ñòð. 93 |

Im (?U/U ? Ua Ua /U 2 ) = (? ln U ? ? ln U )/2i

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

it is easily to show that the class of the Galilei-invariant equations (14) contains the

equation

id

U ?|U |2 /|U |2 + U [d1 (Re (?U/U ) + |?U |2 /|U |2 ) +

iUt + ?U =

2

+ d2 Im (?U/U ? (?U/U )2 ) + d3 (Re (?U/U )2 ) + |?U |2 /|U |2 )],

where ?U = (?U/?x1 , . . . , ?U/?xn ), d1 , d2 , d3 ? R, proposed in [9] from certain

physical considerations. By the way, a nonlinear generalization of the Schr?dinger

o

equation [8]

iUt = (id1 ? h)?U + id1 U |?U |2 /|U |2 + U B(|U |),

does not preserve Galilean symmetry of the linear Schr?dinger equation. Instead it

o

would be appropriate to propose Galilei-invariant nonlinear equations of the class (14)

?

iUt = c?U + (h ? c) U (?U )2 /|U |2 + U B(|U |),

and [13]

iUt = ?h?U + cU ?|U |2 /|U |2 + U B(|U |),

where c is arbitrary complex constant and B is an arbitrary complex function.

Corollary 2. In the class of nonlinear equations of the form (4) algebra AG1 (1, n)

(2a), (2b), (2c), (6) is admitted only for equations given by

(i) In the case ? = 0

?

iUt + h?U = U [D1 ? ln U + D2 ? ln U +|U |?2/? B] +

(16)

?

+ U |U | [D3 |U |a |U |b (ln U )ab + D4 |U |a |U |b (ln U )ab ],

2/??2

where Dj , j = 1, 2, 3, 4 and B are arbitrary complex functions of the argument

|U |2/??2 |U |a |U |a ;

(ii) In the case ? = 0

?

iUt + h?U = U [D1 ? ln U + D2 ? ln U +|U |a |U |a B] +

(17)

?

?1

+ U (|U |a1 |U |a1 ) [D3 |U |a |U |b (ln U )ab + D4 |U |a |U |b (ln U )ab ],

where Dj = Dj (|U |), j = 1, 2, 3, 4 and B = B(|U |) are arbitrary complex functions.

It is easily seen that the class of the AG1 (1, n)-invariant equations (14) contains

the well-known nonlinear Schr?dinger equation

o

iUt + h?U + cU |U |2 = 0 (18)

which in the case n = 1 is integrated by inverse scattering method [14]. Note that in

the case n = 2 equation (17) is invariant under the AG2 (1, 2) algebra [12, 15].

Corollary 3. Within the class of nonlinear equations of the form (4) algebra

AG2 (1, n) (2), (6) for ? = ?n/2 of the linear Schr?dinger equation (5) is conserved

o

only for equations given by

iUt + h?U = U E1 ? ln |U | + U |U |4/n B + U |U |?4/n?2 E2 |U |a |U |b (ln |U |)ab . (19)

Galilei-invariant nonlinear systems of evolution equations 387

In equation (19) E1 , E2 and B are arbitrary complex functions of the argument

|U |?4/n?2 |U |a |U |a , which is an absolute invariant of the generalized Galilei algebra

AG2 (1, n).

If we consider a representation of AG2 (1, n) algebra with basic operators (2), (6)

for ? = 0, a principally different class of quasilinear second-order equations, invariant

with respect to this algebra, namely

iUt + hUa Ua /U = U E1 (|U |)? ln |U | + U |U |a |U |a B(|U |) +

(20)

+ U E2 (|U |)(|U |a1 |U |a1 )?1 |U |a |U |b (ln |U |)ab .

is obtained.

It is easily seen that within the class of equations (20) there is not a single linear

equation, the simplest one among them being Hamilton–Jacobi equation for a complex

function

iUt + hUa Ua /U = 0

which is reduced to a standard form

?W ?W

iWt + hWa Wa = 0, Wa = , Wt =

?xa ?t

by a local substitution U = exp W , W = W (t, x1 , . . . , xn ).

