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4(1 ? m) 3
m+1 m+1
2(m ? 1)? 2 ? + ?? m?1 = 0, ?
? ? ? + ?? m?1 = 0.
?
m?3
The solutions of these equation are:
? ?
2 2
(? ?1 + C), ? ?1 .
? 1?m = ? ? 1?m = ?
(m ? 1) (m ? 1)
2 2

Hence equation (2) has the following solutions:
?
2
u 1?m = ? (10)
ln ?1 (?2 + C?1 ),
(m ? 1)2
?
2
u 1?m = ? (11)
?2 ln ?1 .
(m ? 1)2
Using the groups of invariance of equations (1) and (2) we can duplicate the
solutions (3)–(11). In consequence we obtain multiparametric exact solutions of equa-
tion (1). Write out these solutions for equation (1) in the space R1,3 using the following
notations: a = (a0 , a1 , a2 , a3 ), b = (b0 , b1 , b2 , b3 ), c = (c0 , c1 , c2 , c3 ), yµ = xµ + ?µ
(µ = 0, 1, 2, 3), a · b = a0 b0 ? a1 b1 ? a2 b2 ? a3 b3 , ? = ±1.
?(k ? 1)2
= ?(y · y)(1 + ?(b · y) b · b = 0;
1?k k?2
1) u ), ?= ,
4(k ? 2)
2
3(k?1)
1 k?1
u1?k = ? [(y · y)(1 + ?(b · y)k?2 )] 2 + ?(b · y) 2(k+1) [1 + ?(b · y)k?2 ] 2(k+1)
2) ,
?(k ? 1)2
b · b = 0, ? ? R;
?= ,
4(k ? 2)
2
k(k?1)
1 k?1
u1?k = ? [(y · y)(1 + ?(b · y)k?2 )] 2 + ?(b · y) 2(k+1) (y · y) 2(k+1)
3) ,
?(k ? 1)2
b · b = 0, ? ? R;
?= ,
4(k ? 2)
?(k ? 1)2
u1?k = ??(b · y)k?3 [(y · y) + ?(b · y)3?k ]2 , , b · b = 0;
4) ?=
8(k ? 2)(k + 1)
?(k ? 1)2
k?3
u1?k = ?[(y · y) + (a · y)2 ][1 + ?(b · y)
5) ], ?= ,
2
2(k ? 3)
a · a = ?1, a · b = 0, b · b = 0;
1
k?3
u1?k = ? [((y · y) + (a · y)2 )(1 + ?(b · y)
6) )] 2 +
2

2
k?1
k?1 k?3
+ ?(b · y) k+1 (1 + ?(b · y) ) 2(k+1) ,
2

?(k ? 1)2
a · a = ?1, a · b = 0, b · b = 0, ? ? R;
?= ,
2(k ? 3)
1
k?3
u1?k = ? [((y · y) + (a · y)2 )(1 + ?(b · y)
7) )] 2 +
2

2
(k?1)2 k?1
+ ?(b · y) ((y · y) + (b · y) ) 2
4(k+1) 2(k+1) ,
370 W.I. Fushchych, A.F. Barannyk, Yu.D. Moskalenko

?(k ? 1)2
a · a = ?1, a · b = 0, b · b = 0, ? ? R;
?= ,
2(k ? 3)
5?k
k?5
u1?k = ??(b · y) 2 [(y · y) + (a · y)2 + ?(b · y) 2 ]2 ;
8)
?(k ? 1)2
, a · a = ?1, a · b = 0, b · b = 0;
?=
4(k ? 3)(k + 1)
?
u?1 = ? (y · y) ln(b · y), b · b = 0, k = 2;
9)
4
??
u?1 = ? (b · y)?1 [(y · y) ? ?(b · y) ln(b · y)]2 , b · b = 0,
10) k = 2;
24
?2
u = ?? ln(b · y)[(y · y) + (a · y)2 ],
11)
a · a = ?1, a · b = 0, b · b = 0, k = 3;
??
u?2 = (b · y)?1 [(y · y) + (a · y)2 ? ?(b · y)]2 ,
12)
8
a · a = ?1, a · b = 0, b · b = 0, k = 3.

1. Фущич В.И., Штелень В.М., Серов Н.И., Симметрийный анализ и точные решения нелинейных
уравнений математической физики, Киев, Наук. думка, 1989, 336 c.
2. Fushchych W.I., Serov N.I., The symmetry and some exact solutions of the nonlinear many-
dimensional Liouville, d’Alembert and eikonal equations, J. Phys. A, 1983, 16, 3645–3658.
3. Grundland A.M., Harnad J., Winternitz P., Symmetry reduction for nonlinear relativistically inva-
riant equations, J. Math. Phys., 1984, 25, № 4, 791–806.
4. Фущич В.И., Баранник Л.Ф., Баранник А.Ф., Подгрупповой анализ групп Галилея, Пуанкаре и
редукция нелинейных уравнений, Киев, Наук. думка, 1991, 304 c.
5. Фущич В.И., Баранник А.Ф., Максимальные подалгебры ранга n ? 1 алгебры AP (1, n) и редук-
ция нелинейных волновых уравнений. I, Укр. мат. журн., 1990, 42, № 11, 1250–1256.
6. Фущич В.И., Баранник А.Ф., Максимальные подалгебры ранга n ? 1 алгебры AP (1, n) и редук-
ция нелинейных волновых уравнений, II, Укр. мат. журн., 1990, 42, № 12, 1693–1700.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 371–380.

