ñòð. 39 |

1

Ga = t?xa ? (8)

xa u?u .

2?

For operator (8), the function u is changed as follows:

t(?a )2

xa ?a

u > u = u exp ? ? (9)

,

2? 4?

Symmetry of equations with convection terms 173

Thus, the operators Qa and Ga are essentially di?erent representations of the Galilei

operator.

Let us now investigate the symmetry of systems including equation (2) and addi-

tional conditions for the potentials. Note that in [3], the authors ?nd a nontrivial

symmetry of the nonlinear Fokker–Planck equation by imposing the additional condi-

tions for coe?cient functions.

Let the additional conditions for the potentials Vk be the Euler equations. In other

words, consider the following system:

?u ?u

? ? u = Vk ,

?t ?xk

(10)

?Vk ?Vk

? ?1 Vl = 0, k = 1, n.

?t ?xl

Symmetry of the nonlinear system (10) essentially depends on the value of the

parameter ?1 . There are two di?erent cases.

The ?rst case. ?1 = 1. In this case, system (10) in the class of operators (3) is

invariant under the Lie algebra with the basis operators

Jab = xa ?xb ? xb ?xa + Va ?Vb ? Vb ?Va ,

P 0 = ?t , Pa = ?xa ,

Ga = t?xa ? ?Va , D = 2t?t + xk ?xk ? Vk ?Vk , (11)

A = t2 ?t + txk ?xk ? (xk + tVk )?Vk , Z1 = u?u , Z2 = ?u .

The Galilei operator Ga generates the following ?nite transformations:

t > t = t,

xb > xb = xb + t?b ?ab ,

(12)

Vb > V b = Vb ? ?b ?ab ,

u > u = u,

where we mean summation from 1 to n over the repeated index b.

Conclusion 1. Thus, the scalar function u, unlike the heat equation, is not changed

under the Galilei transformations.

The second case. ?1 = 1. In this case, the invariance algebra of system (10) is

essentially more restricted and does not include the Galilei operator and the projective

one. In other words, for ?1 = 1 in the class of operators (3), system (10) is invariant

under the Lie algebra with basis elements P0 , Pa , Jab , D, Z1 , Z2 of the form (11).

The ?rst case is essentially more interesting and important that the second one.

Therefore, in what follows, we consider system (10) in the case where ?1 = 1.

Consider now system (10), where the Euler equations have the right-hand sides of

?u

the form F (u) ?xk , i.e., the following nonlinear system:

?u ?u

? ? u = Vk ,

?t ?xk

(13)

?Vk ?Vk ?u

? Vl = F (u) , k = 1, n,

?t ?xl ?xk

where F (u) is a smooth function of u. Let us carry out symmetry classi?cation of

system (13), i.e., determine all classes of functions F (u), which admit a nontrivial

symmetry of system (13). We consider the following six cases:

174 W.I. Fushchych, Z.I. Symenoh

Case 1. F (u) is an arbitrary smooth function. System (13) is invariant under the

Galilei algebra

(14)

AG(1, n) = P0 , Pa , Jab , Ga ,

where the basis operators have the form (11).

Case 2. F = C exp(?u) (? and C are arbitrary constants, ? = 0, C = 0). In this

case, the symmetry of system (13) is more extended and includes algebra (14) and

the dilation operator

2

D(1) = 2t?t + xk ?xk ? Vk ?Vk ? ?u .

?

Case 3. F = Cu? (? and C are arbitrary constants, ? = 0, ? = 1, C = 0). In

this case, system (13) is invariant under the extended Galilei algebra (14) with the

dilation operator

2

D(2) = 2t?t + xk ?xk ? Vk ?Vk ? u?u .

?+1

C

Case 4. F = (C is an arbitrary constant, C = 0). The maximal invariance

u

algebra is

P0 , Pa , Jab , Ga , Z1 ,

where Z1 = u?u .

