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p?3
x > x = x exp ? ? bt exp (?),
?
p?1
u > u = (u + b) exp ? p?1 ? ? b.
2
?

If b = 0, then R can be again represented as a linear combination of dilatation and
Galilei operators.
Case 1.4. F (u) = au + b, a, b = const, a = 0. In consequence of a change of
variables one can always set a ? 1 or a ? ?1. Let us consider these cases.
a) The invariance algebra of the equation
(12)
L(Lu) = u + b
is a seven-dimensional algebra, whose basis elements are given by the operators
P0 = ?t , P1 = ?x ,
Y1 = (x + bt)?x + (u + b)?u ,
Y2 = cosh t?x + sinh t?u ,
(13)
Y3 = sinh t?x + cosh t?u ,
Y4 = cosh t?t + (x + bt) sinh t?x + ((x + bt) cosh t + b sinh t)?u ,
Y5 = sinh t?t + (x + bt) cosh t?x + ((x + bt) sinh t + b cosh t)?u .
The operators Y1 –Y3 generate the following finite transformations (because the trans-
formations for Y4 and Y5 are cumbersome we omit their explicit form):
t > t = t,
?
Y1 :
x > x = (x + bt) exp(?) ? bt,
?
u > u = (u + b) exp(?) ? b.
?

t > t = t,
?
Y2 :
x > x = x + ? cosh t,
?
u > u = u + ? sinh t.
?
374 W.I. Fushchych, V.M. Boyko

t > t = t,
?
Y3 :
x > x = x + ? sinh t,
?
u > u = u + ? cosh t.
?
The operator Y1 in (13) can be again represented as a linear combination of the
dilatation and Galilei operators.
b) The invariance algebra of the equation
L(Lu) = ?u + b (14)
is a seven-dimensional algebra, whose basis elements are given by the operators
P 0 = ? t , P1 = ?x ,
R1 = (x ? bt)?x + (u ? b)?u ,
R2 = cos t?x ? sin t?u ,
(15)
R3 = sin t?x + cos t?u ,
R4 = ? cos t?t + (x ? bt) sin t?x + ((x ? bt) cos t ? b sin t)?u ,
R5 = sin t?t + (x ? bt) cos t?x ? ((x ? bt) sin t + b cos t)?u .
The operators R1 –R3 generate the following finite transformations (because the
transformations for R4 and R5 are cumbersome we omit their explicit form):
R1 : t > t = t,
?
x > x = (x ? bt) exp(?) + bt,
?
u > u = (u ? b) exp(?) + b.
?
R2 : t > t = t,
?
x > x = x + ? cos t,
?
u > u = u ? ? sin t.
?
R3 : t > t = t,
?
x > x = x + ? sin t,
?
u > u = u + ? cos t.
?
The operator R1 in (15) can be again represented as a linear combination of dilatation
and Galilei operators.
Case 1.5. F (u) = a, a = const. In the case a = 0 (there exists a change of
variables) without loss of generality we can admit that a ? 1. Thus we consider the
cases a = 0 and a = 1 separately.
a) The maximal invariance algebra of the equation
(16)
L(Lu) = 0
is a ten-dimensional algebra, whose basis elements are given by the operators
P 0 = ? t , P1 = ?x ,
G = t?x + ?u , D = t?t + x?x , D1 = x?x + u?u ,
1 1 1
A1 = t2 ?t + tx?x + x?u , A2 = t2 ?x + t?u , A3 = u?t + u2 ?x ,
2 2 2 (17)
12 12
A4 = (tu ? x)?t + tu ?x + u ?u ,
2 2
122
A5 = t2 u ? 2tx ?t + t u ? 2x2 ?x + tu2 ? 2xu ?u .
2
Symmetry classification of the one-dimensional second order equation 375

We note, that subalgebras P0 , P1 , G and A1 , ?A2 , G in the representation (17)
define two different representations of the Galilei algebra AG(1, 1) [3].
The finite transformations which are generated by the operators (17) have the form
(because the transformations for A4 and A5 are cumbersome we omit their explicit
form):

G : t > t = t,
?
x > x = x + ?t,
?
u > u = u + ?.
?

