38 2. COMPATIBILITY
and substitution of (2.218) into (2.210b) gives (2.217) directly! Furthermore, as kS D S”, we
can write (2.218)
u D S
00
.x/e
3kt
G
S
0
.x/e
3kt
(2.219)
or
u D
N
S
0
.x/e
3kt
G
N
S.x/e
3kt
; (2.220)
where
N
S satisfies the same equation as S. us, we again obtain solutions in terms of Jacobi El-
liptic functions u D
N
S
0
.x/e
3kt
p
q
cn.;
1
p
2
/ and u D
N
S
0
.x/e
3kt
p
q
nc.;
1
p
2
/, D
N
S.x/e
3kt
de-
pending on the sign of q. We note that these exact solutions appear in EqWorld [42].
2.3 COMPATIBILITY IN .2 C 1/ DIMENSIONS
Up until now, we have considered compatibility of PDEs in .1 C 1/ dimensions. We now extend
this idea and consider PDEs in .2 C 1/ dimensions. In this section we consider the compatibility
between the .2 C 1/ dimensional reaction–diffusion equation
u
t
D u
xx
C u
yy
C Q.u; u
x
; u
y
/; (2.221)
and the first-order partial differential equation
u
t
D F
t; x; y; u; u
x
; u
y
: (2.222)
is section is based on the work of Arrigo and Suazo [39]. We will assume that F in (2.222)
is nonlinear in the first derivatives u
x
and u
y
. e case where F is linear in the first derivatives
u
x
and u
y
, is left as an exercise to the reader.
Compatibility between (2.221) and (2.222) gives rise to the compatibility equation con-
straints
F
pp
C F
qq
D 0; (2.223a)
F
xp
F
yq
C pF
up
qF
uq
C .F Q/F
pp
D 0; (2.223b)
F
xq
C F
yp
C qF
up
C pF
uq
C .F Q/F
pq
D 0; (2.223c)
F
t
C F
xx
C F
yy
C 2pF
xu
C 2qF
yu
C 2.F Q/F
yq
C
p
2
C q
2
F
uu
C 2q.F Q/F
uq
C
.
F Q
/
2
F
qq
C
Q
p
F
x
C Q
q
F
y
C
pQ
p
C qQ
q
Q
F
u
pQ
u
F
p
qQ
u
F
q
C FQ
u
D 0: (2.223d)
Eliminating the x and y derivatives in (2.223b) and (2.223c) by (i ) cross differentiation and (i i)
imposing (2.223a) gives
2F
up
C .F
p
Q
p
/F
pp
C .F
q
Q
q
/F
pq
D 0; (2.224a)
2F
uq
C .F
p
Q
p
/F
pq
C .F
q
Q
q
/F
qq
D 0: (2.224b)
2.3. COMPATIBILITY IN .2 C 1/ DIMENSIONS 39
Further, eliminating F
up
and F
uq
by again (i) cross differentiation and (i i) imposing (2.223a)
gives rise to
.2F
pp
Q
pp
C Q
qq
/F
pp
C 2.F
pq
Q
pq
/F
pq
D 0; (2.225a)
.Q
pp
Q
qq
/F
pq
C 2Q
pq
F
qq
D 0: (2.225b)
Solving (2.223a), (2.225a) and (2.225b) for F
pp
, F
pq
and F
qq
gives rise to two cases:
.i/ F
pp
D F
pq
D F
qq
D 0; (2.226a)
.i i/ F
pp
D
1
2
.Q
pp
Q
qq
/; F
pq
D Q
pq
; F
qq
D
1
2
.Q
qq
Q
pp
/: (2.226b)
As we are primarily interested in compatible equations that are more general than quasilinear,
we omit the first case. If we require that the three equations in (2.226b) be compatible, then to
within equivalence transformations of the original equation, Q satisfies
Q
pp
C Q
qq
D 0: (2.227)
Using (2.227), we find that (2.226b) becomes
F
pp
D Q
pp
; F
pq
D Q
pq
; F
qq
D Q
qq
; (2.228)
from which we find that
F D Q.u; p; q/ C X.t; x; y; u/p C Y .t; x; y; u/q CU.t; x; y; u/; (2.229)
where X, Y and U are arbitrary functions. Substituting (2.229) into (2.224a) and (2.224b) gives
2Q
up
C XQ
pp
C YQ
pq
C 2X
u
D 0; (2.230a)
2Q
uq
C XQ
pq
C YQ
qq
C 2Y
u
D 0; (2.230b)
while (2.223b) and (2.223c) become (using (2.227) and (2.230))
.
