ñòð. 92 |

AG(1, n), and its extension by using the operator (2c) will be refereed to as AG1 (1, n)

[1, 2].

Clearly, the unit operators I? and Q? are linearly dependent only in the case when

the determinant

?1 ?2

?= = 0.

?1 ?2

As a result we obtain two essentially different representations of algebras AG1 (1, n)

and AG2 (1, n) for ? = 0 and ? = 0, in contrast to the case of a single diffusion

equation (the nonlinear diffusion equation invariant with respect of a set of AG2 (1, n)

subalgebras was studied in [2, 3]).

J. Phys. A: Math. Gen., 1995, 28, P. 5569–5579; Preprint LiTH-MAT-R-95-18, Department of Mathe-

matics, Link?ping University, Sweden, 12 p.

o

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

Note that in the case when the system (1) is a pair of complex conjugate Schr?-

o

?

?

dinger equations, i.e. U = V , ?1 = ?1 = i, the operators I? and Q? are linearly

independent. This results in the fact that nonlinear generalizations of Schr?dinger

o

equations, preserving its symmetry [1], differ essentiallyfrom nonlinear generaliza-

tions of the diffusion system (1) at ? = 0.

Now consider a system of quasilinear generalizations of diffusion equations (1) of

the form

?1 Ut = Aab Uab + Cab Vab + B1 ,

(3)

?1 Vt = Dab Uab + Eab Vab + B2 ,

Aab , Cab , Dab , Eab , B1 , B2 being arbitrary real or complex differentiable functions of

2n + 2 variables U, V, U1 , . . . , Un , V1 , . . . , Vn . The indices a = 1, . . . , n and b = 1, . . . , n

of functions U and V denote differentiating with respect to xa and xb .

The system (3) generalizes practically all the known nonlinear systems of first-

and second-order evolution equations, describing various processes in physics, chemi-

stry and biology (heat and mass transfer, filtration of two-phase liquid, diffusion in

chemical reactions etc.) [4–7].

?

? ? ? ?

In the case of complex U = V , Aab = E ab , Cab = Dab , B1 = B 2 = B, ?1 = ?2 = i

the system (3) is transformed into a pair of complex conjugate equations. We treat

them as a class of nonlinear generalizations of Schr?dinger equations, namely:

o

? ?

(4a)

iUt = Aab Uab + Dab U ab + B,

? ? ? ?

?i U t = Aab U ab + Dab Uab + B (4b)

(hereinafter complex conjugate equations (4b) are omitted).

For Aab = Dab = Daa = 0, a = b, Aaa = ?h equation (4a) is obviously

transformed into a Schr?dinger equation with nonlinear potential B:

o

(4 )

iUt + h?U = B.

? ? ?

By choice of the corresponding potential B = B(U, U , U1 , . . . , Un , U 1 , . . . , U n ) a great

variety of Schr?dinger equation generalizations, known from the literature (see e.g. [1,

o

2, 8, 9, 10]) can be obtained.

In case of zero potential B a classical Schr?dinger equation is obtained

o

(5)

iUt + h?U = 0

invariant under AG2 (1, n) algebra with the basic operators (2) [11], where

? ?

i

Q? = ? (U ?U ? U ? ? ), (6)

I? = ?(U ?U + U ? ? ).

h U U

Note that the algebra AG2 (1, n) in the case of the Schr?dinger equations is called

o

the Schr?dinger algebra [11].

o

In the present paper all the systems of evolution equations of the form (3), invari-

ant under the chain of algebras AG(1, n) ? AG1 (1, n) ? AG2 (1, n), are described.

The results obtained are illustrated by the examples of the nonlinear Schr?dinger

o

equations, reaction-diffusion equations and Hamilton–Jacobi type systems.

Galilei-invariant nonlinear systems of evolution equations 383

2. Description of systems (3) with Galilean symmetry

The algebra of symmetries for the system of equations (1) contains the Galilei

operators Ga , a = 1, . . . , n, being a mathematical expression of the Galilei relativistic

principle for equations (1). The Galilei operators are also known [3] to be closely

related with the fundamental solution of the diffusion equation. We recall that if some

system of PDEs is invariant with respect to the Galilei algebra or its extention, then it

gives a wide range of possibilities for the construction of multiparametric families of

exact solutions [1, 12, 22]. Moveover the Galilei operators and the projective operator

(2d) generate non-trivial formulae of multiplication of solutions. These formulae can

be used to convert stationary (time-independent) into non-stationary ones with a

different structure.

In view of this it seems reasonable to search for Galilei-invariant nonlinear

generalizations of system (1) in the class of system (3).