Case 5. F = C (C is an arbitrary constant, C = 0). The maximal invariance

algebra is

P0 , Pa , Jab , Ga , D(2) , Z2 ,

where Z2 = ?u . In this case, the dilation operator D(2) has the form

D(2) = 2t?t + xk ?xk ? Vk ?Vk ? 2u?u .

Case 6. F = 0. In this case, system (13) admits the widest invariance algebra,

namely,

P0 , Pa , Jab , Ga , D, A, Z1 , Z2 ,

where the dilation operator D and the projective operator A have the form (11).

Conclusion 2. It is important that system (13) is invariant under the Galilei transfor-

mations for an arbitrary smooth function F (u). It should be stressed once more that,

unlike the standard heat equation, the function u is not changed under the Galilei

transformations.

Consider other examples of systems of the equation of convection di?usion and

additional conditions for the potentials Vk .

Let the functions Vk satisfy the heat equation, i.e., we investigate the following

system:

?u ?u

? ? u = Vk ,

?t ?xk (15)

?Vk

? ?1 Vk = 0, k = 1, n,

?t

where ?1 = 0 is an arbitrary real parameter.

Symmetry of equations with convection terms 175

Theorem 2. System (14) in the class of operators (3) is invariant under the Lie

algebra with the basis operators

P0 , Pa , Jab , D, Z1 , Z2

of the form (11).

The case where the functions Vk satisfy the Laplace equation is more important:

?u ?u

? ? u = Vk ,

?t ?xk (16)

Vk = 0, k = 1, n.

Theorem 3. System of equations (16) in the class of operators (3) is invariant under

the in?nite-dimensional Lie algebra with the basis operators

QA , Qkl , Qa , Z1 , Z2

of the form (5).

Note that the symmetry of system (16) is the same as the symmetry of equa-

tion (2). In other words, the conditions Vk = 0 do not contract the symmetry of the

equation of convection di?usion.

2 The Schr?dinger equation with convection term

o

Consider the Schr?dinger equation with convection term

o

?? ??

(17)

i + ??? = Vk ,

?t ?xk

where ? = ?(t, x) and Vk = Vk (t, x) (k = 1, n) are complex functions. For exten-

sion of symmetry, we regard the functions Vk as dependent variables. Note that the

requirement that the functions Vk are complex is essential for the symmetry of (17).

Let us investigate the symmetry of (17) in the class of ?rst-order di?erential

operators

X = ? µ ?xµ + ??? + ? ? ??? + ?k ?Vk + ??k ?Vk , (18)

?

where ? µ , ?, ? ? , ?k , ??k are functions of t, x, ?, ? ? , V , V ? .

Theorem 4. Equation (17) is invariant under the in?nite-dimensional Lie algebra

with the in?nitesimal operators

QA = 2A?t + Axr ?xr ? iAxr (?Vr ? ?Vr? ) ? A(Vr ?Vr + Vr? ?Vr? ),

? ? ?

Qkl = Bkl (xl ?xk ? xk ?xl + Vl ?Vk ? Vk ?Vl + Vl? ?Vk ? Vk ?Vl? ) ?

?

?

?

? iBkl (xl ?V ? xk ?V ? xl ?V ? + xk ?V ? ), (19)

k l k l

?

Qa = U a ?xa ? iU a (?Va ? ?Va ),

?

Z1 = ??? , Z2 = ? ? ??? , Z3 = ?? , Z4 = ??? ,

where A, B kl (k < l, k, l = 1, n) , U a (a = 1, n) are arbitrary smooth functions of t,

B kl = ?B lk , we mean summation over the index r and no summation over indices

a, k, and l.

176 W.I. Fushchych, Z.I. Symenoh

This theorem is proved by the standard Lie algorithm in the class of operators (18).

Note that algebra (19) includes as a particular case the Galilei operator of the

form:

Ga = t?xa ? i?Va + i?Va . (20)

?

