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?= + b1 .
? 3b3
The reduced equation is given by
d2 ?
(35)
=0
d? 2
so that the two nontrivial exact solution of (32) take the form
v v
u = [±2{6S1 S2 2S2 c?1 exp[(3 2S2 x1 b3 + S1 S2 x0 )/(2S1 b3 )] +
c
v
+ 3S2 ??2 exp[( 2S2 x1 b3 + S1 S2 x0 )/(S1 b3 )] +
c1
v
+ 18S2 b3 c2 exp[(2 2S2 x1 /S1 ]}1/2 +
2
v v v
+ 2 ??2 (b2 ? 3S2 ) exp[(3 2S2 x1 b3 + S1 S2 x0 )/(2S1 b3 )] ?
c
v v v
? 2 6S2 b3 c(2b2 ? 3S2 ) exp[ 2S2 x1 /S1 ] ?
v v
? 2 ?b2 c1 exp[( 2S2 x1 b3 + S1 S2 x0 )/(2S1 b3 )] ?
?
v
1
? [2 6S2 b3 c exp[ 2S2 x1 /S1 ] + ??1 ? c
6b3
v
? exp[( 2S2 x1 b3 + S1 S2 x0 )/(2S1 b3 )] ?
v v
? ??1 exp[(3 2S2 x1 b3 + S1 S2 x0 )/(2S1 b3 )]?1 ,
c
Q-symmetry generators and exact solutions 161

where

S2 = 3b1 b3 + b2 .
S1 = 3?b3 , 2

Note. The case ? = ?1 and f (u) = ?3 u3 + ?1 u + ?0 , i.e. the equation
b b b

?2u ? 3 ?
?u ? (36)
+ 2 = b 3 u + b1 u + b0
?x0 ?x1
has been studied by Fushchych et al. [14]. This case can be obtained from theorem 2
by considering

c 2 c 2 c 2
k3 = ? ?0 k4 = ?1
k1 = 0, k2 = , b , b ,
?3 ?3 ?3
3 2 2
b b b

i.e.
? 3 ? 3 ?
2?3 u + (?3 u3 + ?1 u + ?0 ) .
Q= + b b b b
?x0 2 ?x1 2 ?u
Case 3: Consider the equation

?2u
?
= au3 . (37)
+
?x2
?x0 1

From theorem 3, with ? = x?2 , it follows that
1

? ? ?
Q = x2 + 3x1 + 3u .
1
?x0 ?x1 ?u
The similarity ansatz is then given by
1
? = x0 ? x2
u = x1 ?(?),
61
so that the reduced equation takes the form

d2 ?
= 9a?3 . (38)
2
d?
The general solution, in terms of an elliptic integral, is given by

3v
?
d?
v = 3a(? + c2 )
2
c1 + ? 4
0

so that a solution of (37) can be given in the form

3v
u/x1
d? 1
v 2a x0 ? x2 + c2 .
=
61
2
c1 + ? 4
0

Case 4(a): From theorem 4.1 we obtain the implicit ansatz
df x1
= ?(x0 ) + v
du x0
162 N. Euler, A. K?hler, W.I. Fushchych
o

for (1) where f satisfies (26) and ? = ?1. The reduced equation takes the following
form
d? 1
??2=0 (39)
+
dx0 2x0
so that an exact solution of the nonlinear partial differential equation is
df x1 4
= v + x0 .
du x0 3
Case 4(b): From theorem 4.2 we obtain the implicit ansatz
df
= x2 ?(x0 ) + 1
1
du
for (1) where f satisfies (26) and ? = ?1. The reduced equation takes the following
form
d?
? 2? + 2?2 = 0 (40)
dx0
so that an exact solution of the nonlinear partial differential ansatz equation is given
by
x2
df 1
= +1
du 1 + c1 exp(?2x0 )
4. Generalization to m + 1 dimensions
For a generalization for m space dimensions we consider the equations
?u 1
?u = au3 , (41)
+
?x0 2n
?u 1
(42)
+ ?u = f (u),
?x0 2n
where a and n are real constants, f satisfies
d2 f ?2 ?2
and ? ? + ··· + 2 .
f 2 = 2,
?x2
du 2xm
1

