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u = a2 u (x0 a3 ; x1 a), (22)

где ?, ?0 , ?1 , a — групповые параметры.

Объединяя формулу (2) с формулами (20), (21), (22), легко построить широкие

классы точных (солитонных и несолитонных) решений уравнения KdV. В частно-

сти, решение

2

u = ?1 +

ch2 (x1 + 2x0 )

после размножения с помощью формул (20)–(22), где ? = ?6, ?0 = 0, принимает

вид классического солитонного решения

2a2

u= .

ch2 a(x1 ? 4a2 x0 + ?1 )

Очевидно, что формула (2) не является единственной для уравнения KdV. На-

пример, формулы вида

(1) (1)

(2) (1) (1) (1) (2) (1)

u = u ?2( z 2 + k ), = z 2+ k + u,

z1

(1) (1)

(1) (2) (1)

? 6( z 2 + k ) z + z 111 = 0

z0

также дают размножение решений уравнения (1).

W.I. Fushchych, Scientific Works 2002, Vol. 4, 504–523.

Second-order differential invariants

of the rotation group O(n)

and of its extensions: E(n), P (1, n), G(1, n)

W.I. FUSHCHYCH, I.A. YEGORCHENKO

Functional bases of second-order differential invariants of the Euclid, Poincar?, Galilei,

e

conformal, and projective algebras are constructed. The results obtained allow us to

describe new classes of nonlinear many-dimensional invariant equations.

0. Introduction

The concept of the invariant is widely used in various domains of mathematics. In

this paper, we investigate the differential invariants within the framework of symmetry

analysis of differential equations.

Differential invariants and construction of invariant equations were considered by

S. Lie [1] and his followers [2, 3]. Tresse [2] had proved the theorem on the existence

and finiteness of a functional basis of differential invariants. However, there exist quite

a few papers devoted to the construction in explicit form of differential invariants for

specific groups involved in mechanics and mathematical physics.

Knowledge of differential invariants of a certain algebra or group facilitates clas-

sification of equations invariant with respect to this algebra or group. There are also

some general methods for the investigation of differential equations which need tide

explicit form of differential invariants for these equations’ symmetry groups (see, e.g.,

[3, 4]).

A brief review of our investigation of second-order differential invariants for the

Poincar? and Galilei groups is given in [5, 6]. Our results on functional bases

e

of differential invariants are founded on the Lemma about functionally independent

warrants for the proper orthogonal group and two n-dimensional symmetric tensors

of the order 2.

We should like to stress that we consider functionally independent invariants of

but not irreducible ones, as in the classical theory of invariants.

Bases of irreducible invariants for the group O(3) and three-dimensional symmetric

tensors and vectors are adduced in [7].

The definitions of differential invariants differ in various domains of mathematics,

e.g. in differential geometry and symmetry analysis of differential equations. Thus,

we believe that some preliminary notes are necessary, though these formulae and

definitions can be found in [8, 9, 10].

We deal with Lie algebras consisting of the infinitesimal operators

X = ? i (x, u)?xi + ? r (x, u)?ur . (0.1)

Here x = (x1 , x2 , . . . , xn ), u = (u1 , . . . , um ). We usually mean the summation over

the repeating indices.

Acta Appl. Math., 1992, 28, 69–92.

Second-order differential invariants of the rotation group O(n) 505

Definition 1. The function

F = F (x, u, u, . . . , u),

1 l

where u is the set of all kth-order partial derivatives of the function u is called

k

a differential invariant for the Lie algebra L with basis elements Xi of the form

(0.1) (L = Xi ) if it is an invariant of the lth prolongation of this algebra:

l

X s F (x, u, u, . . . , u) = ?s (x, u, u, . . . , u)F, (0.2)

1 1

l l

where the ?s are some functions; when ?i = 0, F is called an absolute invariant;

when ?i = 0, it is a relative invariant.

Further, we deal mostly with absolute differential invariants and when writing

‘differential invariant’ we mean ‘absolute differential invariant’.

Definition 2. A maximal set of functionally independent invariants of order r ? l of

the Lie algebra L is called a functional basis of the lth-order differential invariants

for the algebra L.

We consider invariants of order 1 and 2 and need the first and second prolongations

of the operator X (0.1) (see, e.g., [8–11])

1 2 1

r r

X = X + ?i ?ur , X = X +?ij ?ur

i ij

r r

the coefficients ?i and ?ij taking the form

?i = (?xi + us ?us )? r ? ur (?xi + us ?us )? k ,

r

i i

k

?ij = (?xi + uj ?us + ujk ?us )?i ? ur (?xj + us ?us )? k .

r s s r

j

ik

k

While writing out lists of invariants, we shall use the following designations

?2u

?u

ua ? , uab ? ,

?xa ?xa ?xb

Sk (uab ) ? ua1 a2 ua2 a3 · · · uak?1 ak uak a1 , (0.3)

Sjk (uab , vab ) ? ua1 a2 · · · uaj?1 aj vaj aj+1 · · · vak a1 ,

Rk (ua , uab ) ? ua1 uak ua1 a2 ua2 a3 · · · uak?1 ak uak a1 .

Here and further we mean summation over the repeated indices from 1 to n. In all

the lists of invariants, k takes on the values from 1 to n and j takes the values from

0 to k. We shall not discern the upper and lower indices with respect to summation:

for all Latin indices

xa xa ? xa xa ? xa xa = x2 + x2 + · · · + x2 .

1 2 n

1. Differential invariants for the Euclid algebra

The Euclid algebra AE(n) is defined by basis operators

?

?a ? Jab = xa ?b ? xb ?a . (1.1)

,

?xa

Here and further, the letters a, b, c, d, when used as indices, take on the values from

1 to n, n being the number of space variables (n ? 3).

