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Q6 2x0 x0


Таблица 2
Редуцированные
Операторы Анзацы
уравнения
x0 + x = ?(u)
2
xa ?0 + nu?a ? =0
2n
(?x)2
x0 u ? = ?(u) ? =0
?a ?x?0 u?a 2
2ux0
?a (?x)2 ?0 + 2(?x)u?a + 2?a u2 ?u ? ?x = ? ? =0
u
?x ?x
v
v 2
1 ?
x0 ?a + ?a 2u?u u= + ?(x0 ) ?+ =0
?x
v
2 2x0
x0
v
L?1 (u)du = ?
2x0 ?a + ?a L(u)?u + ?(x0 ) ?+ =0
?x
v
2x0
2x0

?a + ?a ln u?u = ?x + ?(x0 ) ? +1=0
du
ln u
x2 ?
nx0 ?a + xa ?u u ? 2nx = ?(x0 ) ?+ =0
x0
0
2
(?x) ?
x0 ?a + ?a ?x?u u? = ?(x0 ) ?+ =0
2x0 x0



Проинтегрировав редуцированные уравнения и подставив найденную функцию
? в соответствующий анзац, получаем решения уравнения (1)
x2 + a du x1
uL (u) = v
= x1 ? x0 ,
u= 1 , ,
2x0 ln u 2x0
где Q = const.
Замечание 1. Используя лиевскую симметрию уравнения (1), можно установить
формулу размножения его решений u = ? 2 f (?2 x0 + a0 , ??x1 + a1 ), где ?, ?, a0 ,
a1 — произвольные постоянные.
Условная инвариантность нелинейного уравнения теплопроводности 95

Замечание 2. Полученные результаты можно обобщить на случай, когда x =
(x0 , x) ? R1+n и вместо уравнения (1) рассмотреть уравнение
(8)
u0 + u?u = 0.
В табл. 2 ? — постоянный единичный вектор. Отметим, что операторы O1 и
Q6 были обобщены двумя разными способами.
Аналогично, как и для одномерного уравнения, получим решения уравнения (8):
x2 + c (?x)2 + c du
= (?x) ? x0 ,
u= , u= ,
2nx0 2x0 ln u
?x
uL (u) = v , c = const.
2x0
В заключение приведем следующий результат:
Теорема 3. Уравнение
u0 + ?(eu ?u) + ?e?u = 0, (9)
где u = u(x) ? R1 , x = (x0 , x) ? R1+n , инвариантно относительно конформной
алгебры AC(1, n)
? ?
Jab = xa ?b ? xb ?a ,
?0 = , ?a = ,
?x0 ?xa
eu xa
J0a = x0 + ?a + ?0 , D = x0 ?0 + xa ?a + ?u ,
? ?n
(10)
2
eu eu e2u
xa xa
D? ? ? 2 ?0 ,
K0 = 2 x0 + x0 +
? ? ?n ?
2
eu e2u
xa xa xa
Ka = ?2 D ? ? ? 2 ?a ,
x0 + a, b = 1, n,
?n ? ?n ?
при дополнительном условии
nu ?
e (?u)2 + e?u = 0. (11)
u0 +
2 2
В формулах (10) по повторяющимся индексам предполагается суммирование
от 1 до n.
Доказательство. Для доказательства теоремы достаточно доказать, что системаv
уравнений (9), (11) конформно инвариантна. Локальной заменой x0 = (y0 ? w)/ 2,
v
xa = ?n/2ya , u = ln(?w/ 2) данная система сводится к системе уравнений
Даламбера – Гамильтона:
n
2w = ,
w (12)
?
w? w = 1,
где w = w(y), y = (y0 , y) ? R1+n , ? = 0, n, w? = ?y? , w? = g µ? w? , g µ? —
?w

метрический тензор с сигнатурой (+, ?, . . . , ?).
Поскольку система уравнений (12) конформно инвариантна (см. [4]), то теорема
доказана.
96 В.И. Фущич, Н.И. Серов, Т.К. Амеров

1. Фущич В.И., Как расширить симметрию дифференциальных уравнений?, в сб. Симметрия и
решения нелинейных уравнений математической физики, Киев, Ин-т математики АН УССР,
1987, 4–16.
2. Фущич В.И., Штелень В.М., Серов Н.И., Симметрийный анализ и точные решения нелинейных
уравнений математической физики, Киев, Наук. думка, 1989, 336 с.
3. Фущич В.И., Серов Н.И., Чопик В.И., Условная инвариантность и нелинейные уравнения те-
плопроводности, Докл. АН УССР, Сер. А., 1988, № 9, 17–21.
4. Fushchych W.I., Zhdanov R.Z., On some new exact solutions of nonlinear d’Alembert and Hamilton
equations, Preprint N 468, Minneapolis, Institute for Mathematics and its Applications, 1988, 5 p.
W.I. Fushchych, Scientific Works 2002, Vol. 4, 97–100.

