54 3. DIFFERENTIAL SUBSTITUTIONS
From (3.31) and (3.30a) we find
A.F / D c
1
F Cc
2
; F D F
1
.v/p CF
2
.v/; (3.32)
where c
1
and c
2
are arbitrary constants and F
1
and F
2
are arbitrary functions. With these as-
signments, returning to (3.30b) and isolating coefficients with respect to p gives
2F
0
1
c
1
F
2
1
D 0; (3.33a)
F
1
.
B c
1
F
2
c
2
/
D 0; (3.33b)
which we conveniently solve as
F
1
D
2
c
1
v Cc
3
; B D c
1
F Cc
2
; (3.34)
where c
3
is an additional arbitrary constant. e remaining equation in (3.30c) becomes
F
00
2
D 0; (3.35)
which we solve as
F
2
D c
4
v Cc
5
: (3.36)
With appropriate translation and scaling of variables we can set the following: c
1
D 1, c
2
D
c
4
D c
5
D 0, and suppress the subscript in c
3
. us, we have the following: solutions of
u
t
C u u
x
D u
xx
(3.37)
can be obtained via
u D 2
v
x
v
C c v (3.38)
where
v
satisfies
v
t
C c v v
x
D v
xx
: (3.39)
If we set c D 0 we get the Hopf–Cole transformation. It is interesting to note that if we set
c D 1, we get a transformation which gives rise to solutions of the same equation.
3.2 KDV-MKDV CONNECTION
e Korteweg–deVries equation (KdV)
u
t
C 6uu
x
C u
xxx
D 0; (3.40)
first introduced by Korteweg and DeVries [56] to model shallow water waves, is a remarkable
NLPDE. It has a number of applications and possesses a number of special properties (see, for
example, Miura [57]). In 1968, Robert Miura found this remarkable transformation [58]. He
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