84 4. POINT AND CONTACT TRANSFORMATIONS
while second-order derivatives transform as
u
xx
D
v
ss
v
rr
v
ss
v
2
rs
; u
xy
D
v
rs
v
rr
v
ss
v
2
rs
; u
yy
D
v
rr
v
rr
v
ss
v
2
rs
; (4.91)
thus transforming (4.85) to the linear PDE
1 C r
2
v
rr
C 2rsv
rs
C
1 C s
2
v
ss
D 0: (4.92)
If we denote A D 1 C r
2
, B D 2rs and C D 1 C s
2
then B
2
4AC < 0 so that the PDE (4.92)
is elliptic. We now show that (4.92) is transformable to Laplace’s equation. Introducing polar
coordinates
r D cos b; s D sin b (4.93)
transforms (4.92) to
4
C
2
v
C v
bb
C v
D 0: (4.94)
Further, if we let
D
1
sinh a
; (4.95)
then (4.94) becomes
v
aa
C v
bb
C
2v
a
sinh a cosh a
D 0: (4.96)
Next we transform the dependent variable v as
v D
w
tanh a
; (4.97)
leading to
w
aa
C w
bb
C
2w
cosh
2
a
D 0: (4.98)
Finally, if we introduce the first-order Darboux transformation
w D W
a
tanh a W; (4.99)
then equation (4.98) becomes
@
@a
r
2
W
tanh
a
r
2
W
D
0;
(4.100)
where r
2
W D 0 is Laplace’s equation. Integrating (4.100) gives
r
2
W D F .b/ cosh a; (4.101)
where F .b/ is arbitrary, but we can set F .b/ D 0 without loss of generality. e rationale is as
follows. If F .b/ is arbitrary, then we can set F D G
00
C G so
r
2
W D
G
00
.b/ C G.b/
cosh a: (4.102)