36 2. COMPATIBILITY
which involves the second member of the Riccati hierarchy. Under the substitution u D S
x
=S,
(2.203) becomes
S
xxx
C kS
x
D 0: (2.204)
e solution of (2.204) differs depending on the sign of k and is given as:
S.t /
D
s
2
.t/e
!x
C s
1
.t/e
!x
C s
0
.t/ .k D !
2
/;
(2.205a)
S.t / D s
2
.t/x
2
C s
1
.t/x C s
0
.t/ .k D 0/; (2.205b)
S.t / D s
2
.t/ sin.!x/ C s
1
.t/ cos.!x/ C s
0
.t/ .k D !
2
/; (2.205c)
where s
1
; s
2
and s
3
are arbitrary functions of integration. Substitution of any of these into
(2.202a) leads to the following ODEs for s
1
; s
2
and s
3
:
s
1
s
0
2
s
2
s
0
1
D 0; (2.206a)
s
2
s
0
0
s
0
s
0
2
3ks
0
s
2
D 0; (2.206b)
for k ¤ 0 and
s
1
s
0
2
s
2
s
0
1
D 0; (2.207a)
s
2
s
0
0
s
0
s
0
2
6s
2
2
D 0; (2.207b)
for k D 0. ese are easily solved, giving
S.t / D
c
1
e
!x
C c
2
e
!x
C c
3
e
3!
2
t
s.t/ .k D !
2
/; (2.208a)
S.t / D
c
1
.x
2
C 6t/ C c
2
x C c
3
s.t/ .k D 0/; (2.208b)
S.t / D
c
1
sin.!x/ C c
2
cos.!x/ C c
3
e
3!
2
t
s.t/ .k D !
2
/; (2.208c)
where c
1
c
3
are constant and via u D S
x
=S gives the exact solutions
u D
!
.
c
1
e
!x
c
2
e
!x
/
c
1
e
!x
C c
2
e
!x
C c
3
e
3!
2
t
.k D !
2
/;
u D
2c
1
x C c
2
c
1
.x
2
C 6t/ C c
2
x C c
3
.k D 0/;
u D
!
.
c
1
cos.!x/ c
2
sin.!x/
/
c
1
cos.!x/ C c
2
sin.!x/ C c
3
e
3!
2
t
.k D !
2
/:
(2.209)
e second place we saw cubic source terms was in (2.197). We will set k
1
D 4k and
k
2
D q for convenience, so (2.197) becomes
u
t
3
S
0
S
u
x
D 3
S
0
S
0
u; (2.210a)
u
t
D u
xx
C qu
3
2ku; (2.210b)
2.2. SECOND-ORDER PDES 37
where S satisfies S
00
C kS D 0. We will distinguish between two different cases:
.i/ k D 0;
.i i/ k ¤ 0:
In the case of k D 0, then S D c
1
x C c
2
. However, we can set c
1
D 1 and c
2
= 0 without
loss of generality. In this case, (2.210a) becomes
u
t
3
x
u
x
D
3
x
2
u; (2.211)
which is solved giving
u D xF .x
2
C 6t/; (2.212)
where F is an arbitrary function of its argument. Substitution of (2.212) into (2.210b) (with
k D 0) gives
F
00
C
q
4
F
3
D 0; (2.213)
where F D F ./;  D x
2
C 6t. e solution of (2.213) can be expressed in terms of the Jacobi
elliptic functions so
F D
2
p
q
cn
;
1
p
2
and F D
2
p
q
nc
;
1
p
2
depending on the sign of q.
In the case of k ¤ 0, we integrate (2.210a), giving
u D
kS.x/
S
0
.x/
F
3kt ln.S
0
/
; (2.214)
where F is again an arbitrary function of its argument; substitution into (2.210b) gives rise to
the ODE
F
00
C 3F
0
C 2F C qF
3
D 0; (2.215)
where F D F ./; D 3kt ln.S
0
/. If we introduce the change of variables
D ln ; F D G; (2.216)
where G D G./, then (2.215) becomes
G
00
C qG
3
D 0; (2.217)
which is essentially (2.213). If we impose the change of variables (2.216) on the solution (2.214),
then we obtain
u D kS.x/e
3kt
G
S
0
.x/e
3kt
; (2.218)
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