<< Предыдущая

стр. 128
(из 145 стр.)

ОГЛАВЛЕНИЕ

Следующая >>

Proof. Using Lie’s algorithm [5, 4] we find from the condition of invariance that the
generator of scale transformations should have the form

D = x? ? w?w + ? 2 (v, w)?v

provided (following from the invariance of the second equation of system (3)) that

?
2 2 2
? 2 = kv + bw + c,
?vv = ?ww = ?wv = 0
(6)
dF
2?bw3 + 3?cw2 + (2 ? k)F ? w = 0.
dw
The general solution of equations (6) is given in (4). Thus the theorem is proved.
In particular, as follows from Theorem 1, the equation

2u + ?u3 + ?u = 0 (7)

is approximately scale invariant and the corresponding generator has the form D =
x? ? w?w + v?v . This statement holds true even if ? = 0.
Theorem 2. Equation (1) is approximately conformally invariant if and only if

F (u) = ?3??u2 + au3 (8)

with the generator of conformal transformations having the form

K = 2cx[x? ? w?w ? (v ? ?)?v ] ? x2 c?, (9)

where ?, a, cµ are arbitrary constants.
The proof of Theorem 2 is performed in the same spirit as that of Theorem 1.
Suppose that in (2)

(10)
v = f (w),

where f is an arbitrary differentiable function. In this case the system (3) takes the
form

2w + ?w3 = 0, (11)

wµ wµ f + 2wf? + 3?w2 f + F (w) = 0,
? wµ ? ?w/?xµ . (12)

From the condition of splitting of equation (12) one has to put

wµ wµ = A(w), (13)

where A is some function of w. Equation (13) is compatible with (11) if A(w) = ?w4 ,
i.e.

wµ wµ = ?w4 . (14)
554 W.I. Fushchych, W.M. Shtelen

(For more details see [1, 2].) Taking account of (11) and (14) we rewrite (12) as
?(w2 f ? wf? + 3f ) + w?2 F (w) = 0.
? (15)
So, if we find function f (w) as a solution of equation (15), we thereby obtain by means
of expressions (2) and (10) approximate solutions of equation (1). It will be noted that
a subset of such solutions of equation (1) is approximately conformally invariant since
the corresponding approximate system (11) and (14) is conformally invariant [1, 2].
Solutions of equation (15) for functions F (w) given in (4) have the form
?
a b c
??
? w?k ? w ? , k = 0, ?1,
?
? ?[k(k + 2) + 3]
? k+1 k
1 (16)
f (w) =
??c ln w ? bw ? (2c + a/?), k = 0,
?
?
? 3
?
?w(a/2? + b ln w) + c, k = ?1.
The solution of the system (11) and (14) is the function
w = ±[?(x? + a? )(x? + a? )]?1/2 , (17)
where a? are arbitrary constants.
When ? = 0, the non-trivial condition of splitting of equation (12) compatible with
the equation 2w = 0 is
wµ wµ = 1. (18)
So, in this case we find approximate solutions of equation (1) by means of expressions
(2) and (10), where function f (w) is determined from the equation
? (19)
f + F (w) = 0
and w, in turn, is determined from the system
2w = 0, wµ wµ = 1. (20)
?
The system (20) is invariant under the extended Poincar? group P (1, 4) and has
e
solution [1]
?? ?? = 1, (21)
w = ?x + a,
where a, ?? are arbitrary constants.
In particular, equation
2u + ?u = 0 (22)
?
is approximately invariant under the group P (1, 4) on the subset of solutions
13
u=w?? (23)
w + a1 w + a2 ,
6
where w is given in (21) and a1 , a2 are arbitrary constants.
In conclusion, let us note some generalisations of the concept of approximate
symmetry studied in this paper. First of all, obviously, one can consider higher orders
of approximation of u in ?, i.e. u = w + ?v (1) + ?2 v (2) + · · ·, and can study the
On approximate symmetry and approximate solutions 555

symmetry of the corresponding approximate system of PDE for functions w, v (1) ,
v (2) , and so on. Secondly, one can expand in ?-series not only dependent variables,
but also independent ones, e.g. x0 ? t = x + ?z (1) + ?2 z (2) + · · ·, and can construct in
this way the corresponding approximate system and then study its symmetry. Another
approach to the study of approximate symmetry it to use some special approximations,
say the two-point Pad? approximants
e
? ??1
m n
? fk ? ?j gj ? m, n < ?,
k
(24)
u= ,
j=0
k=0

where functions fk , gj are determined from the condition: when ? > 0 expression
(24) coincides with the expansion

u = v (0) + ?v (1) + ?2 v (2) + · · · , ? 1

and when ? > ? (24) coinsides with the expansion

u = w(0) + ??1 w(1) + ??2 w(2) + · · · , ? 1.

We also note that the symmetry of a system of PDE which approximates the
non-linear wave equation was studied by Shulga [6]. Using symmetry properties,
Mitropolsky and Shulga [3] obtained some asymptotic solutions of the non-linear
wave equation.
Note added. Readers who are less well acquinted with work in this might refer
to the related work of Winternitz et al [7] which is also concerned with this type of
non-linear wave equation from a symmetry point of view.

1. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of nonlinear
equations of mathematical physics, Kyiv, Naukova Dumka, 1989.
2. Fushchych W.I., Zhdanov R.Z., Preprint N 468, University of Minnesota, 1988.
3. Mitropolsky Yu.A., Shulga M.W., Dokl. Akad. Nauk, 1987, 295, № 1, 30–33.
4. Olver P., Applications of Lie groups to differential equations, Berlin, Springer, 1986.
5. Ovsyannikov L.V., Group analysis of differential equations, Moscow, Nauka, 1978.
6. Shulga M.W., in Symmetry and Solutions of Nonlinear Equations of Mathematical Physics, Kyiv,
Institute of Mathematics, 1987, 96–99.
7. Winternutz P., Grundland A.M., Tuszynski J.A., J. Math. Phys., 1987, 28, 2194–2212.
W.I. Fushchych, Scientific Works 2001, Vol. 3, 556–560.

Дифференциальные инварианты
алгебры Галилея
В.И. ФУЩИЧ, И.А. ЕГОРЧЕНКО
Bases of the second-order differential invariants of the Galilei algebra are constructed
for n-dimensional real and complex scalar functions. New classes of the non-linear
nonrelativistic equations are found.

Хорошо известно, что уравнение теплопроводности

2µut + uaa = 0, uaa = ?u,
(1)
n?3
u = u(t, x), x = (x1 , x2 , . . . , xn ),

инвариантно относительно обобщенной алгебры Галилея AGI (1, n) с базисными
2
операторами [1]:
? ?
Jab = xa ?b ? xb ?a ,
?t = , ?a = ,
?t ?xa
?
(2)
Ga = t?a + µxa u?u ?u = , u?u , D = 2t?t + xa ?a + ?u?u ,
?u
1 n
A = tD ? t2 ?t + µx2 u?u , ?=?
2 2
(по повторяющимся индексам подразумевается суммирование от 1 до n).
Уравнение Шредингера

(3)
2im?t + ?aa = 0, ?aa = ??,

? = ?(t, x) — комплекснозначная функция, инвариантно относительно алгебры
Галилея с базисными операторами [2]:
? ?
, pa = ?i , Jab = xa pb ? xb pa ,
p0 = i
?t ?xa
J = i(??? ? ? ? ??? ), Ga = tpa ? imxa J,
(4)
D = 2tp0 ? xa pa + ?I, где I = i(??? + ? ? ??? ),
imx2 n
A = t2 p0 ? txa pa + J + ?tI, ? = ? ,
2 2
звездочка означает комплексное сопряжение.
Алгебру (4) будем в дальнейшем обозначать символом AII (1, n).
2
В настоящей работе построены функциональные базисы дифференциальных
инвариантов второго порядка алгебр AGI (1, n) и AGII (1, n) Найденные инвариан-
2 2
ты дают возможность строить широкие классы многомерных нелинейных уравне-
ний параболического типа.
Доклады АН УССР, Сер. А, Физ.-мат. и техн. науки, 1989, № 4, 29–32.
Дифференциальные инварианты алгебры Галилея 557

Определение абсолютного дифференциального инварианта m-го порядка и фун-
кционального базиса инвариантов группы и алгебры Ли см. например, в [3, 4].
Обозначим символом AGI (1, n) алгебру Галилея с базисными элементами
AGI (1, n) = ?t , ?a , u?u , Ga , Jab (2);
AGII (1, n) = p0 , pa , J, Ga , Jab (4).
Символами AGI (1, n) и AGII (1, n) обозначим расширенные алгебры Галилея:
1 1

? ?
AGI (1, n) = AGI (1, n) + D, AGII (1, n) = AGII (1, n) + D.
1 1

Для упрощения записи инвариантов введем замену
Re ?
(5)
u = exp ?, ? = exp ?, Im ? = arctg .
Im ?
Далее будут использоваться следующие обозначения:
Sj (?ab ) = ?a1 a2 · · · ?aj?1 aj ?aj a1 = Sj ,
Sjk (?ab , ?? ) = ?a1 a2 · · · ?ak?1 ak ??k ak+1 · · · ??j?1 aj ??j a1 = Sjk ,
a a a
ab (6)
?2? ??
Rj (?a , ?ab ) = ?a1 ?aj ?a1 a2 · · · ?aj?1 aj , ?ab = , ?a = .
?xa ?xb ?xa
Все индексы j принимают значения от 1 до n, индексы k — от 0 до j.
Инварианты строятся из ковариантных тензоров. Для алгебры AGI (1, n) эти
тензоры имеют вид
(7)
?a = µ?at + ?b ?ab , ?ab .
Теорема 1. Функциональный базис абсолютных дифференциальных инвариан-
тов алгебры AGI (1, n) при µ = 0 состоит из 2n + 2 инвариантов:
M1 = 2µ?t + ?a ?a , M2 = µ2 ?tt + 2µ?a ?at + ?a ?b ?ab ,
(8)
Sj , Rj = Rj (?a , ?ab ).
Для алгебры AGI (1, n) (µ = 0):
1
M2 Rj Sj
2, , .
3+j 1+j
M1 M1 M1
Для алгебры AGI (1, n) (µ = 0):
2

? ?
N2 Rj Sj
2, , (j = 2, . . . , n),
3+j 1+j
N1 N1 N1
где
N1 = 2µ?t + ?a ?a + ?aa ,
1 1 12
N2 = µ2 ?tt + 2µ ?t ?aa + ?a ?at + ?a ?b ?ab + ?a ?a ?bb + ?,
2n bb
n n
j j
(?n)l (j ? 1)!(j + 1)!
(?n)l j!

<< Предыдущая

стр. 128
(из 145 стр.)

ОГЛАВЛЕНИЕ

Следующая >>