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,
u1 (1.29a)
r 1 k(2/??1)?1
Rk (?a , ?ab ) (r = 2, . . . , m);
(2) when ? = 0:
ur (r = 1, . . . , m), (u1 u1 )?2k Sjk (wab , wab ),
1 r
dd
(1.29b)
(u1 u1 )1?2k Rk (ur , wab ) (r = 2, . . . , m)
1
a
dd

(for the set of invariants (u1 u1 )?2k Sk (wab ), k does not take the value n); the tensors
dd
r r
?ab , wab are constructed similarly to (1.24) and
ur u1
? a.
a
r
?a =
ur u1
Theorem 4 is proved similarly to Theorem 3.
Second-order differential invariants of the rotation group O(n) 513

The functional independence of the sets of invariants follows from Lemma 2 and 3
taking into account the fact that transformations ur > ?ab , ur > wab (r = 1, . . . , m)
r r
ab ab
and ur > ?a (r = 2, . . . , m) are nondegenerate.
r
a
1.4. Differential invariants of the rotation algebra. The rotation algebra is
defined by the basis operators Jab (1.1).
The second-order invariants of this algebra for m scalar functions ur are construc-
ted with xa , ur , ur , wab similarly to invariants of the Euclid algebra.
r
a
Theorem 5. There is a functional basis of the second-order differential invariants
for the algebra AO(n) that has the form

ur , Sjk (u1 , ur ), Rk (ur , u1 ), Rk (xa , u1 ), r = 1, . . . , m;
ab ab a ab ab

the corresponding basis of invariants for the algebra Jab , D , where D is defined
by (1.28), consists of the expressions

ur Sjk (u1 , ur )
, Rk (ur , u1 )(u1 )2k/??1?k ,
ab ab
(r = 2, . . . , m), a ab
1 1 )k(1?2/?)
u (u
Rk (xa , u1 )(u1 )2/?(k?2)?k+1 , when ? = 0;
ab
u , Rk (ur , u1 )(u1 )?k , Sjk (u1 , ur )(u1 )?k (k = 1 when r = 1),
r
a aa aa
ab ab ab
1 1 2?k
Rk (xa , uab )(uaa ) when ? = 0.

A basis of invariants for the algebra Jab , D, Ka when ? = 0, consists of the
expressions (1.29a) and
1
Rk (xa , ?ab )
, k = 2, . . . , n + 1;
x2 (u1 )(k?1)(1?2/?)
when ? = 0 it consists of the expressions (1.29b) and
1
Rk (xa , wab )
(x2 = xa xa ).
x2 (waa )k?1
1

The proof of this theorem is similar to the proofs of Theorems 2 and 3; notice that
(xa ) is a co variant tensor with respect to the conformal operators.
2. Differential invariants of the Poincar? and conformal algebra
e
In this section, we consider differential invariants of the second order for a set
of m scalar functions

n ? 3.
ur = ur (x0 , x1 , . . . , xn ), (2.1)

The Poincar? algebra AP (1, n) is defined by the basis operators
e
?
Jµ? = xµ p? ? x? pµ , (2.2)
pµ = igµ? ,
?xµ

where µ, ? take the values 0, 1, . . . , n; the summation is implied over the repeated
indices (if they are small Greek letters) in the following way:

x? x? ? x? x? ? x? x? = x2 ? x2 ? · · · ? x2 , gµ? = diag (1, ?1, . . . , ?1). (2.3)
0 1 n
514 W.I. Fushchych, I.A. Yegorchenko

We consider x? and x? equal with respect to summation not to mix signs of
derivatives and numbers of functions.
The quasilinear second-order invariants of the Poincar? algebra were described
e
in [12].
Theorem 6. There is a functional basis of the second-order differential invariants
of the Poincar? algebra AP (l, n) for a set of m scalar functions ur consisting of
e
n(n + 1)
m(2n + 3) + (m ? 1)
2
invariants
ur , Rk (ur , u1 ), Sjk (ur , u1 ).
µ µ? µ? µ?

In this section, everywhere k = 1, . . . , n + 1; j = 0, . . . , k; r = 1, . . . , m.
?
For the extended Poincar? algebra AP (l, n) = pµ , Jµ? , D , where
e
D = xµ pµ + ?ur pur (2.4)
(pur = i(?/?ur ), the summation over r from 1 to m is implied) the corresponding
basis has the following form:
(1) when ? = 0:
Sjk (ur , u1 )(u1 )?k , Rk (ur , u1 )(u1 )?k ;
ur , µ? µ? ?? µ µ? ??

