ñòð. 38 |

dimensional space (t, x).

(iii) We describe nonlinear equations of type (4) invariant under the algebra

AG(1, 3), the extended Galilei algebra AG1 (1, 3) = AG(1, 3), D , and the full Galilei

algebra AG2 (1, 3) = AG1 (1, 3), A . D and A are the dilation and projective operators,

respectively.

(i) For solving the above problems we use the method described in [1, 2, 5, 6, 7].

Theorem 2. A: Among linear partial di?erential equations (PDE) of arbitrary even

order 2n

L? = 0,

2n

(5)

L = A + B µ ?µ + C µ? ?µ? + Dµ?? ?µ?? + · · · + E µ??...? ?µ??...? ,

2n

there exists the unique equation

(?1 S + ?2 S 2 + · · · + ?n S n )? = ?? (6)

invariant under the algebra AG(1, 3).

B: There are no linear PDE of arbitrary odd order 2n + 1

L? = 0,

L = A + B µ ?µ + C µ? ?µ? + Dµ?? ?µ?? + · · ·

(7)

2n+1

2n

· · · + E µ??...? ?µ??...? + Gµ??...?? ?µ??...?? ,

2n 2n+1

with one non-zero coe?cient of the highest derivatives at least, invariant under

AG(1, 3).

2n+1

2n

Here, A, B µ , C µ? , Dµ?? , . . . , E µ??...? , Gµ??...?? are arbitrary functions of t and x;

?1 , ?2 , . . . , ?n , ? are arbitrary constants, ?n = 0; ?µ ? ?/?xµ , ?µ? ? ? 2 /?xµ ?x? , . . .

(µ, ?, . . . , ? = 0, 3).

168 W.I. Fushchych, Z.I. Symenoh

Proof. The scheme and idea of the proof of the theorem is very simple but the

concrete realization is not simple. We describe in more details the proof of part A.

Part B is proved in the same way as the ?rst part of the theorem.

According to the Lie method [1, 3, 4], we ?nd the 2nth prolongations of the

operators (2) and consider the system of determining equations

? X ? AG(1, 3). (8)

X L? = 0,

L?=0

(2n)

Writing equations (8) in the explicit form and equating coe?cients for equal deri-

vatives, we solve the system of partial di?erential equations to obtain functions A,

2n

B µ , C µ? , Dµ?? , . . ., E µ??...? .

Invariance of equation (5) under the operators P0 , Pa results in the fact that functi-

2n

ons A, B µ , C µ? , Dµ?? , . . ., E µ??...? do not depend on t and x, i.e. these coe?cients are

arbitrary constants. In other words, our PDE has the form L? ? Q(1) (p0 , pa )? = 0,

where Q(1) is a polynomial in (p0 , pa ) with constant coe?cients.

After taking into account the invariance under the operators Jab , we ?nd that the

equation has the form L? ? Q(2) (p0 , p2 )? = 0, where Q(2) is a polynomial in (p0 , p2 ).

a a

After considering the invariance under the Galilei operators Ga , we obtain that the

equation has the form L? ? Q(3) (p0 ? 2m p2 )? = 0, where Q(3) is a polynomial in

1

a

(p0 ? 2m pa ). In other words, the equation has the form (6). The theorem is proved.

12

Consequence. Among fourth-order linear PDE there exists the unique equation in-

variant under the algebra AG(1, 3) with basic operators (2). This equation has the

form

(?1 S + ?2 S 2 )? = ??,

where ?2 = 0.

(ii) Now, we consider equation (4) in two dimensions t, x and carry out symmetry

classi?cation of potentials V = V (x) of this equation, i.e., we ?nd all functions V =

V (x) admitting an extension of symmetry of (4). The following statement is true.

Theorem 3. Two-dimensional equation (4) with ?n = 0, n = 1 is invariant under

the following algebras:

(1) P0 , I , i? V (x) is an arbitrary di?erentiable function;

(2) AG(1, 1) = P0 , P1 , G, I , i? V = const;

?

(3) AG2 (1, 1) = P0 , P1 , G, D, A, I , i? V = V1 = const the following equalities

are true:

(n?k)/n

?k n V1

k = 1, . . . , n ? 1; (9)

= ,

?n k ?n

? n

(4) P0 , D, A, I , i? V = V1 + C/x2n , V1 , C are constants and (9) are true; k

are the binomial coe?cients.

The operators in Theorem 3 have the following representation:

?

