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(1.5)

Ga = t?a , a = 1, n

being different from the Ga operators (1.4a).

The operators (1.5) generate the following transformations:

t > t = t, xa > xa = xa + va t, U > U = U. (1.6)

We call (1.2) and (1.3) the projective Galilean transformations (PGT) and (1.6) the

Galilean transformations (GT).

Equation (1.1) admits operators (1.4a) but does not admit operators (1.5).

In § 2 we describe the nonlinear second-order PDE

F (t, x, U, U0 , U , U ) ? ??U + A(t, x, U )Ut + B(t, x, U, U ) = 0, (1.7)

I II I

where

U = (U1 , . . . , Un ), U = (U11 , U12 , . . . , Unn ),

I II

Uab = ? 2 U/?xa ?xb ,

Ua = ?U/?xa , a, b = 1, n,

F , A, B being arbitrary differentiable functions, invariant under the PGT (1.2) and

(1.3) as well as projective and scale transformations generated by operators (1.4c) and

(1.4d).

In § 3 we construct the most general nonlinear PDE of the form

? 2 U/?t?xa = U0a (1.8)

F (t, x, U, U0 , U , U00 , U01 , . . . , U0n , U ) = 0,

I II

which are invariant under the GT (1.6) and the translation group generated by the

operators (1.4e). In particular, it is established that a set of equations of the form

(1.8) does not contain linear equations (except, obviously U0 = 0, U00 = 0) invariant

under the GT (1.6) and the group of time and space translations.

The Galilean relativistic principle and nonlinear partial differential equation 487

In the final part of § 3 we give several examples of Galilei invariant equations in

independent variables (t, x1 ) space, for which general solutions are constructed.

It is to be noted that equations of the class (1.7) are widely used to describe nonli-

near diffusion, heat and other processes. In particular, this class includes diffusion

equation of the form

?U ? ?U

(1.9)

= C(U )

?t ?xa ?xa

as well as nonlinear Sch?dinger equations (if U is a complex function) and Hamilton–

o

Jacobi equations. The group classification of (1.9) for the one-dimensional case was

carried out by Ovsyannikov [12] and for the three-dimensional case by Dorodnitsyn

et al [3] and Fushchych [4].

2. Equation invariant under the projective Galilean transformations

First of all in this section we are going to find the conditions to be imposed on

the functions A and B under which (1.7) is invariant under the PGT (1.2) and (1.3).

The complete solution of this problem is given by the following theorem.

Theorem 1. Equation (1.7) is invariant under the PGT if and only if

(2.1)

A(t, x, U ) = f (t, w),

?|x|2

xa Ua

B(t, x, U, U ) = U g(t, w, w1 , . . . , wn ) + (f (t, w) ? ?) (2.2)

+ U ,

4t2

t

I

where

?|x|2

|x|2 = xa xa , (2.3)

w = U exp ,

4t

?|x|2

1 xa

(2.4)

wa = Ua + ? U exp , a = 1, n,

2t 4t

and f , g are arbitrary differentiable functions.

Proof. To prove the theorem let us use the Lie method (for a modern account,

see Bluman and Cole [1] and Ibragimov [13]). According to Lie’s approach, (1.7) is

considered as a manifold in the space of the following variables: t, x, U , U , U . (1.7)

I II

is invariant under the transformations generated by an infinitesimal operator

? ?

X = ? µ (t, x, U ) + ?(t, x, U ) , µ = 0, n

?xµ ?U

when the following invariance condition is fulfilled:

2 2

(2.5)

X F = X (??U + AUt + B)|F =0 = 0,

2

where X is the second prolongation of the infinitesimal operator X, i.e.

? ?

2

µ

+ ? µ? (t, x, U ) (2.6)

X = X + ? (t, x, U ) , µ, ? = 0, n,

?Uµ ?Uµ?

488 W.I. Fushchych, R.M. Cherniha

?µ = ?µ + Uµ ?U ? Ui (?µ + Uµ ?U ),

i i

i = 0, n,

? µ? = ?µ? + U? ?µU + Uµ ??U + Uµ U? ?U U + Uµ? ?U ? Ui (?µ? + U? ?µU )?

i i

?Uµ Ui (??U + U? ?U U ) ? Uµi (?? + U? ?U )?

i i i i

?Ui? (?µ + Uµ ?U ) ? Uµ? Ui ?U ,

i i i

i = 0, n.

Substituting (2.6) into (2.5), we obtain

?A ?B ?A ?B

?(? 11 + . . . + ? nn ) + ? µ U0 + +? U0 + +

?xµ ?xµ ?U ?U

(2.7)

?B

+?0 A + ?a = 0, a = 1, n.