(2) when ? = 0:
ur
, Sjk (ur , u1 )(u1 )k(2/??1) , Rk (ur , u1 )(u1 )2k/??k?1 ,
µ? µ? µ µ?
u1
where Sjk , Rk are defined similarly to (0.3) and the summation over small Greek
indices is of the type (2.2).
For the conformal algebra AC(1, n) = pµ , Jµ? , D, Kµ , where
Kµ = 2xµ D ? x? x? pµ
(D being the dilation operator (2.3)), the corresponding basis consists of the expres-
sions
ur
r 1 1 k(2/??1)
, Rk (?µ , ?µ? )(u1 )k(2/??1)?1 ;
r 1
Sjk (?µ? , ?µ? )(u ) ,
u1
when ? = 0; r = 2, . . . , m, there is no summation over r; the conformally covariant
tensors have the form
ur ur
ur u1 ur ur
µ µ µ? ??
= r ? 1 , ?µ? = ?uµ? + (1 ? ?) r ? gµ?
r r r
?µ .
2ur
u u u
When ? = 0, the corresponding basis of invariants for the conformal algebra has
the form
Sjk (wµ? , wµ? )(u1 u1 )?2k ,
ur , 1
Rk (ur , wµ? )(u1 u1 )1?2k ,
1
r = 2, . . . , m;
?? µ ??
r
the tensors (wµ? ),
gµ? r
wµ? = ur ur ur ? ? ur (ur ur + ur ur )
r
u
1 ? n ??
?? µ? ? µ ?? ? ?µ

are conformally invariant (there is no summation over r).
Second-order differential invariants of the rotation group O(n) 515

The proof of Theorem 6 follows from those of Theorems 2, 3 for x = (x1 , . . . , xn+1 )
if we substitute ix0 instead of xn+1 .
Similarly to the results of Paragraph 1.4, it is possible to construct the invariants
of the algebras Jµ? , Jµ? , D , Jµ? , D, Kµ .
The obtained results allow us to construct new nonlinear many-dimensional equa-
tions, e.g. the equation
u? u ?
u?? ? uµ u? uµ? = (u? u? )2 F (u),
1?n
where F is an arbitrary function, is invariant under the algebra AC(1, n), ? = 0. The
left member of the above equation is equal to wµµ .
There is another quasi-linear relativistic equation with rich symmetry properties
(1 ? u? u? )uµµ ? u? uµ u?µ = 0,
that is, the Born–Infeld equation. The symmetry and solutions of this equation were
investigated in [10, 13]. This equation is invariant under the algebra AP (1, n+1) with
the basis operators
JAB = xA pB ? xB pA ,
A, B = 1, . . . , n + 1, xn+1 ? u.
Let us consider the class of equations
uµ? uµ? = F (uµµ , uµ u? uµ? , uµ uµ , u).
It is evident that they are invariant with respect to the Poincar? algebra AP (1, n)
e
out the straightforward search the conformally invariant equations from this class
with the standard Lie technique requires a lot of cumbersome calculations. The use of
differential invariants turns this problem into one of elementary algebra, e.g. if ? = 0
1
F ? uµ? uµ? = ? S2 (?µ? ) + u2(1?2/?) ?(S1 (?µ? )u2/??1 ),
?
where ?µ? is of the form (1.24) and ? is an arbitrary function. Whence
? + n u? u?
F = u2(1?2/?) ? u2/??1 uµµ ? ?
? u
2(1 ? ?)
1 2uµµ u? u?
? (?2 + n2 )(u? u? )2 ? uµ u? uµ? + .
?2 u2 ?u ?u
It is useful to note that besides the traces of matrix powers (0.3), one can utilize
r r
all possible invariants of covariant tensors ?µ? , wµ? to construct conformally invariant
equations.
3. Differential invariants of an infinite-dimensional algebra
It is well-known that the simplest first-order relativistic equation — the eikonal or
Hamilton equation
u? u? ? u2 ? u 2 ? · · · ? u2 = 0 (3.1)
0 1 n

is invariant under the infinite-dimensional algebra AP ? (1, n) generated by the opera-
tors [10, 14]
X = (bµ? x? + aµ )?µ + ?(u)?u , (3.2)
516 W.I. Fushchych, I.A. Yegorchenko

?bµ? = b?µ , aµ , ? being arbitrary differentiate functions on u. Equation (3.1) is widely
used in geometrical optics.
In this section, we describe a class of second-order equations invariant under the
algebra (3.2).
It is easy to show that the tensor of the rank 2
?µ? = uµ u?? u? + u? u?µ u? ? 2uµ u? u?? (3.3)
is covariant under the algebra AP ? (1, n) (3.2).
Theorem 7. The equations of the form
(3.4)
Sk (?µ? ) = 0, k = 1, 2, . . . ,
Sk being defined as (0.3), are invariant with respect to the algebra AP ? (1, n) (3.2).
The problem of the description of all such equations is more difficult and we do
not consider it here.
Let us investigate in more detail the quasi-linear second-order equation of the form
uµ uµ? u? ? uµ uµ u?? = 0. (3.5)
Theorem 8. When n ? 2, equation (3.5) is invariant with respect to the algebra
AP ? (1, n) with generators of the form
?

X + d(u)xµ ?µ ,
X is of the form (3.2), d(u) is an arbitrary function on u.
The proofs of Theorems 7 and 8 can be easily obtained with the Lie technique
using the criterion of invariance
2
X Sk (?µ? ) = 0,
Sk (?µ? )=0

2
where X is the second prolongation of the operator X [8–10].
4. Differential invariants of the Galilei algebra
4.1. It is well-known that the heat equation
2µut + ?u = 0, ?u ? uaa ,
(4.1)
n?3
u = u(t, x), x = (x1 , . . . , xn ),

is invariant under the generalized Galilei algebra AGI (1, n) with the basis operators
2

? ?
Jab = xa ?b ? xb ?a ,
?t = , ?a = ,
?t ?xa
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