ñòð. 91 |

(23)

L(Lu) + Lu = au + b

is a five-dimensional algebra, whose basis elements are given by the operators

b b

P0 = ? t , P1 = ?x , Z1 = x + t ?x + u + ?u ,

a a

and two other operators depending on constant a have the form

a) a = ? 1

4

1 1 1 1

Z2 = exp ? t ? x ? ?u , Z3 = exp ? t t?x + 1 ? t ?u ,

2 2 2 2

b) a > ? 1 , a = 0

4

Z4 = exp(?t)(?x + ??u ), Z5 = exp(?t)(?x + ??u ),

where

v v

?1 ? ?1 +

4a + 1 4a + 1

?= , ?= ,

2 2

c) a < ? 1

4

Z6 = exp(?t)(sin ?t?x + (? sin ?t + ? cos ?t)?u ),

Z7 = exp(?t)(cos ?t?x + (? cos ?t ? ? sin ?t)?u ),

where

?(4a + 1)

1

?=? , ?= .

2 2

The corresponding finite transformations have the form:

t > t = t,

?

Z1 :

b b

x>x= x + t exp(?) ? t,

?

a a

b b

u>u= exp(?) ? .

? u+

a a

t > t = t,

?

Z4 :

x > x = x + ? exp(?t),

?

u > u = u + ?? exp(?t).

?

378 W.I. Fushchych, V.M. Boyko

t > t = t,

?

Z3 :

1

x > x = x + ?t exp ? t ,

?

2

1 1

u > u = u + ? 1 ? t exp ? t .

?

2 2

t > t = t,

?

Z6 :

x > x = x + ? sin ?t exp(?t),

?

u > u = u + ?(? sin ?t + ? cos ?t) exp(?t).

?

t > t = t,

?

Z7 :

x > x = x + ? cos ?t exp(?t),

?

u > u = u + ?(? cos ?t ? ? sin ?t) exp(?t).

?

Case 2.4. F (u) = a, a = const. The invariance algebra of the equation

(24)

L(Lu) + Lu = a

is a five-dimensional algebra, whose basis elements are given by the operators

P0 = ? t ,

P1 = ?x , G = t?x + ?u ,

(25)

a

Q1 = x ? t2 ?x + (u ? at)?u , Q2 = exp(?t)(?x ? ?u ).

2

The finite transformations for Q1 , Q2 have the form:

Q1 : t > t = t,

?

a a

x > x = x ? t2 exp(?) + t2 ,

?

2 2

u > u = (u ? at) exp(?) + at.

?

Q2 : t > t = t,

?

x > x = x + ? exp(?t),

?

u > u = u ? ? exp(?t).

?

Construction of solutions

In the case when the equation (4) has the form

(26)

L(Lu) + ?Lu = a, a, ? = const

the change of variables

(27)

t = ?, x = ? + u?, u=u

enable us to construct the general solution of (26). In consequence of the change of

variables (27) we obtain:

? ?

> ?? ,

L= +u

?t ?x

?u ?u u?

>

Lu = +u .

?t ?x 1 + ? u?

Symmetry classification of the one-dimensional second order equation 379

After the change of variables the equation (26) has the form

u? u?

(28)

?? +? = a.

1 + ? u? 1 + ? u?

Integrating (28) one time, we get the linear nonhomogeneous partial differential

equation. Finding first integrals of the corresponding system of characteristic equa-

tions and doing the inverse change of variables we find the solutions of (26).

Remark. We notice that the solution of equation 1 + ? u? = 0 in variables (t, x, u) is

x = f (t), where f (t) is an arbitrary function. Thus (26) is equivalent to an ordinary

differential equation in this singular case.

Let us illustrate it on the example of equations (16). After the change of variables

(27), equation (16) is rewritten in the form:

u?

(29)

?? = 0.

1 + ? u?

Integrating (29) we obtain

u?

(30)

= g(?),

1 + ? u?

where g(?) is an arbitrary function.

If g(?) ? 0, then u? = 0 and we get the solution of type (3) (because, it is

obvious that the solution of equation Lu = 0 is a solution of (16)). When g(?) = 0,

?1

in accordance with arbitrary choice of g(?) we can set g(?) = ?2(dh(?)/d?) .

Therefore (30) has the form

2? 2

u? = ? (31)

u? + .

h (?) h (?)