506 W.I. Fushchych, I.A. Yegorchenko

The algebra AE(n) is an invariance algebra for a wide class of many-dimensional

scalar equations involved in mathematical physics — the Schr?dinger, heat, d’Alembert

o

equations, etc.

In this section, we shall explain in detail how to construct a functional basis of the

second-order differential invariants for the algebra AE(n). This basis will be further

used to find invariant bases for various algebras containing the Euclid algebra as

a subalgebra — the Poincar?, Galilei, conformal, projective algebras, etc.

e

1.1. The main results. Let us first formulate the main results of the section in

the form of theorems.

Theorem 1. There is a functional basis of second-order differential invariants for

the Euclid algebra AE(n) with the basis operators (1.1) for the scalar function

u = u(x1 , . . . , xn ) consisting of these 2n + 1 invariants

(1.2)

u, Sk (uab ), Rk (ua , uab ).

Theorem 2. The second-order differential invariants of the algebra AE(n) (1.1) for

the set of scalar functions ur , r = 1, . . . , m, can be represented as functions of the

following expressions:

ur , Sjk (u1 , ur ), Rk (ur , u1 ). (1.3)

ab ab a ab

1.2. Proofs of the theorems. Absolute differential invariants are obtained as

solutions of a linear system of first-order partial differential equations (PDE). Thus,

the number of elements of a functional basis is equal to the number of independent

integrals of this system. This number is equal to the difference between the number

of variables on which the functions being sought depend, and the rank of the corres-

ponding system of PDE (in our case, this rank is equal to the generic rank of the

prolonged operator algebra [8, 9].

To prove the fact that N invariants which have been found, F i = F i (x, u, u, . . . , u),

1 l

form a functional basis, it is necessary and sufficient to prove the following state-

ments:

(1) the F i are invariants;

(2) the F i are functionally independent;

(3) the set of invariants F i is complete or N is equal to the difference of the number

of variables (x, u, u, . . . , u) and the rank of the system of defining operators.

1 l

We seek second-order differential invariants in the form

F = F (x, u, u, u).

12

It follows from the condition of invariance with respect to translation operators ?a

that F does not depend on xa ; evidently, u is an invariant of the operators (1.1). Thus,

it is sufficient to seek invariants depending on u and u only. The criterion of the

1 2

absolute invariance (0.1) in this case has the form

?

Jab F (u, u) = 0, (1.4)

12

where

?

Jab = ur ?ur ? ur ?ur + 2(ur ?ur ? ur ?ur ), (1.5)

a b ac bc

a ac

b bc

the summation over r from 1 to m being implied.

Second-order differential invariants of the rotation group O(n) 507

In that way, the problem of finding the second-order differential invariants of the

algebra AE(n) is reduced to the construction of a functional basis for the rotational

algebra AO(n) with the basis operators (1.5) for m vectors and m symmetric tensors

of order 2.

Lemma 1. The rank of the algebra AO(n) is equal to (n(n ? 1))/2.

Proof. It is sufficient to prove the lemma for m = 1. The basis of the algebra (1.5)

consists of (n(n ? l))/2 operators. According to definition [8], its rank is equal to the

generic rank of the coefficient matrix of these operators. Let us put uab = 0 when

a = b and write down the coefficient columns by ?uab of the operators (1.5):

? ?

u11 ? u22 ···

0 0

? ?

u11 ? u33 · · ·

0 0

? ?. (1.6)

? ?

··· ··· ··· ···

· · · un?1,n?1 ? unn

0 0

When uaa = ubb for a = b and all uaa = 0, the determinant of the matrix (1.6)

does not vanish, therefore its generic rank (that is, the generic rank the algebra being

considered) cannot be less than (n(n ? 1))/2. The lemma is proved.

Lemma 2. The expressions

(1.7)

Sk (uab ), Rk (ua , uab )

are functionally independent.

Proof. To establish independence of expressions (1.7), it is sufficient to consider the

case when uab = 0 if a = b and uaa = 0. Let us write down the Jacobian of the

invariants

···

1 1

· · · 2unn 0

2u11

··· ··· ···

· · · nun?1

nun?1 (1.8)

rr

11

···

2u1 2un

··· ··· ··· ···

· · · 2un un?1

2u1 un?1 nn

11

The Jacobian (1.8) is equal up to a coefficient to the product of two Vandermonde

determinants and is not equal to zero if uaa = ubb whenever a = b. Thus, the

expressions (1.17) are functionally independent.

Proof of Theorem 1. The fact that expressions (1.2) are invariants of AO(n) can be

easily proved by direct substitution of these expressions into the invariance conditions.

Nevertheless, it is useful to note that Sk (uab ) are traces of the symmetric matrix

(uab ) = U and its powers, Rk (ua , uab ) are the scalar products of the vector (ua ) =

(u1 , . . . , un ), the matrix U k?1 and the vector (ua )T .

The invariants for the vector (ua ) and the symmetric tensor (uab ) depend on their

(n(n + 3))/2 elements. Thus, it follows from Lemma 1 that a functional basis of the

algebra AO(n) for (ua ) and (uab ) must consist of

n(n + 3) n(n ? 1)

? = 2n

2 2

invariants.

508 W.I. Fushchych, I.A. Yegorchenko

Therefore the set (1.7) is a complete set of functionally independent invariants

of the form F = F (u, u) and (1.2) represents a functional basis of the second-order

12

invariants for the algebra AE(n). The theorem is proved.

Let us consider the case of two vectors (ua ), (va ) and two symmetric tensors of

the second order (uab ), (vab ). The operators of the rotation algebra have the form

(1.5), u ? u1 , v ? u2 .

In this case, a functional basis of invariants contains

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