3.5. DARBOUX TRANSFORMATIONS 67
As the process is now identical to that presented in (3.117)–(3.124), we simply state our results.
Solutions of
u
t
D u
xx
C
.
g.t; x/ C 2.ln /
xx
/
u (3.152)
can be obtained via the Darboux transformation
u D v
xx
ˇ
ˇ
ˇ
ˇ
!
1
!
2
!
1
xx
!
2
xx
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
!
1
!
2
!
1
x
!
2
x
ˇ
ˇ
ˇ
ˇ
v
x
C
ˇ
ˇ
ˇ
ˇ
!
1
x
!
2
x
!
1
xx
!
2
xx
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
!
1
!
2
!
1
x
!
2
x
ˇ
ˇ
ˇ
ˇ
v; (3.153)
where D !
1
!
0
2
!
2
!
0
1
, !
1
, !
2
, and v are all independent solutions of
w
t
D w
xx
C g.t; x/w: (3.154)
ese results also generalize to nth order Darboux transformations; we refer the reader to Arrigo
and Hickling [62] for futher details.
3.5.3 DARBOUX TRANSFORMATIONS BETWEEN TWO WAVE
EQUATIONS
In this section we wish to connect solutions to two different wave equations. In particular, find
wave equations of the form
u
t t
D u
xx
C f .x/u; (3.155)
that admit Darboux transformations where v satisfies
v
tt
D v
xx
: (3.156)
Again, we will consider first- and second-order transformations
First-Order Darboux Transformations
We will start with first-order Darboux transformations as we did earlier and seek solutions of
(3.155) via the Darboux transformations of the form
u D v
x
C A.x/v; (3.157)
where v satisfies (3.156). As we did previously, substituting (3.157) into (3.155), imposing
(3.156) and isolating coefficients with respect to v and v
x
, gives
2A
0
C f D 0; (3.158a)
A
00
C fA D 0; (3.158b)
where prime denotes differentiation with respect to the argument. From (3.158a) we see that
f D 2A
0
; (3.159)
68 3. DIFFERENTIAL SUBSTITUTIONS
which in turn gives (3.158b) as
A
00
2AA
0
D 0: (3.160)
is can be integrated once giving
A
0
A
2
D c; (3.161)
where c is a constant of integration. If we introduce a Hopf–Cole transformation
A D
0
=; (3.162)
where D .x/; then (3.161) becomes
00
C c D 0: (3.163)
Combining the results, solutions of
u
t t
D u
xx
C 2.ln /
00
u (3.164)
are obtained via the Darboux transformation
u D v
x
0
v; (3.165)
where v is a solution of the wave equation (3.156) and a solution of (3.163).
Second-Order Darboux Transformations
We now extend the Darboux transformation to second order and seek solutions of
u
t
D u
xx
C f .x/u (3.166)
via the Darboux transformations of the form
u D v
xx
C A.x/v
x
C B.x/v; (3.167)
where v satisfies (3.156). As we did previously, substituting (3.167) into (3.166), imposing
(3.156) and isolating coefficients with respect to v, v
x
, and v
xx
, gives
2A
0
C f D 0; (3.168a)
A
00
C 2B
0
C fA D 0: (3.168b)
B
00
C f B D 0: (3.168c)
From (3.168a) we again see that
f D 2A
0
; (3.169)
which, in turn, gives (3.168b) and (3.168c) as
A
00
2AA
0
C 2B
0
D 0; (3.170a)
B
00
2BA
0
D 0: (3.170b)
3.5. DARBOUX TRANSFORMATIONS 69
If we introduce the matrix
D
0 1
B A
; (3.171)
the system of equations (3.170) conveniently becomes
00
2
0
D 0: (3.172)
Again, the Matrix Hopf–Cole tranformation
D ˆ
0
ˆ
1
(3.173)
works well here leading to the linear Matrix equation
ˆ
1
ˆ
xx
0
D 0; (3.174)
which we integrate giving
ˆ
xx
D ˆ c; (3.175)
where c is a 2 by 2 matrix of arbitrary constants, namely
c D
c
11
c
12
c
21
c
22
: (3.176)
As the process is now identical to that presented in (3.173)–(3.124), we simply state our results.
Solutions of
u
t t
D u
xx
C 2.ln /
00
u (3.177)
are obtained via the Darboux transformation
u D v
xx
ˇ
ˇ
ˇ
ˇ
!
1
!
2
!
1
00
!
2
00
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
!
1
!
2
!
1
0
!
2
0
ˇ
ˇ
ˇ
ˇ
v
x
C
ˇ
ˇ
ˇ
ˇ
!
1
0
!
2
0
!
1
0
!
2
0
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
!
1
!
2
!
1
0
!
2
0
ˇ
ˇ
ˇ
ˇ
v; (3.178)
where D !
1
!
0
2
!
2
!
0
1
, with !
1
, !
2
and v satisfying the ODEs
!
00
1
D c
11
!
1
C c
21
!
2
;
!
00
2
D c
12
!
1
C c
22
!
2
;
(3.179)
and the wave equation (3.156).
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