G = tp1 ? mx, n

P 0 = p0 , P1 = p1 , P 0 = p0 = P 0 +

? V1 /?n ,

(10)

D = 2tp0 ? xp1 ? (i/2)(2n ? 3), A = t2 p0 ? tD ? (1/2)mx2 ,

? ?

I is the unit operator.

High-order equations of motion in quantum mechanics and Galilean relativity 169

Consequence. The 2nth-order PDE

(S n + V (x))? = 0

is invariant under the following algebras:

(1) P0 , I , i? V (x) is an arbitrary di?erentiable function;

(2) AG(1, 1) = P0 , P1 , G, I , i? V = const;

(3) AG2 (1, 1) = P0 , P1 , G, D, A, I , i? V = 0;

(4) P0 , D, A, I , i? V = C/x2n , where C is an arbitrary constant.

The above operators have representation (10) with V1 = 0.

Note that symmetry classi?cation of potentials for the fourth-order PDE of the

form

(?1 S + ?2 S 2 + V (x))? = 0

was carried out in [8]. In this case, symmetry operators have representation (10) with

?2

V1 = 4?12 and n = 2.

(iii) Now, let us consider nonlinear PDE of type (4) in (r + 1)-dimensional space:

S n ? + F (??? )? = 0, (11)

where ?? is complex conjugated function, n is an arbitrary integer power, F is an

arbitrary complex function of ??? .

We study symmetry classi?cation of (11), i.e. we ?nd all functions F (??? ) which

admit an extension of symmetry of equation (11).

Theorem 4. Equation (11) is invariant under the following algebras:

(1) P0 , Pa , Jab , Ga , Q1 , i? F is an arbitrary di?erentiable function;

(2) P0 , Pa , Jab , Ga , Q1 , Q2 , i? F = const = 0;

(3) P0 , Pa , Jab , Ga , Q1 , D , i? F = C(??? )k , k = 0;

?

(4) P0 , Pa , Jab , Ga , Q1 , D, A , i? F = C(??? )(2n)/(r+2?2n) ;

(5) P0 , Pa , Jab , Ga , Q1 , Q2 , D, A , i? F = 0.

Here, indices a, b are from 1 to r, a = b, k is an arbitrary number (k = 0), and

the above operators have the following representation:

P0 = p0 , Pa = pa , Jab = xa pb ? xb pa , Ga = t?xa + imxa Q1 ,

Q1 = ??? ? ?? ??? , Q2 = ??? + ?? ??? ,

r + 2 ? 2n

?

D = 2t?t + xc ?xc ? (n/k)Q2 , D = 2t?t + xc ?xc ? Q2 ,

2

r + 2 ? 2n

A = t2 ?t + txc ?xc + (i/2)mxc xc Q1 ? tQ2 ,

2

where summation from 1 to r over the repeated indices c is understood.

Thus, in the present paper, we have described the unique linear PDE of arbit-

rary even order which is invariant under the Galilei group. We have investigated the

exhaustive symmetry classi?cation of potentials V (x) of (4) and functions F (??? )

of the nonlinear equation (11), i.e. we have pointed out all functions admitting an

extension of the invariance algebra.

The authors would like to thank an anonymous reviewer for kindness and helpful

suggestions. The paper is partly supported by INTAS, Royal Society, and the Inter-

national Soros Science Education Program (grant No PSU061097).

170 W.I. Fushchych, Z.I. Symenoh

1. Fushchych W., Shtelen W., Serov N., Symmetry analysis and exact solutions of equations of

nonlinear mathematical physics, Dordrecht, Kluwer Academic Publishers, 1993, 436 p.

2. Fushchych W.I., Nikitin A.G., Symmetry of equations of quantum mechanics, New York,

Allerton Press, Inc., 1994, 465 p.

3. Ovsyannikov L.V., Group analysis of di?erential equations, New York, Academic Press, 1982,

400 p.

4. Olver P., Application of Lie qroups to di?erential equations, New York, Springer, 1986, 497 p.

5. Fushchych W.I., Cherniha R.M., The Galilean relativistic principle and nonlinear partial diffe-

rential equations, J. Phys. A: Math. Gen., 1985, 18, 3491–3503.

6. Fushchych W.I., Symmetry in problems of mathematical physics, in Algebra-Theoretic Analysis

in Mathematical Physics, Kiev, Institute of Mathematics, 1981, 6–28 (in Russian).