In case E1 = E2 = 0 equation

iUt + h?U = U |U |4/n B (21)

is obtained from the class of equations (19) which had been obtained in [1, 12]. Note

that at B = c = const equation (21) is transformed into an equation with fixed power

nonlinearity, studied in a series of papers (for n = 1 [16, 17], n = 2 [18] and n = 3 [1,

2, 12, 19]). In [1, 12] multiparametric families of invariant solutions of equation (21)

of the form

|U |a |U |a

iUt + h?U = cU

|U |2

are also constructed and systematized.

Being written in the case of one spatial variable (n = 1), after simple transformati-

ons the class of equations (19) is given by

iUt + hUxx = U E1 (ln |U |)xx + U |U |4 B, (22)

U = U (t, x), x = x1 ,

E1 and B being arbitrary complex functions of the argument |U |?3 |U |x .

Obviously, a specific case of equation (22) is given by

iUt + hUxx + c1 U |U |4 + c2 U |U ||U |x = 0 (23)

which at h = 1, c1 = 1, c2 = 4 is known as Eckhaus equation [20, 21]. Equation

(23) has been studied in detail for arbitrary constant values of c1 and c2 in [22]. A

multidimensional generalization of equation (23), posessing AG2 (1, n) symmetry, can

be proposed

iUt + h?U + c1 U |U |4/n + c2 U |U |?1+2/n (|U |a |U |a )1/2 = 0. (24)

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

4. Galilei-invariant systems of Hamilton–Jacobi-type

? ?

It should be noted that the local substitution U = M (U ), V = N (V ), where

M , N are arbitrary differentiable functions, reduces any equation system with the

symmetry AG(1, n), AG1 (1, n) or AG2 (1, n) to a locally equivalent system with the

same symmetry, but with different representation of operators Q? and I? , namely

?1 ?1

dM dN

?

Q? = ?1 M ?U + ? 2 N ?V ,

? ?

? ?

dU dV

?1 ?1

dM dN

I? = ?1 M ?U + ?2 N ?V .

? ?

? ?

dU dV

? ?

In the particular case where M = exp(U ), N = exp(V ), we obtain

? (25)

Q? = ?1 ?U + ?2 ?V , I? = ?1 ?U + ?2 ?V .

? ? ? ?

In this case the class of equation systems, invariant with respect to AG2 (1, n) algebra

in the representation (2), (25), at ? = 0 is given by

? ?? ? ?

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

? ?? ?

+ Ua Ua + C(? )(? a ?a )?1 ?a ?b [?2 Uab ? ?1 Vab ],

?? ? ?

??? ??

1 1

(26)

? ?? ? ?

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

? ?? ?

+ Va Va + E(? )(? a ?a )?1 ?a ?b [?2 Uab ? ?1 Vab ],

?? ? ?

??? ??

1 1

? ?? ? ?

where ? = ?2 U ? ?1 V , ?a = ?2 Ua ? ?1 Va and A, B1 , B2 , C, D, E are arbitrary

?

differentiable functions.

In case where ?1 = ?2 = 0, A = C = D = E = 0 the system of equations (26) is

? ?

·

reduced to the systems of the form (the symbols ? being omitted below)

?1 Ut = Ua Ua + ?a ?a B1 (?),

(27)

?1 Vt = Va Va + ?a ?a B2 (?), ?1 ?2 = 0

It is natural to call system (27) a generalization of the noncoupled system of the

Hamilton–Jacobi (HJ) equations

(28)

?1 Ut = Ua Ua , ?1 V t = V a V a .

In contrast to the symmetry of a single HJ equation [2, 23], the local symmetry of

the system (28) is exhausted by AG2 (1, n) algebra (2), (25) at ?1 = ?2 = 0 with

additional operators

D1 = ?t?t + U ?U + V ?V . (29)

P V = ?V ,

Thus, all the nonlinear generalizations of the form

?1 Ut = Ua Ua + B1 (U, V, U1 , . . . , Un , V1 , . . . , Vn ),

(30)

?1 Vt = Va Va + B2 (U, V, U1 , . . . , Un , V1 , . . . , Vn )

of HJ system, preserving its symmetry AG2 (1, n), are exhaused by system (27).