This operator generates the following ?nite transformations:

xb > xb = xb + ?b t?ab ,

t > t = t,

? > ? = ?, ? ? > ? ? = ? ? ,

Vb > Vb = Vb ? i?b ?ab , Vb? > Vb? = Vb? + i?b ?ab ,

where ?b is an arbitrary real parameter and we mean summation from 1 to n over the

repeated index b. Note that the wave function ? is not changed for these transforma-

tions. Operator (20) is essentially di?erent from the standard Galilei operator

i

xa (??? ? ? ? ??? ). (21)

Ga = t?xa +

2?

of the free Schr?dinger equation (Vk = 0). Note that we cannot derive operator (21)

o

from algebra (19). Thus, we have two essentially di?erent representations of the Galilei

operator: (20) for the Schr?dinger equation with convection term and (21) for the free

o

Schr?dinger equation.

o

Remark 2. If we assume that the functions Vk are real in equation (17) and study

symmetry in the class of operators

X = ? µ ?xµ + ??? + ? ? ??? + ?a ?Va , (22)

where the unknown functions ? µ , ?, ? ? , ?a depend on t, x, ?, ? ? , V , then the maxi-

mal invariance algebra of equation (17) is su?ciently restricted. Namely, in the class

of operators (22), equation (17) is invariant under the Lie algebra with the basis

operators

P0 , Pa , Jab = xa ?xb ? xb ?xa + Va ?Vb ? Vb ?Va ,

D = 2t?t + xr ?xr ? Vr ?Vr , Z1 = ??? , Z2 = ? ? ??? , Z3 = ?? , Z4 = ??? .

Thus, in the case of real functions Vk , equation (17) is not invariant under the Galilei

transformations.

Consider now the system of equation (17) with the additional condition for the

potentials Vk , namely, the complex Euler equations:

?? ??

i + ??? = Vk ,

?t ?xk

(23)

?Vk ?Vk ??

? Vl

i = F (|?|) .

?t ?xl ?xk

Here, ? and Vk are complex dependent variables of t and x, F is a smooth function

of |?|. The coe?cients of the second equation of (23) provide the broad symmetry of

this system.

Symmetry of equations with convection terms 177

Let us investigate symmetry classi?cation of system (23). Consider the following

?ve cases.

Case 1. F is an arbitrary smooth function. The maximal invariance algebra is

P0 , Pa , Jab , Ga , where

Jab = xa ?xb ? xb ?xa + Va ?Vb ? Vb ?Va + Va ?Vb? ? Vb? ?Va ,

? ?

Ga = t?xa ? i?Va + i?Va .

?

Case 2. F = C|?|k (C is an arbitrary complex constant, C = 0, k is an arbitrary

real number, k = 0 and k = ?1). The maximal invariance algebra is P0 , Pa , Jab , Ga ,

D(1) , where

2

D(1) = 2t?t + xr ?xr ? Vr ?Vr ? Vr? ?Vr? ? (??? + ? ? ??? ).

1+k

C

Case 3. F = (C is an arbitrary complex constant, C = 0). The maximal

|?|

invariance algebra is P0 , Pa , Jab , Ga , Z = Z1 + Z2 , where

Z = ??? + ? ? ??? , Z2 = ? ? ??? .

Z1 = ??? ,

Case 4. F = C = 0 (C is an arbitrary complex constant). The maximal invariance

algebra is P0 , Pa , Jab , Ga , D(1) , Z3 , Z4 , where

Z3 = ?? , Z4 = ??? .

Case 5. F = 0. The maximal invariance algebra is P0 , Pa , Jab , Ga , D, A, Z1 , Z2 , Z3 ,

Z4 , where

D = 2t?t + xr ?xr ? Vr ?Vr ? Vr? ?Vr? ,

A = t2 ?t + txr ?xr ? (ixr + tVr )?Vr + (ixr ? tVr? )?Vr? .

ñòð. 39 |