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i=1

where Si are determined by the recurrent relations
x
d2 Si (x) dV
Si+1 (x) = ? i = 1, . . . , n ? 1,
V (y)Si (y)dy + 4V 2 Si (x),
+4
dx2 dx x0

def
with S1 = dV /dx.
The above formulae (14) give the general solution of the ?rst n equations from
Eq. (13). Inserting these into the last equation yields equation for the function V (x)
of the form
n+1
(15)
Ck Sn+2?k = 0.
k=1

As S1 = dV /dx, Eq. (15) is nothing else than an equation of the stationary mKdV
hierarchy, which is obtained from the higher mKdV equations by setting v(t, x) =
v(x + Ct), C = const.
Integrating Eqs. (11) yields
x
k
? (16)
rk (x) = 2 Ci V (y)Sk+1?i (y)dy + Ck , k = 1, . . . , n,
i=1 x0

?
where Ci are arbitrary complex constants.
Stationary mKdV hierarchy and integrability of the Dirac equations 289

Thus, the formulae (10), (14), (15), (16) give the general solution of the system of
determining equations (11), (12) within the class of functions of the form (10). This
means, in particular, that provided the function V (x) is a solution of Eq. (15) with
some ?xed n and C1 , . . . , Cn , the Dirac equation possesses a Lie symmetry. Hence we
conclude that it is integrable by quadratures due to Lemma 1. Consequently, we have
proved the following remarkable fact.
Theorem 1. Let V (x) be a solution of an equation of the mKdV hierarchy of the
form (15). Then the Dirac equation (1) is integrable by quadratures.
Note that the equations of the stationary mKdV hierarchy are transformed to the
equations of the stationary KdV hierarchy with the help of the Miura transformation
and the latter are integrated in ?-functions with any n ? N [7].
There is a deep relationship of the above results with those obtained by Novikov
in Ref. [8], where it was established, in particular, that periodical solutions of the
stationary KdV hierarchy give rise to the integrable stationary Schr?dinger equa-
o
tions (3). This relationship is established via the Lax representation for higher KdV
equations. Since we consider the stationary KdV equations, the Lax representation
reduces to the condition that there exists an N th-order di?erential operator
N
di
Q= qi (x) ,
dxi
i=0

commuting with the Schr?dinger operator d2 /dx2 ?W (x), provided W (x) is a solution
o
of the corresponding higher stationary KdV equation. Consequently, Q is the higher
symmetry of the Schr?dinger equation in a sense of [3].
o
On the set of solutions of the Schr?dinger equation (3) we can reduce the opera-
o
tor Q to a ?rst-order Lie symmetry of the form (for more details, see Ref. [9])
d
?
Q = ?(x, ?) + ?(x, ?),
dx
where ?, ? are polynomials in ?. This gives us the ansatz for a Lie symmetry used at
the beginning of this Letter.
Thus, the approach to integrating ordinary di?erential equations suggested here is
based on their high-order Lie symmetry. To the best of our knowledge, the high-order
Lie symmetries were not used until now for integrating ordinary di?erential equations.
It is important to note that within the framework of the Lie approach one always
deals with the set of solutions as a whole. This means that speci?c properties of subsets
of solutions (like periodicity) are not taken into account. To study these one needs
more subtle analytic methods. On the other hand, the Lie approach has the merit
of being a universal tool applicable to a wide range of ordinary di?erential equations
having the same algebraic-theoretical properties. For example, it is not di?cult to
generalize the technique developed for integrating the Dirac equation (1) in order to
integrate an arbitrary system of ordinary di?erential equations of the form
du
? (V (x)?3 + ?)u = 0, (17)
i?1
dx
where ?1 , ?2 are arbitrary ?nite- or in?nite-dimensional constant matrices forming,
together with the matrix ?3 = ?i[?1 , ?2 ], a basis of the Lie algebra su(2). The result
290 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