The system of characteristic equation for (31) is

d? h (?)d? h (?)du

(32)

= = .

?2

1 2?

Hence, we obtain two first integrals:

d?

? 2 ? h(?) = C1 , u± (33)

= C2 .

h(?) + C1

Integrating (33) and expressing C1 and C2 by (?, ?, u) we find a solution of (30) in

the form

(34)

? (C1 , C2 ) = 0,

where ? is an arbitrary function. Performing in (34) the inverse change of variables

we get a solution of (16). For instance, we set h(?) = ?. Then the expression

x ? ut ? t2 = ?(u + 2t), (35)

defines the class of implicit solutions of equation (16), where ? is an arbitrary function.

380 W.I. Fushchych, V.M. Boyko

The same results we can obtain for other cases of (26). If F (u) = const in (4)

then this method does not lead to solutions. Below we give some classes of solutions

of equations (26):

1. L(Lu) = 0

C2

1.1. x ? ut + t = ?(u ? Ct);

2

1.2. u ± ln(x ? ut ? t) = ? t2 ? (x ? ut)2 ;

t(x ? ut)3 1

= ? t2 ?

1.3. u + 2 ;

t (x ? ut)2?1 (x ? ut)2

x ? ut x ? ut

? exp t2 dt;

1.4. u = ?

exp (t2 ) exp (t2 )

2. L(Lu) = a

a C a

x ? ut + t3 + t2 = ? u ? t2 ? Ct ;

3 2 2

3. L(Lu) + Lu = a

a

x ? ut ? C(t + 1) exp(?t) + t2 = ? (u + C exp(?t) ? at) ,

2

C = const, ? is an arbitrary function.

Thus, we have investigated the symmetry classifications of (4) and pointed out

all functions F (u) under which the invariance algebra of (4) admits the extension.

The new representations which may have an interesting physical interpretation are

obtained. In the case F (u) = const we described the algorithm of construction of

the general solution of (4) and pointed out some solutions. The symmetry properties

of (4) can be used for a symmetry reduction and construction of the solutions and for

their generation by finite group transformations [3, 4, 5].

Acknowledgement

The main part of this work for the authors was made by the financial support by

Soros Grant, Grant of the Ukrainian Foundation for Fundamental Research and the

Swedish Institute.

1. Fushchych W.I., New nonlinear equation for electromagnetic field having the velocity different

from c, Dopovidi Akademii Nauk Ukrainy (Proceedings of the Academy of Sciences of Ukrainy),

1992, ¹ 1, 24–27.

2. Fushchych W., Symmetry analysis. Preface, in “Symmetry Analysis of Equations of Mathematical

Physics”, Kiev, Inst. of Math., 1992, 5–6.

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

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

4. Olver P., Application of Lie groups to differential equations, New York, Springer, 1986, 497 p.

5. Ovsyannikov L.V., Group analysis of differential equations, New York, Academic Press, 1982, 400 p.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 381–392.

Galilei-invariant nonlinear systems

of evolution equations

W.I. FUSHCHYCH, R.M. CHERNIHA

All systems of (n + 1)-dimensional quasilinear evolution second-order equations invari-

ant under 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 equations,

o

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

1. Introduction

The (n + 1)-dimensional diffusion (heat) system of equations

?1 Ut = ?U,

?2 Vt = ?V,

where U = U (t, x), V = V (t, x) are unknown differentiable real functions, Ut =

?U/?t, Vt = ?V /?t, x = (x1 , . . . xn ), ?1 , ?2 ? R, is known to be invariant under the

generalized Galilei algebra AG2 (1, n) [1, 2]

(2a)

P t = ?t , Pa = ?a ,

xa

Ga = tPa ? Jab = xa Pb ? xb Pa , (2b)

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

2

(2c)

D = 2tPt + xa Pa + I? ,

1 1

? = t2 Pt + txa Pa ? |x|2 Q? + tI? , ?k = ? n. (2d)

4 2

In relations (2) and elsewhere hereinafter I? = ?1 U ?U + ?2 V ?V , ?U ? ?/?U , ?V ?

?/?V , ?t ? ?/?t, ?a ? ?/?xa , ?k ? R, k = 1, 2 and a summation is assumed from 1

to n over repeated indices.

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