7. Fushchych W., Ansatz’95, J. Nonlinear Math. Phys., 1995, 2, ¹ 3–4, 216–235.

8. Symenoh Z.I., Symmetry classi?cation of potentials for the generalized Schr?dinger equation,

o

Pros. of NAS of Ukraine, 1995, ¹ 3, 20–22.

W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 171–180.

Symmetry of equations

with convection terms

W.I. FUSHCHYCH, Z.I. SYMENOH

We study symmetry properties of the heat equation with convection term (the equa-

tion of convection di?usion) and the Schr?dinger equation with convection term. We

o

also investigate the symmetry of systems of these equations with additional conditions

for potentials. The obtained results are applied to construction of exact solutions of

the system of the Schr?dinger equation with convection term and the Euler equations

o

for potentials.

Study of symmetry properties of evolution equations is an important problem in

mathematical physics. These equations are thoroughly investigated by a number of

authors (see, e.g., [1, 2, 3]). The fundamental property of these equations is the fact

that they are invariant under the Galilei transformations.

It is known [4] that the nonlinear heat equation

?u

? ? u = F (u) (1)

?t

is not invariant under the Galilei transformations if F (u) = 0. It is Galilei–invariant

only in the case of linear equation, i.e., in the case where F (u) = 0 (up to equivalence

transformations). Therefore, it is important to consider nonlinear evolution equations

which admit the Galilei operator.

In the present paper, we study symmetry properties of equations with convection

terms, namely, the heat equation with convection term (the equation of convection

di?usion) and the Schr?dinger equation with convection term. We also investigate the

o

symmetry of systems of these equations with additional conditions for potentials Vk .

The results of symmetry classi?cation are applied to constructing exact solutions of

the system of the Schr?dinger equation with convection term and the Euler equations

o

for potentials.

1 Symmetry of the equation of convection di?usion

The equation of convection di?usion has the form

?u ?u

? ? u = Vk (2)

,

?t ?xk

where u = u(t, x) is a real function, ? is a real parameter, the index k varies from 1

to n.

To extend the symmetry of equation (2), we apply the idea proposed in [4, 5, 6].

Namely, we assume that the functions Vk = Vk (t, x) are new dependent variables on

J. Nonlinear Math. Phys., 1997, 4, ¹ 3–4, P. 470–479.

172 W.I. Fushchych, Z.I. Symenoh

equal conditions with the function u. In other words, we seek for symmetry operators

of equation (2) in the form

X = ? µ ?xµ + ??u + ?k ?Vk , (3)

where ? µ , ?, ?k are real functions of t, x, u, V . Applying the Lie algorithm [7, 8, 9],

we ?nd that the unknown functions ? µ , ?, ?k have the form

?

? 0 = 2A(t), ? k = A(t)xk + B kl (t)xl + U k (t),

(4)

? ? ? ?

?k = B kl (t)Vl ? A(t)xk ? B kl (t)xl ? U k (t) ? A(t)Vk , ? = C1 u + C2 ,

where A, B kl , (k, l = 1, n, k = l), B kl = ?B lk , U k (k = 1, n) are arbitrary smooth

real functions of t; C1 , C2 are arbitrary constants. Thus, the following assertion is

true:

Theorem 1. The equation of convection di?usion (2) in the class of operators (3) is

invariant under the in?nite-dimensional Lie algebra with in?nitesimal operators

? ? ?

QA = 2A(t)?t + A(t)xr ?xr ? [A(t)xr + A(t)Vr ]?Vr ,

?

Qkl = B kl (t) [xl ?xk ? xk ?xl + Vl ?Vk ? Vk ?Vl ] ? B kl (t)(xl ?Vk ? xk ?Vl ),

(5)

?

Qa = U a (t)?x ? U a (t)?V , a = 1, n,

a a

Z1 = u?u , Z2 = ?u ,

where we mean summation from 1 to n over the repeated index r and no summation

over indices k, l, and a.

Remark 1. In?nite-dimensional algebra (5) includes the Galilei operator Qa . This

operator generates the following transformations:

t > t = t,

xb > xb = xb + ?b U b (t)?ab ,

(6)

u > u = u,

?

V b > V b = Vb ? ?b U b (t)?ab ,

where ?b is an arbitrary real parameter of transformations, ?ab is the Kronecker

symbol, there is summation from 1 to n over the repeated index b and no summation

over the repeated index a. We see that the function u is not changed under the action

of this operator. This fact is essentially di?erent from the Galilei transformations for

the standard free heat equation

?u

? ? u = 0, (7)

?t

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