will be the same, namely, if V = V (x) is a solution of an equation of the stationary
mKdV hierarchy, then the system of ordinary di?erential equations (17) is integrable
by quadratures.
In conclusion let us demonstrate how the above procedure works for the simplest
case n = 1. With this choice of n, Eq. (15) reads
d3 V
C2 dV dV
? + 6V 2 (18)
= 0,
dx3
C1 dx dx
which is exactly the stationary mKdV equation and is obtained from Eq. (2) via the
ansatz v(t, x) = V (C2 x ? C1 t).
A simple computation yield the form of the coe?cients of the Lie symmetry (5),
d2 V
i
f (x) = ? ? 2C1 V 3 ? C2 ? 4C1 ?2 ) ,
C1
2
4? dx
1 dV 1
? C1 V 2 ? C2 ? 2C1 ?2 , (19)
g(x) = C1
2 dx 2
1 dV 1
+ C1 V 2 + C2 + 2C1 ?2 ,
h(x) = C1
2 dx 2
which satisfy the determining equations (6) inasmuch as the function V (x) is a solution
of the stationary mKdV equation.
Thus, the Dirac equation with potential V (x) satisfying the stationary mKdV
equation (18) is integrable by quadratures and its general solution is given by formu-
lae (7) and (19).
Note that due to the remark made at the very beginning of the paper components
of the function u ful?ll the stationary Schr?dinger equation (3). This is in a good
o
accordance with results of Ref. papers [9].
One of the authors (R.Zh.) is supported by the Ukrainain DFFD Foundation
(project 1.4/356).

1. Zakharov V.E., Manakov S.V., Novikov S.P., Pitaevski L.P., Theory of solitons: the inverse
scattering method, New York, Consultants Bureau, 1980.
2. Lamb G.L., Jr., Elements of soliton theory, New York, Wiley, 1980.
3. Fushchych W.I., Nikitin A.G., Symmetry of equations of quantum mechanics, New York,
Allerton Press, 1994.
4. Cartan E., Les Syst?mes Di?erentiels Ext`rieurs et Leur Applications G?ometriques, Paris,
e e e
Hermann, 1945.
5. Ibragimov N.Kh., Transformation groups applied to mathematical physics, Dordrecht, Reidel,
1985.
6. Olver P.J., Applications of Lie groups to di?erential equations, New York, Springer, 1986.
7. Dubrovin B.A., Matveev V.B., Novikov S.P., Uspekhi Matem. Nauk, 1976, 31, 55.
8. Novikov S.P., Functional Anal. Appl., 1974, 8, 54.
9. Zhdanov R.Z., J. Math. Phys., 1996, 37, 3198.
W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 291–294.

Конформна iнварiантнiсть системи
рiвнянь ейконалу
В.I. ФУЩИЧ, М.I. СЄРОВ, Ю.Г. ПОДОШВЄЛЕВ

The conformal symmetry of the system of eikonal equations u? u?,µ = g ?? , u? u?,µ =
µ µ
???? , where g ?? is the metric tensor with the signature (+, ?) and ??? is the
Kronecker symbol, is studied. The symmetry of the above system is used for ?nding
its exact solutions at n = 1. The isomorphism of the algebras AC(2, 2) and AO(3, 3)
is used to construct invariants of the conformal algebra. Formulas concerning the
multiplication of solutions are presented.

Одним з основних рiвнянь геометричної оптики є рiвняння ейконалу

uµ uµ = m, (1)

де uµ = ?xµ , u = u(x), x = (x0 , x) ? R1+n , µ = 0, n; m — довiльна стала. В фор-
?u

мулi (1) i скрiзь нижче пiд iндексами, якi повторюються, слiд розумiти суму.
В роботах [1–6] детально вивченi симетрiйнi властивостi цього рiвняння, прове-
дена редукцiя та побудованi класи його точних розв’язкiв. Зокрема в [5] вста-
новлено, що при m = 1 рiвняння (1) iнварiантне вiдносно конформної алгебри
AC(1, n + 1), а при m = ?1 — вiдносно алгебри AC(2, n). Дiя цих алгебр визначе-
на в n+1-вимiрному просторi Пуанкаре–Мiнковського R(1, n+1) з координатами
x = (x0 , x1 , . . . , xn , xn+1 ? u).
Поставимо задачу узагальнити рiвняння (1) на випадок системи рiвнянь для
функцiй u1 i u2 , яка була б iнварiантною вiдносно алгебри AC(1 + 1, n + 1), або
AC(1 + 2, n) в просторi R(1, n + 2) з координатами x = (x0 , x1 , . . . , xn , xn+1 ?
u1 , xn+2 ? u2 ). Розв’язком поставленої задачi е таке твердження.
Теорема. 1. Максимальною алгеброю iнварiантностi системи рiвнянь

u? u?,µ = g ?? , (2)
µ

є конформна алгебра AC(1 + 1, n + 1), де g ?? — метричний тензор з сигнатурою
(+, ?).
2. Максимальною алгеброю iнварiантностi системи рiвнянь

u? u?,µ = ???? ,
µ

є конформна алгебра AC(1 + 2, n), де ??? — символ Кронекера, ?, ? = 1, 2.
Теорема доводиться стандартним методом С. Лi [8].
У випадку n = 1 система рiвнянь має вигляд
(u1 )2 ? (u1 )2 = 1,
0 1
(u0 ) ? (u2 )2 = ?1,
22
(3)
2
u0 u0 ? u1 u1 = 0.
12 12