?xa

F =0

After explicit expressions for ?µ , ? µ? have been substituted into (2.7) and the

obtained relation being split into separate parts for coefficients at U0a and Uab , a = b,

the conditions for ? µ are found:

µ

?a ? ?? 0 /?xa = 0, ?U ? ?? µ /?U = 0,

0 a b

?b + ?a = 0,

(2.8)

a = b, a, b = 1, n, µ = 0, n.

After taking into account (2.8) the invariance condition, written in its complete

form, is given by

?A ?B ?A ?B µ

+ (?0 + ?U U0 ? Uµ ?0 )A+

?µ U0 + +? U0 +

?xµ ?xµ ?U ?U

?U

+(?a + ?U Ua ? Ub ?a ) ? ?? ? Ua Ua ?U U ?

b

(2.9)

?Ua

?2Ua ?aU ? ?u ?U + 2Uaa ?a + Ua ?? µ

a

= 0.

F =0

In our case, taking into consideration the explicit form of the operators (1.4a) the

coefficient functions ? µ , ? of the operator X are written in the form

1

? = ? ?ga xa U,

? 0 = 0, ? a = ga t,

2

where ga , a = 1, n are arbitrary parameters.

Having used the explicit form of ? µ and ? as well as the arbitrary nature and

independence of the parameters ga (2.9) is reduced to the following linear differential

equation system, which enables one to find the functions A(t, x, U ) and B(t, x, U, U ):

I

?A 1 ?A

? ?xa U (2.10)

t = 0, a = 1, n,

?xa 2 ?U

2 ?B ?B ?B ?B ?B

? xa U ?U ? xa U1 ? · · · ? xa Un

t +

? ?xa ?U ?Ua ?U1 ?Un

(2.11)

2

+xa B ? Ua (A ? ?) = 0, a = 1, n.

?

The Galilean relativistic principle and nonlinear partial differential equation 489

Thus, the proof of the theorem is reduced to the construction of the general

solution of the strongly overdetermined system (2.10) and (2.11) consisting of 2n

equations for the functions A and B.

Now let us proceed in using the standard method to find the solutions of the

first-order PDE (see, e.g., Courant and Hilbert [2]).

Let us write the system of characteristic ordinary differential equations (ODE)

corresponding to the system (2.10)

dxa dU

(2.12)

=1 , a = 1, n.

? 2 ?xa U

t

From (2.12) we obtain two invariants necessary for the construction of the general

solution of the system (2.10):

?|x|2

(2.13)

w = U exp , w0 = t.

4t

Consequently, the general solution of (2.10) is determined by invariants (2.13) and has

the form

(2.14)

A(t, x, U ) = f (w, w0 ),

where f is an arbitrary differentiable function.

Now let us write the characteristic system of ODE (2.11):

dxa dU dUa dU1 dUa?1

? = ··· =

= = = =

(2/?)t xa U U + xa Ua xa U1 xa Ua?1

(2.15)

dUa+1 dUn dB

= ··· =

= = , a = 1, n.

xa B + (2/?)(? ? f (w, w0 ))

xa Ua+1 xa Un

In (2.15), contrary to all the previous ones, the repeated indices do not mean summa-

tion.

Having solved the system (2.15) we obtain the following system of invariants

necessary for the determination of the function B:

?|x|2

w = U exp , w0 = t,

4t

?|x|2

?xa

(2.16)

wa = Ua + U exp , a = 1, n,

2t 4t

?|x|2 ?|x|2

xa Ua

I = B + (? ? f (w, w0 )) + U exp .

4t2

t 4t

The function B is, consequently, determined from the functional equation

(2.17)

?(w, w0 , w1 , . . . , wn , I) = 0

which gives us the general solution of (2.11):

?|x|2

xa Ua

B = U g(w, w0 , w1 , . . . , wn ) + (f (w, w0 ) ? ?) (2.18)

+ U ,

4t2

t

where g is an arbitrary differentiable function.

490 W.I. Fushchych, R.M. Cherniha

Thus, we are able to construct all the equations of the form (1.7), which are

invariant under pot, completing by this the proof of the theorem.

Consequence 1. If one supposes the coefficient B in (1.7) to be independent of the

derivatives U , then

I

(2.19)

?U = ?U0 + U g(w, t)

is the most general equation, invariant under the PGT, g being here an arbitrary

differentiable function.

A class of equations (1.7) with coefficients (2.1) and (2.2) contains as a subclass a

set of equations which are invariant under the operators (1.4b) of the rotation group.

The complete description of (1.7) which admits both operators (1.4a) and (1.4b) is

given by the following theorem.

Theorem 2. Equations from the class (1.7) are invariant under the operators (1.4a)

and (1.4b) if and only if they have the form

?|x|2

xa Ua

?U = f (w, t)Ut + U g(w, wa wa , t) + (f (w, t) ? ?) (2.20)

+ U ,

4t2

t

where

2

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