3.5. DARBOUX TRANSFORMATIONS 65
admit a Darboux transformation of the form
u D v
xx
3 sinh x
cosh x
v
x
C
2 cosh
2
x 3
cosh
2
x
v: (3.132)
ese results generalize to nth order Darboux transformations; we refer the reader to Arrigo and
Hickling [61] for futher details.
3.5.2 DARBOUX TRANSFORMATIONS BETWEEN TWO DIFFUSION
EQUATIONS
In this section we wish to extend the previous section’s results and now have v satisfy
v
t
D v
xx
C g.t; x/v: (3.133)
First-Order Darboux Transformations
We will start with first-order Darboux transformations as we did earlier and seek solutions of
u
t
D u
xx
C f .t; x/u (3.134)
via the Darboux transformations of the form
u D v
x
C A.t; x/v; (3.135)
where v satisfies (3.133). As we did previously, substituting (3.135) into (3.134), imposing
(3.133) and isolating coefficients with respect to v and v
x
gives
2A
x
C f g D 0; (3.136a)
A
t
A
xx
fA C g
x
C gA D 0: (3.136b)
From (3.136a) we see that
f D g 2A
x
; (3.137)
which in turn gives (3.136b) as
A
t
C 2AA
x
A
xx
C g
x
D 0: (3.138)
e Hopf–Cole transformation (3.99), i.e.,
A D
x
= (3.139)
works well here, where substitution and integrating (3.138) gives
t
xx
C g.t; x/ D 0: (3.140)
Note that the function of integration is omitted without loss of generality. Combining the re-
sults, solutions of
u
t
D u
xx
C .g.t; x/ C 2.ln /
xx
/u (3.141)
66 3. DIFFERENTIAL SUBSTITUTIONS
are obtained via the Darboux transformation
u D v
x
x
v; (3.142)
where v and satisfy (3.133) and (3.140), respectively.
Second-Order Darboux Transformations
We now extend the Darboux transformation to second order and seek solutions of
u
t
D u
xx
C f .t; x/u (3.143)
via the Darboux transformations of the form
u D v
xx
C A.t; x/v
x
C B.t; x/v; (3.144)
where v satisfies (3.133). As we did previously, substituting (3.144) into (3.143), imposing
(3.133) and isolating coefficients with respect to v, v
x
, and v
xx
, gives
2A
x
C f g D 0; (3.145a)
A
t
A
xx
2B
x
fA C 2g
x
C gA D 0; (3.145b)
B
t
B
xx
C gB f B C Ag
x
C g
xx
D 0: (3.145c)
From (3.145a) we again see that
f D g 2A
x
; (3.146)
which, in turn, gives (3.145b) and (3.145c) as
A
t
C 2AA
x
A
xx
2B
x
C 2g
x
D 0: (3.147a)
B
t
C 2BA
x
B
xx
C g
xx
C Ag
x
D 0: (3.147b)
If we introduce the matrices
D
0 1
B A
; G D
g 0
g
x
g
; (3.148)
the system of equations (3.147) conveniently becomes
t
C 2
x
xx
C G G C G
x
D 0: (3.149)
Again, the Matrix Hopf–Cole transformation
D ˆ
x
ˆ
1
(3.150)
works well here, leading to the linear Matrix equation
ˆ
t
ˆ
xx
Gˆ D 0: (3.151)
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