Доповiдi НАН України, 1999, № 1, C. 43–47.
292 В.I. Фущич, М.I. Сєров, Ю.Г. Подошвєлев

Використаємо симетрiю системи рiвнянь (3) для знаходження її точних розв’яз-
кiв, якi будемо шукати у виглядi
v = ?1 (?), w = ?2 (w) (4)
(див., наприклад, [5]), де ?1 (?) i ?2 (?) — невiдомi функцiї, якi потрiбно визначи-
ти, а ? = ?(x, u1 , u2 ), v = (x, u1 , u2 ) та w = w(x, u1 , u2 ) — iнварiанти конформної
алгебри. Для знаходження iнварiантiв конформної алгебри необхiдно проiнтегру-
вати нелiнiйну систему звичайних диференцiальних рiвнянь. Основна складнiсть
полягає в тому, що не iснує загальних методiв розв’язування таких систем. Але
дану систему можна звести до лiнiйної, використовуючи iзоморфiзм мiж кон-
формною алгеброю AC(m, k) та алгеброю Лоренца AO(m + 1, k + 1). У випадку
m = k = 2 даний iзоморфiзм здiйснюється за допомогою замiни (бiльш детально
про це див., наприклад, [7]):
z2 z5 z4 z3
x0 = , x1 = , x2 = , x3 = ,
z6 ? z1 z6 ? z1 z6 ? z1 z6 ? z1
x2 ? 1
z6 + z1
x2 = x2 ? x2 ? x2 + x2 = (5)
, z1 = 2 z6 ,
z6 ? z1
0 1 2 3
x +1
2x0 2x3 2x2 2x1
z2 = 2 z 6 , z3 = 2 z 6 , z4 = 2 z 6 , z5 = 2 z6 ,
x +1 x +1 x +1 x +1
i дiє на конусi z1 + z2 + z3 ? z4 ? z5 ? z6 = 0 точно. Зв’язок мiж операторами
2 2 2 2 2 2

конформної алгебри AC(2, 2) та алгебри Лоренца AO(3, 3) = {Jab }, a, b = 1, 6,
задається формулами
P? = f (J1?+2 ? J?+2n+3 ), D = ?f (J1n+3 ), J?? = f (J?+2?+2 ),
K? = f (J1?+2 + J?+2n+3 ).
Вiдповiдна система Лагранжа–Ейлера є лiнiйна, однорiдна i в матричнiй формi
має вигляд
?
Z = AZ,
де A — числова матриця розмiрностi 6 ? 6. Вигляд розв’язкiв системи (6) визна-
чається виглядом коренiв характеристичного рiвняння
det(A ? ?E) = 0, (6)
(E — одинична матриця). Розкриваючи визначник шостого порядку i виконуючи
елементарнi перетворення, (7) матиме вигляд
?6 + M ?4 + T ?2 + P = 0,
де M , T , P — числа, якi визначаються через елементи матрицi A. Залежно вiд
значень M , T , P та рангу матрицi (A??E) знайдено 15 рiзних випадкiв розв’язку
системи (6). Для кожного з цих випадкiв при використаннi замiн (5) знайдено шу-
канi iнварiанти w, v i w. Не наводячи громiздких обчислень, кiнцевий результат
зобразимо за допомогою табл. 1.
В табл. 1 введенi позначення: ax = aA xA , x2 = xA xA , a, b, c, d, a , b , c , d
— довiльнi сталi вектори, якi задовольняють умови a2 = ?b2 = ?c2 = d2 = 1,
ab = ac = ad = bc = bd = cd = 0, A = 0, 3.
Конформна iнварiантнiсть системи рiвнянь ейконалу 293

Таблиця 1. Iнварiантнi змiннi групи C(2, 2).
№ ? v w
1 ax cx dx
x2
2 bx cx
x2 +1
3 ax bx
dx dx dx
x2 (dx+bx)+dx?bx (x2 +1)2 +4(bx)2
4 ax
(cx)2 (cx)2
cx
(x ?1)2 ?4(cx)2

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