Among the non-linear generalizations of HJ system (27), a system of equations

with unique symmetry properties exists, namely for B1 = 0, B2 = ?1/(?2 )2 (in the

following ?1 = 1, ?2 = ?).

Galilei-invariant nonlinear systems of evolution equations 389

Theorem 4. The maximal (in the sense of Lie) algebra of the invariance for the

system of equations

Ut = Ua Ua ,

(31)

Vt = ??Ua Ua + 2Ua Va

is generated by the basic operators

xa

Q? = ??U ? ?V , Ga = tPa ?

Pt , Pa , Jab , X = W ?V , Q? ,

2

1

D = 2tPt + xa Pa , ? = t2 Pt + txa Pa ? |x|2 Q? ,

4

xa

G1 = U Pa ? Pt , D1 = 2U PU + xa Pa , (32)

a

2

1

?1 = U 2 PU + U xa Pa ? |x|2 Pt ,

4

1

Ka = xa tPt ? 2tU + |x|2 Pa + xa xb Pb + xa U Q? ,

2

where W are an arbitrary differentiable function of ?U ? V .

Note that the presence of the operatorX including an arbitrary function W in the

invariance algebra for the system (31) is natural, since the second equation of the

system is linear with respect to the function V . Much more interesting is the fact

the system (31) can be considered as a generalization of classical HJ equation to the

case of two unknown functions, since for W = 1 the operators (32) generate the

same algebra as the HJ equation. We consider this fact to be very important, since a

trivial generalization of the above-mentioned equation to the system of (28) does not

preserve the symmetry of the HJ equation.

5. Galilei-invariant reaction-diffusion systems

Now consider a nonlinear system of evolution equations, given by

?1 Ut = ?U + f (U, V ),

(33)

?2 Vt = ?V + g(U, V ),

where f , g are arbitrary differentiable functions. The systems of reaction-diffusion

equations (33) has been studied intensively of late (see, e.g., [4, 6, 7]). As follows

from theorems 1, 2 and 3, the class of systems (33) contains systems with broad

symmetry. In particular, all the systems of equations of the form

? = U ?2 V ??1 ,

?1 Ut = ?U + U f (?),

(34)

?2 Vt = ?V + V g(?)

will be invariant under the Galilei algebra AG(1, n).

Note 1. In the case, where ?2 = ?1 = ?, f = d1 ((U + V )/V )d0 ? 1, g = d2 ((U +

V )/V )d0 ? d3 and d0 , d1 , d2 , d3 ? R the system (34) is the particular case of the

conservation equations for normal and mutant cells [7, 24].

In case where f = ?1 ? ?2/? , g = ?2 ? ?2/? , ? = 0 (? is defined in the introduction)

there will be invariance under the algebra AG1 (1, n). Finally, for ? = ?n(?2 ? ?1 )/2,

i.e. ?1 = ?2 = ?n/2, the system of equations

?1 Ut = ?U + ?1 U 1+?2 ? V ??1 ? ,

(35)

?2 Vt = ?V + ?2 V 1??1 ? U ?2 ?

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

is obtained (where ? = 4/(n(?2 ? ?1 )), ?2 = ?1 , ?k ? R), preserving the AG2 (1, n)-

symmetry of the linear system (1).

Note 2. For ?2 = ??1 = ? the diffusion system (33) is reduced by substitution

V = Y ? Z, (36)

U = Y + Z, Y = Y (t, x), Z = Z(t, x)

to the system of equations

??Yt = ?Z + f1 (Y, Z),

?Zt = ?Y + g1 (Y, Z),

whose invariance under the chain of algebras AG(1, n) ? AG1 (1, n) ? AG2 (1, n)

with the unit operator Q? = ?Y ?Z + ?Z?Y is described by the substitution (36)

being applied to the system of equations of the form (33) with the corresponding

symmetry.

It is interesting to consider system (33) in case where one of the equations

degenerates into an elliptical one. Without reducing generality we consider ?2 = 0,

?1 = 1. Then according to the theorem 1, all systems of the form (33) for ?2 = 0,

?1 = 1 and posessing AG(1, n) symmetry are given by

(37a)

Ut = ?U + U f (V ),

(37b)

0 = ?V + g(V ),

where f and g being arbitrary functions.

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