An exact solution for (41) is found to be
2? · x
u= ,
3ax0 ? (? · x)2
m
where ? · x = ?j xj , etc., with ?j arbitrary real constants.
j=1
This solution is obtained from the Q-symmetries
? ? ?
+ 3a?j u3 ,
Qj = 2?j + 3au
?x0 ?xj ?u
where ?2 = an and j = 1, . . . , m. This leads to the ansatz
2 1
?=? ? 2ax0
u= ,
?(?) ? 2? · x u2
Q-symmetry generators and exact solutions 163

from which the reduced equation
3
d2 ? d?
2 2=
d? d?
follows. From the Q-symmetry
? ? ?
Qj = ?j (? · x)2 + 3? · x + 3?j u
?x0 ?xj ?u

with ?2 = 1 and j = 1, . . . , m, the ansatz
1
u = ? · x?(?), ? = x0 ? (? · x)2
6
reduced (41) to the equation
d2 ?
= 9a?3 . (43)
2
d?
An exact solution for (41) is then given by

3v
u/(? · x)
d? 1
v 2a x0 ? (? · x)2 + c2 .
=
2 6
c1 + ? 4
0

For (42) we obtain the Q-symmetries
v ? ?
Qj = 2 x0 + ?j f (u) ,
?x0 ?u
where ? 2 = 2n and j = 1, . . . , m. This leads to the implicit ansatz
?·x
df
= v + ?(x0 ),
du x0
and the reduces equation
d? ?
? 2 = 0. (44)
+
dx0 2x0
An exact solution of (42) is then given by
? · x + c1
df 4
v
= + (x0 ).
du x0 3
5. Concluding remarks
From the above results it is clear that the study of Q-symmetries provides a
useful method for obtaining exact solutions for nonlinear partial differential equations.
Note that all the reduced equations: (31), (33)–(35), (38)–(40), (43), (44) that were
obtained by Q-symmetry reductions are integrable and we were able to solve these
reduced equations in general.
Generalized Q-symmetries, in the form of Q-B?cklund symmetries, defined by
a
?
(45)
QB = g(x0 , . . . , xm , u, u0 , . . . , uj1 ···jq ) ,
?u
164 N. Euler, A. K?hler, W.I. Fushchych
o

can be considered for eq. (2). Here q > r. This will extend the number of exact
solutions for (2). We could find no Q-B?cklund symmetry for eq. (1) with nonlinear f .
a
In [1, 17] an example is given to demonstrate the method by which one can obtain
exact solutions with Lie–B?cklund generators. Note that, in the case of Lie–B?cklund
a a
or Q-B?cklund symmetries for (2), one can, in general, not find the general solution
a
of the equation
js? (QB ?) = 0. (46)
This is due to the fact that (46) is usually more complicated, in that it has a higher
order of derivatives and of nonlinearity, than (2). By, however, considering linear
combinations of symmetries in the contraction (46), one can combine (2) and (46) to
eliminate some derivatives or non-linearities (see [1, 17]).
In the study of conditional symmetries one can consider additional differential
equations as conditions for (2), and then study the symmetry properties of the combi-
ned two equations. However, one then has to consider the compatibility problem
between (2) and the additional equation. This approach was studied, and exact soluti-
ons were obtained, for the multi-dimensional d’Alembert equation [18, 19] and some
nonlinear equations of acoustics [20].
1. Euler N., Steeb W.-H., Continuous symmetries, Lie algebras and differential equations, Mannheim,
B.I. Wissenschaftsverlag, 1992.
2. Euler N., Shul’ga M.W., Steeb W.H., J. Phys. A: Math. Gen., 1993, to appear.

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