D : t > t = t exp(?),
?
x > x = x exp(?),
?
u > u = u.
?

D1 : t > t = t,
?
x > x = x exp(?),
?
u > u = u exp(?).
?

2t
t>t=
?
A1 : ,
2 ? ?t
4x
x>x=
? 2,
(2 ? ?t)
2x?
u>u=u+
? .
2 ? ?t

t > t = t,
?
A2 :
1
x > x = x + ?t2 ,
?
2
u > u = u + ?t.
?

t > t = t + ?u,
?
A3 :
1
x > x = x + ?u2 ,
?
2
u > u = u.
?

b) The maximal invariance algebra of the equation

(18)
L(Lu) = 1

is a ten-dimensional algebra, whose basis elements are given by the operators

P0 = ?t , P1 = ?x , G = t?x + ?u ,
1 1
x ? t 3 ?x + u ? t 2 ?u ,
B1 = t?t + 3x?x + 2u?u , B2 =
6 2
(19)
1 1 1 1
B3 = t2 ?t + tx + t4 ?x + x + t3 ?u , A2 = t2 ?x + t?u ,
2 12 3 2
1 1 2 14 1
B4 = u ? t2 ?t + u ? t ?x + tu ? t3 ?u ,
2 2 8 2
376 W.I. Fushchych, V.M. Boyko

1 1 2 12 1
tu ? x ? t3 ?t + tu ? t x ? t5 ?x +
B5 =
3 2 2 24
1 2 12 5
u + t u ? tx ? t4 ?u ,
+
2 2 24
1 122 1 1
B6 = t2 u ? 2tx ? t4 ?t + t u ? 2x2 ? t3 x ? t6 ?x +
6 2 3 72
1 1
+ tu2 ? 2xu + t3 u ? t2 x ? t5 ?u .
3 12
The algebra, generated by the operators (19), includes again two different Galilei
algebras P0 , P1 , G and B3 , ?A2 , G as subalgebras.
The finite transformations which are generated by the operators (19) have the
form (because the transformations for B4 , B5 and B6 are cumbersome we omit their
explicit form):
t > t = t exp(?),
?
B1 :
x > x = x exp(3?),
?
u > u = u exp(2?).
?

t > t = t,
?
B2 :
1 1
x>x= x ? t3 exp(?) + t3 ,
?
6 6
1 1
u>u= u ? t2 exp(?) + t2 .
?
2 2
2t
t>t=
?
B3 : ,
2 ? ?t
12x ? 2t3 4t3
x>x=
? 2+ 3,
3(2 ? ?t) 3(2 ? ?t)
12x ? 2t3
2t2 12x + t3
u>u=u+ 2 + 3t(2 ? ?t) ?
? .
(2 ? ?t) 6t

II. Let us consider equation (4) for ? = 0. As it was noticed above, we can set
? ? 1. Symmetry classification gives in this case four principally distinct cases.
Case 2.1. F (u) is an arbitrary continuously differentiable function. The maximal
invariance algebra of the equation
(20)
L(Lu) + Lu = F (u),
is the two-dimensional algebra (5).
Case 2.2. F (u) = au3 ? 2 u, a = const, a = 0. The maximal invariance algebra of
9
the equation
2
L(Lu) + Lu = au3 ? u (21)
9
is a three-dimensional algebra, whose basis elements are given by the operators
1 1
?t ? u?u . (22)
P 0 = ?t , P1 = ?x , Z = exp t
3 3
Symmetry classification of the one-dimensional second order equation 377

The operator Z generates the following finite transformations:
1 ?
t > t = ?3 ln exp ? t ?
? ,
3 3
x > x = x,
?
1 1
u > u = u 1 ? ? exp
? t .
3 3
Case 2.3. F (u) = au + b, a, b = const, a = 0. The invariance algebra of the
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