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?=0
R(u)
?
?1 = ?t ? R(u), ? 2 = ??x + A2 (R(? ))d?,
1
0
u ?
где R(u) = A1 (? )d? , R(? ) — функция, обратная к R(? ).
0
Несимметрийный подход к построению точных решений 353

Подстановка полученных результатов в формулу (13) дает два класса точных
решений нелинейного ДУЧП

+ ?2 A1 (u)A?3 (u),
?
utt = A4 (u)ux
1 1
x
(15)
R(u)
? = ?(?t ? R(u)),
tA2 (u)), A2 (R(? ))d?
?u = ?(?x + ? ??x +
1 1
0

где ? = 0, 1; ? — произвольная дважды непрерывно-дифференцируемая функция.
Интегрируя ДУЧП 2) из (12), получаем классы точных решений нелинейного
ДУЧП
? µA2 (u)A?3 (u).
?
utt = A4 ux
2 2
x

1) A2 (u) задается формулой (6а),
µ=0
?A2 (u) = (?x + C1 )?(??t ? A?2 (?x + C1 )),
2

µ<0
?A2 (u) = ch (|µ|1/2 ?x + C1 )?(2?|µ|1/2 ?t + A?2 (u) sh 2(|µ|1/2 ?x + C1 ),
2

µ<0
?A2 (u) = cos(µ1/2 ?x + C1 )?(2?µ1/2 ? + A?2 (u) sin 2(µ1/2 ?x + C1 );
2

2) A2 (u) задается формулой (6б)
µ=0
?A2 = (?t + C1 )?1 ?(??x ? A2 (?t + C1 )),
2

µ<0
?A2 (u) = ch?1 (?|µ|1/2 t + C1 )?(2?|µ|1/2 ?x + A2 sh 2(|µ|1/2 ?tx + C1 )),
2

µ>0
?A2 (u) = cos?1 (µ1/2 ?t + C1 )?(2?µ1/2 ?x + A2 sin 2(µ1/2 ?tx + C1 ));
2

3) A2 (u) задается одной из формул (6в)
µ=0
?(?t + ?x + C1 )A2 (u) = (??A2 (u) + ?)?(t ? (?t + ?x + C1 )(??A2 (u) + ?)?1 );
2 2

µ<0
?A2 (u) ch (|µ|1/2 (?t + ?x) + C1 ) = (??A2 (u) + ?) ?
2

? ?([??A2 (u) sh 2|µ|1/2 ?t + ? sh 2(|µ|1/2 ?x + C1 )](??A2 (u) + ?)?1 );
2 2

µ>0
?A2 (u) cos(µ1/2 (?t + ?x) + C1 ) = (??A2 (u) + ?) ?
2

? ?([??A2 (u) sin 2µ1/2 ?t + ? sin 2(µ1/2 ?x + C1 )](??A2 (u) + ?)?1 ).
2 2
354 В.И. Фущич, И.В. Ревенко, Р.З. Жданов

В приведенных формулах ? ? C 2 (R1 , R1 ) — произвольная функция; ?, ?, C1 —
произвольные действительные параметры, причем ? = 0, ? = 0, ? = 0, 1, ? = ±1.
Ввиду того, что построенные нами точные решении содержат произвольную
функцию, они могут использоваться при анализе довольно широкого класса крае-
вых задач для нелинейного ДУЧП (1). Например, решение задачи Коши
utt = a2 (u)ux , u(0, x) = U0 (x), ut (0, x) = U1 (x),
x

где U0 (x), U1 (x) — произвольные действительные функции, связанные соотноше-
?
нием U1 (x) = ±a(U0 (x))U0 (x), задается неявной фopмyлой u = U0 (x ± ta(u)).