Theorem 1. The system of nonlinear equations (3) is invariant under the Galilei

algebra in the represention (2a), (2b) if and only if it has the form:

?1 Ut = ?U + U [A1 ? ln U + C1 ? ln V + B1 ] +

+ U [A2 ?a ?b (ln U )ab + C2 ?a ?b (ln V )ab ],

(7)

?2 Vt = ?V + V [D1 ? ln U + E1 ? ln V + B2 ] +

+ V [D2 ?a ?b (ln U )ab + E2 ?a ?b (ln V )ab ],

where (ln U )ab ? ? 2 ln U/?xa ?xb , (ln V )ab ? ? 2 ln V /?xa ?xb , ? ln U ? (ln U )11 +

· · · + (ln U )nn , ? ln V ? (ln V )11 + · · · + (ln V )nn , ? = U ?2 V ??1 , ?a = ??/?xa ?

(?2 Ua /U ? ?1 Va /V )? and Ak , Bk , Ck , Dk , Ek , k = 1, 2 are arbitrary functions of

absolute invariants of the AG(1, n) algebra ? and ? = ?a ?a .

The proof of this and the following theorems is based on the classical Lie scheme,

which is realized in [3, 12] for obtaining the Galilei invariant equations. The detailed

cumbersome calculations are omitted.

Note that in case where ?1 = 0, i.e. the first equation of system (3) being elliptical,

the absolute invariants of the Galilei algebra simplify considerebly: ? = U , ? = Ua Ua .

In case of systems of the form (3) being AG1 (1, n)- and AG2 (1, n)-invariant the

structure of such systems essentially depends on the determinant ?.

Theorem 2. The nonlinear system (3) is invariant with respect to algebra AG1 (1, n)

with basic operators (2a)–(2c) if and only if it has the form:

(i) In case when ? = 0

?1 Ut = ?U + U [A1 (?)? ln U + A2 (?)? ln V + ? ?2/? B1 (?)] +

? ? ?

? ?

+ U ? 2/??2 [C1 (?)?a ?b (ln U )ab + C2 (?)?a ?b (ln V )ab ],

(8)

?2 Vt = ?V + V [D1 (?)? ln U + D2 (?)? ln V + ? ?2/? B2 (?)] +

? ? ?

? ?

+ V ? 2/??2 [E1 (?)?a ?b (ln U )ab + E2 (?)?a ?b (ln V )ab ].

(ii) In case when ? = 0

?1 Ut = ?U + U [A1 (?)? ln V + A2 (?)? ln V + ?a ?a B1 (?)] +

+ U (?a1 ?a1 )?1 ?a ?b [C1 (?)(ln U )ab + C2 (?)(ln V )ab ],

(9)

?2 Vt = ?V + V [D1 (?)? ln U + D2 (?)? ln V + ?a ?a B2 (?)] +

+ V (?a1 ?a1 )?1 ?? ?b [E1 (?)(ln U )ab + E2 (?)(ln U )ab ],

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

?

where Ak , Bk , Ck , Dk , Ek , k = 1, 2 being arbitrary functions, ? = ?a ?a ? 2/??2 and

? are the absolute first-order invariants of the algebra AG1 (1, n), a1 = 1, . . . , n (?a ,

? see theorem 1).

In the case when the first equation of system (3) degenerates into an elliptical

?

(?1 = 0) equation, the absolute invariants in systems (8) and (9) simplify and ? =

Ua Ua U 2/?1 ?2 for ? = 0, ? = U for ? = 0.

Theorem 3. The nonlinear system of equations (3) is invariant with respect to

algebra AG2 (1, n) with basic operators (2) (?1 , ?2 are arbitrary constants) iff it has

the form:

(i) In case when ? = 0

?1 Ut = ?1 ?U + U A(?)(?2 ? ln U ? ?1 ? ln V ) + U ? ?2/? B1 (?) +

? ?

?

?

+ (1 ? ?1 )Ua Ua /U + U ? 2/??2 ?a ?b [?2 (ln U )ab ? ?1 (ln V )ab ]C(?),

?

(10)

?2 Vt = ?2 ?V + V D(?)(?2 ? ln U ? ?1 ? ln V ) + V ? ?2/? B2 (?) +

? ?

?

?

+ (1 ? ?2 )Va Va /V + V ? 2/??2 ?a ?b [?2 (ln U )ab ? ?1 (ln V )ab ]E(?).

?

(ii) In case when ? = 0

?1 Ut = ?1 ?U + U A(?)(?2 ? ln U ? ?1 ? ln V ) + U ?a ?a B1 (?) +

?

+ (1 ? ?1 )Ua Ua /U + U (?a1 ?a1 )?1 ?a ?b [?2 (ln U )ab ? ?1 (ln V )ab ]C(?),

?

(11)

?2 Vt = ?2 ?V + V D(?)(?2 ? ln U ? ?1 ? ln V ) + V ?a ?a B2 (?) +

?

