
F .x C2p ˙ iy; q ˙ ip/ D 0: (5.256)
Here we try
X D x C2p; Y D y; P D p; Q D q: (5.257)
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