Xp C Yq C 2U
/
Q
pp
C
.
Xq Yp
/
Q
pq
C 2
X
x
Y
y
D 0; (2.231a)
.
Xq Yp
/
Q
pp
.
Xp C Yq C 2U
/
Q
pq
2
X
y
C Y
x
D 0: (2.231b)
If we differentiate (2.230a) and (2.230b) with respect to x and y, we obtain
X
x
Q
pp
C Y
x
Q
pq
C 2X
xu
D 0; X
x
Q
pq
C Y
x
Q
qq
C 2Y
xu
D 0; (2.232a)
X
y
Q
pp
C Y
y
Q
pq
C 2X
yu
D 0; X
y
Q
pq
C Y
y
Q
qq
C 2Y
yu
D 0: (2.232b)
If X
2
x
C Y
2
x
¤ 0, then solving (2.227) and (2.232a) for Q
pp
, Q
pq
and Q
qq
gives
Q
pp
D Q
qq
D
2
.
Y
x
Y
xu
X
x
X
xu
/
X
2
x
C Y
2
x
; Q
pq
D
2
.
X
x
Y
xu
C Y
x
X
xu
/
X
2
x
C Y
2
x
:
40 2. COMPATIBILITY
If X
2
y
C Y
2
y
¤ 0, then solving (2.227) and (2.232b) for Q
pp
, Q
pq
and Q
qq
gives
Q
pp
D Q
qq
D
2
Y
y
Y
yu
X
y
X
yu
X
2
y
C Y
2
y
; Q
pq
D
2
X
y
Y
yu
C Y
y
X
yu
X
2
y
C Y
2
y
:
In any case, this shows that Q
pp
, Q
pq
and Q
qq
are, at most, functions of u only. us, if we let
Q
pp
D Q
qq
D 2g
1
.u/; Q
pq
D g
2
.u/;
for arbitrary functions g
1
and g
2
, then Q has the form
Q D g
1
.u/
p
2
q
2
C g
2
.u/p q C g
3
.u/p Cg
4
.u/q C g
5
.u/; (2.233)
where g
3
g
5
are further arbitrary functions. Substituting (2.233) into (2.231) gives
2
.
Xp C Yq C 2U
/
g
1
C
.
Xq Yp
/
g
2
C 2
X
x
Y
y
D 0;
2
.
Xq Yp
/
g
1
.
Xp C Yq C 2U
/
g
2
2
X
y
C Y
x
D 0:
(2.234)
Since both equations in (2.234) must be satisfied for all p and q, this requires that each coefficient
of
p
and
q
must vanish. is leads to
2g
1
X g
2
Y D 0;
g
2
X C 2g
1
Y D 0;
(2.235)
and
2g
1
U C X
x
Y
y
D 0;
g
2
U C X
y
C Y
x
D 0:
(2.236)
From (2.235) we see that either g
1
D g
2
D 0 or X D Y D 0. If g
1
D g
2
D 0, then Q
is quasilinear giving that F is quasilinear, which violates our non-quasilinearity condition. If
X D Y D 0, we are led to a contradiction, as we imposed X
2
x
C Y
2
x
¤ 0 or X
2
y
C Y
2
y
¤ 0. us,
it follows that
X
2
x
C Y
2
x
D 0; X
2
y
C Y
2
y
D 0;
or
X
x
D 0; X
y
D 0; Y
x
D 0; Y
y
D 0:
Furthermore, from (2.236) we obtain U D 0. Since Q is not quasilinear then from (2.231) we
deduce that
.
Xp C Yq
/
2
C
.
Xq Yp
/
2
D 0;
from which we obtain X D Y D 0. With this assignment, we see from (2.229) that F D Q and
from (2.230) that Q satisfies
Q
up
D 0; Q
uq
D 0; (2.237)
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