+ (1 ? ?2 )Va Va /V + V (?a1 ?a1 )?1 ?a ?b [?2 (ln U )ab ? ?1 (ln V )ab ]E(?),

?

where A, B1 , B2 , C, D, E being arbitrary functions, ?k = ?2?k /n, k = 1, 2 (?k see

?

operator I? ).

It can be noticed that in case where ?1 ?2 = 0 systems (10) and (11) can be reduced

by the local substitution U > U ?1 , V > V ?2 to the systems of the same structure,

? ?

but with ?k = 1, i.e. ?k = ?n/2. The specific case of ?1 = ?2 = 0 will be considered

?

in what following.

The classes of AG2 (1, n)-invariant systems (10) and (11) thus obtained contain, in

particular, such genaralizations of equations (1) as (? = 0)

?1 Ut = ?U + e1 U (?2 ? ln U ? ?1 ? ln V ),

?2 Ut = ?V + e2 V (?2 ? ln U ? ?1 ? ln V )

and (? = 0)

?(U V ?1 ) ?(U V ?1 )

Ut = ?U + e1 U ,

?xa ?xa

?(U V ?1 ) ?(U V ?1 )

Vt = ?V + e2 V ,

?xa ?xa

where e1 , e2 ? R.

Galilei-invariant nonlinear systems of evolution equations 385

In the case where the first of equations (3) degenerates into an elliptical one

(?1 = 0), the AG2 (1, n)-invariant systems of equations are simply

0 = A1 (?)?U + A2 (?)(Ua1 Ua1 )?1 Ua Ub Uab + U 1?2/?1 B1 (?) +

? ? ?

+ U C(?)[? ln V ? (Ua Ua )?1 Ua Ub (ln V )ab ],

?

1 1

V V

D1 (?)?U + D2 (?)(Ua1 Ua1 )?1 Ua Ub Uab +

? ? (12)

?2 Vt = ?2 ?V +

?

U U

+ (1 ? ?2 )Va Va /V + V U ?2/?1 B2 (?) +

?

?

+ V E(?)[? ln V ? (Ua Ua )?1 Ua Ub (ln V )ab ]

?

1 1

if ? = 0, and

0 = A1 (U )?U + A2 (U )(Ua1 Ua1 )?1 Ua Ub Uab + Ua Ua B1 (U ) +

+ C(U )[? ln V ? (Ua1 Ua1 )?1 Ua Ub (ln V )ab ],

(13)

?2 Vt = ?2 ?V + V D1 (U )?U + V D2 (U )(Ua1 Ua1 )?1 Ua Ub Uab + V Ua Ua B2 (U ) +

?

+ (1 ? ?2 )Va Va /V + V E(U )[? ln V ? (Ua1 Ua1 )?1 Ua Ub (ln V )ab ],

?

?

if ? = 0. In equations (12), (13) Ak , Bk , Dk , E, C are arbitrary functions, ? =

Ua Ua U 2/?1 ?2 , ?2 = ?2?2 /n. In [13] integration of two-dimensional systems of equati-

?

ons (12), (13) form was reduced to the integration of linear heat equation with a

source.

3. Galilei-invariant nonlinear generalizations of the Schr?dinger equation

o

As noted above, a class of nonlinear generalization of Schr?dinger equation (4)

o

is a specific case of evolution equations (3). On the basis of theorems 1, 2 and 3

this enables one to describe all quasilinear generalizations of Schr?dinger equation

o

(5), which are invariant with respect to a chain of algebras AG(1, n) ? AG1 (1, n) ?

AG2 (1, n).

Corollary 1. In the class of nonlinear equations of the form (4) algebra AG(1,n)

?

(2a),(2b) with Q? = ? h (U ?U ? U ? ? ) is admitted only for equations given by

i

U

?

iUt + h?U = U [A1 ? ln U + A2 ? ln U +B] +

(14)

?

+ U [A3 |U |a |U |b (ln U )ab + A4 |U |a |U |b (ln U )ab ],

where Aj = 0, j = 1, 2, 3, 4 and B are arbitrary complex functions of two arguments

?

|U | and |U |a |U |a ; |U |2 = U U , |U |a = ?|U |/?xa .

In case Aj = 0 the class of equations (14) is reduced to an equation

iUt + h?U = U B(|U |, |U |a |U |a ) (15)

obtained in [1, 12], whose specific case is a Schr?dinger equation with power nonli-

o

nearity U |U | , ? = const.

?

By using the identities

? ln |U |2 = (?|U |2 ? 4|U |a |U |a )/|U |2 ,

Re (?U/U ) + |?U |2 /|U |2 = ? ln |U | + |U |a |U |a /|U |2 ,

ñòð. 92 |