Symmetry classification
of the one-dimensional second order equation
of hydrodynamical type
W.I. FUSHCHYCH, V.M. BOYKO
The paper contains a symmetry classification of the one-dimensional second order equa-
tion of hydrodynamical type L(Lu) + ?Lu = F (u), where L ? ?t + u?x . Some classes
of exact solutions of this equation are pointed out.

In [1, 2] the following generalized Navier–Stokes equation

?1 Lv + ?2 L(Lv) = F v 2 v + ?4 ?p, (1)

was proposed, where
? ?
L? + vl + ?3 , l = 1, 2, 3,
?t ?xl
v = v 1 , v 2 , v 3 , v l = v l (t, x), p = p(t, x), ? is the gradient, is the Laplace operator,
?1 , ?2 , ?3 , ?4 are arbitrary real parameters, F v 2 is an arbitrary differentiable
function.
In the one-dimensional scalar case, when ?3 = 0, ?4 = 0, equation (1) has the
form

(2)
?1 Lu + ?2 L(Lu) = F (u),

where u = u(t, x), L ? ?t + u?x .
In the case when ?2 = 0 and F (u) = 0, equation (2) is known to describe the
simple wave

u = ?(x ? tu), (3)

where ? is an arbitrary function. Formula (3) gives the general solution of the equation
?u ?u
+u = 0.
?t ?x
If ?2 = 0, then equation (2) can be rewritten in the form

(4)
L(Lu) + ?Lu = F (u), ? = const.

Equation (4), in expanded form, is written as follows
2
?2u ?2u 2
?u ?u ?u 2? u ?u ?u
+ 2u + +u +u +? +u = F (u).
2 ?x2
?t ?t?x ?t ?x ?x ?t ?x
Preprint LiTH-MAT-R-95-19, Department of Mathematics, Link?ping University, Sweden, 11 p.
o
372 W.I. Fushchych, V.M. Boyko

This equation with arbitrary F (u) is evidently invariant under the two-dimensional
algebra of translations that is determined by the operators

(5)
P 0 = ?t , P1 = ?x .

In the present paper we carry out a symmetry classification of the equation (4),
i.e., we describe functions F (u), with which the equation (4) admits more extensive
Lie algebras than the two-dimensional algebra of translations (5).
Symmetry classification
Symmetry classification of (4) is performed on the base of the Lie algorithm [3,
4, 5] in the class of first-order differential operators

X = ? 0 (t, x, u)?t + ? 1 (t, x, u)?x + ?(t, x, u)?u . (6)

Remark. In cases 1.4, 2.3, 2.4 we assume that
?? 0 ?? 1
= 0, = 0.
?u ?u
It is obvious, that the cases ? = 0 and ? = 0 will be essentially different for the
investigation of symmetries of the equation (4). If ? = 0, then one can always set
? ? 1 (there exists a change of variables). For this reason we consider the cases
? = 0 and ? = 1 separately.
I. Let us consider equation (4), when ? = 0, i.e., the equation

(7)
L(Lu) = F (u).

Symmetry classification of (7) leads to five distinct cases.
Case 1.1. F (u) is an arbitrary continuously differentiable function. The maximal
invariance algebra in this case is the two-dimensional algebra (5).
Case 1.2. F (u) = a exp (bu), a, b = const, a = 0, b = 0. Without loss of generality
we can put b ? 1 (there exists a change of variables). The maximal invariance algebra
of the equation

(8)
L(Lu) = a exp (u)

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

Y = t?t + (x ? 2t)?x ? 2?u . (9)
P 0 = ?t , P1 = ?x ,

The finite transformations which are generated by the operator Y in (9) have the
form:
t > t = t exp (?),
?
x > x = (x ? 2?t) exp (?),
?
u > u = u ? 2?.
?

Hereafter ? is a real group parameter of the corresponding Lie group.
We note that Y in (9) can be represented as the linear combination of the dilatation
and Galilei operators

Y = (t?t + x?x ) ? 2(t?x + ?u ) = D ? 2G.
Symmetry classification of the one-dimensional second order equation 373

The operators D and G commute, thus the transformations corresponding to Y can
be interpreted as a composition of dilatation and Galilei transformations, i.e., as
a composition of dilatation on t and x with a change of inertial system. On the other
hand, the operators (9) form a subalgebra of extended Galilei algebra, although the
extended Galilei algebra is not the invariance algebra of the equation (8). The same
results are valid for other cases of equation (4).
Case 1.3. F (u) = a(u + b)p , a, b, p = const, a = 0, p = 0, p = 1. The maximal
invariance algebra of the equation
L(Lu) = a(u + b)p (10)
is a three-dimensional algebra, whose basis elements are given by the operators
P 0 = ?t ,P1 = ?x ,
p?3 (11)
2b 2
x? t ?x ?
R = t?t + (u + b)?u .
p?1 p?1 p?1
The operator R generates the following finite transformations:
t > t = t exp (?),
?
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