Merons and instantons as products
?
of self-interaction of the Dirac–Gursey
spinor field
W.I. FUSHCHYCH, W.M. SHTELEN
In this letter we show that the most physically interesting solutions of the SU (2) Yang–
Mills equations, the well known meron and instanton solutions (and some others), are
generated by corresponding solutions of nonlinear Dirac–G?rsey spinor equations.
u

The idea of describing particles (fields) of spin 0, 1, 3 , 2, . . . by means of a field
2
of spin 1/2 was put forward by Louis de Broglie in the 1930s. Later, in the 1950s it
was developed by Heisenberg and Pauli in their unified field theory. Here we consider
another realisation of this idea based on the possibility of constructing from a spinor
field ?, which satisfies some given equation, different bispinor densities, say scalar
? ?
u = ??, vector Aµ = ??µ ?, and so on. The fruitfulness of such an approach is
demonstrated by examples of meron and instanton solutions of SU (2) Yang–Mills
(YM) equations. We hope that it is not simply a mathematical trick but possibly
reveals some important intrinsic features of merons and instantons.
It is well known that a vast class of solutions of SU (2) YM equations
2Y µ ? ?µ ?? Y ? + e[(?? Y ? ) ? Y µ ? 2(?? Y µ ) ? Y ? + (?µ Y ? ) ? Y ? ]+
(1)
+ e2 Y ? ? (Y ? ? Y µ ) = 0,

where Y ? = Y ? (x) = {Y?1 , Y?2 , Y?3 } is the YM potential, µ, ? = 0, 3, x ? R(4)
(Euclidean space), can be constructed by means of a scalar field ? which satisfies the
nonlinear wave equation
2? + ?1 ?3 = 0 (2)
(?1 is an arbitrary constant). In order to do this one has to use the ’t Hooft–Corrigan–
Fairlie–Wilczek ansatz (see, for example, [1])
eY0a = ??a ln ?, a = 1, 2, 3,
(3)
eYja = (?jan ?n ± ?ja ?0 ) ln ?.
The following solutions of equation (2) are of special interest
?1/2
? = ?1 x2 (4)
,
1/2
(a ? b)2
(5)
?= ,
?1 (x ? a)2 (x ? b)2

8 ?
(6)
?= ,
?1 x2 + ?2
J. Phys. A: Math. Gen., 1990, 23, L517–L520.
98 W.I. Fushchych, W.M. Shtelen

where (x ? a) ? (x? ? a? )(x? ? a? ) and a? , b? , ? are arbitrary constants, because
they give rise to the one-meron [2]
xa xn x0
eY0a = ± eYja = ??jan ? ?aj 2 (7)
,
x2 x2 x
to the two-meron [2]
(x ? a)a (x ? b)a
eY0a = ± + ,
(x ? a) (x ? b)2
2
(8)
(x ? a)n (x ? b)n (x ? a)0 (x ? b)0
eYja = ??jan ? ?aj
+ +
(x ? a)2 (x ? b)2 (x ? a)2 (x ? b)2
and to the instanton [3]
2xa 2xa 2x0
eY0a = ? eYja = ??jan ± ?aj 2 (9)
,
x2 + ?2 x2 + ?2 x + ?2
solutions of YM equations (1), respectively. We shall show that scalar fields (4)–(6)
can be constructed in turn from the spinor field ? which satisfies the Dirac–G?rsey
u
equation
?
i?? + ?(??)1/3 ? = 0, (10)

where ?? are 4 ? 4 Dirac matrices, ? = ?(x) is a four-component complex function
?
(column), ? = ? + ?0 and ? is an arbitrary constant. Equation (10) is conformally
invariant as well as (1) and (2), but it has conformal degree 3/2 while the conformal
degree of the scalar field from (2) is 1. (Detailed analysis of conformal symmetry is
given in [4] where, in particular, it was pointed out that the conformal degree is an
important intrinsic characteristic of a field.) So, to construct the scalar field ? from
?
the spinor field ? properly, we should not simply put ? = ?? but
?
? = (??)1/3 . (11)
Further, we consider the following two solutions of equation (10) obtained by Kor-
tel [5] and Merwe [6]
v
3/2
i?x + x2
13
(12)
?(x) = ?
(x2 )5/4
4?
and
3/2
4? i?x + ?
(13)
?(x) = ?,
(x2 + ?2 )2
?
where ? is an arbitrary constant spinor and one can choose, without loss of generality,
?? = 1; ? is an arbitrary constant. These solutions were obtained by means of the
?
Heisenberg ansatz [7]
(14)
?(x) = [f (w) + i?xg(w)]?,
v
where f and g are real scalar functions, w = x2 and ? is a constant spinor.
One can make sure that the substitution of (12) and (13) into (11) gives rise to (4)
Merons and instantons as products of self-interaction 99
v v
and (6) provided ? = 3 ?1 and ? = 2?1 respectively. It should be noted that,
2
generally speaking, scalar field (11), constructed from spinor field (14) and satisfying
equation (10), does not satisfy equation (2), but we do not know the explicit form of
such solutions of equation (10).
Further we note that solution (5) of equation (2) can be obtained as a result of the
following procedure of group multiplication of solutions. Applying to (4) the formulae
of generating solutions by conformal transformations [4]
xµ ? cµ x2
1
?(c, x) = 1 ? 2cx + c2 x2 , (15)
?II (x) = ?I (x), xµ = ,
?(c, x) ?(c, x)
where ?II (x) means a new solution, ?I (x) means an old one, and cµ are arbitrary
constants, then by translational transformations

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