1. Ames W.F., Loher R.J., Adams E., Group properties of utt = [f (u)ux ]x , J. Non-Linear Mechanics,
1981, 16, № 5/6, 439–447.
2. Фущич В.И., Серов Н.И., Репета В.К., Условная симметрия, редукция и точные решения нели-
нейного волнового уравнения, Докл. АН УССР, 1991, № 4, 8–12.
3. 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.
4. Фущич В.И., Штелень В.М., Серов Н.И., Симметрийный анализ и точные решения нелинейных
уравнений математической физики, Киев, Наук. думка, 1989, 336 с.
5. Forsyth A.R., Theory of differential equations, Cambridge, Univ. Press, 1906, Vol. 6, 596 p.
6. Сидоров А.Ф., Шапеев В.П., Яненко Н.Н., Метод дифференциальных связей и его приложения
в газовой динамике, Новосибирск, Наука, 1984, 270 с.
7. Овсянников Л.В., Групповой анализ дифференциальных уравнений, М., Наука, 1978, 400 с.
W.I. Fushchych, Scientific Works 2002, Vol. 4, 355–360.

On the non-Lie reduction
of the nonlinear Dirac equation
W.I. FUSHCHYCH, R.Z. ZHDANOV
The method of construction of exact solutions of nonlinear spinor equations based on
their conditional (non-Lie) symmetry is suggested. With the help of this method new
ans?tze that reduce the nonlinear Poincar?-invariant Dirac equation to ordinary di-
a e
fferential equations are constructed. The new family of exact solutions of the nonlinear
Dirac equation with scalar selfinteraction is found.

1. Introduction
It is common knowledge that the classical Lie approach to the construction of
exact solutions of partial differential equations (PDEs) essentially uses invariance
properties of the set of solutions of the considered equation [1, 2]. In Refs. [3–5]
a natural generalization of the Lie approach was suggested that takes into account
not only the symmetry of the set of solutions of PDEs as a whole, but the symmetry
of their subsets as well. This is achieved by imposing on the solutions of the initial
equation such additional conditions (equations) that the obtained system of PDEs is
compatible and possesses wide symmetry.
Using the above idea, in the present paper we construct a family of the new exact
solutions of the following nonlinear spinor equation:
?
{i?µ ?mu ? ?(??)1/2k }? = 0, (1)
?, k = const,
where ? = ?(x0 , x1 , x2 , x3 ) is the four-component complex function, ? = ? † ?0 and
?
?µ are 4 ? 4 Dirac matrices, ?µ = ?/?xµ , and µ = 0, 3. Hereafter, the summation over
the repeated indices is supposed.
2. Construction of the non-Lie anz?tze for the spinor field
a
The solution of Eq. (1) is found in the form
?(x) = exp{fµ? (x)?µ ?? }?(?), (2)
where ?(?) is a four-component function and fµ? (x) and ? = ?(x) are scala real
functions. The functions fµ? , ? are chosen such that substitution of expression (2)
into Eq. (1) yields an ordinary differential equation (ODE) for ? = ?(?).
We shall describe ans?tze (2) as reducing the PDE (1) to systems of ODEs if the
a
functions fµ? , ? are of the following structure:
1
f00 = ?f11 = ?f22 = ?f33 = ?0 (x0 + x3 , x1 , x2 ),
4
1
f01 = ?f10 = f13 = ?f31 = ?1 (x0 + x3 , x1 , x2 ),
2
1
= ?f20 = f23 = ?f32 = ?2 (x0 + x3 , x1 , x2 ),
f02
2
f03 = f30 = f12 = f21 = 0, ? = ?(x0 + x3 , x1 , x2 ).
J. Math. Phys., 1991, 32, № 12, 3488–3490.
356 W.I. Fushchych, R.Z. Zhdanov

Substituting the ansatz

(3)
?(x) = exp{?0 + (?1 ?1 + ?2 ?2 )(?0 + ?3 )}?(?)

into Eq. (1) and multiplying the obtained equality by

exp{??0 ? (?1 ?1 + ?2 ?2 )(?0 + ?3 )}

one has
i[(?0 + ?3 )?? ?0 + ?a ?a ?0 + ?a ?B (?a ?B )(?0 + ?3 ) ? 2?a (?a ?0 )(?0 + ?3 )]? +
+ i[(?0 + ?3 )(?? ? ? 2?a ?a ?) + ?a ?a ?]? ? ?e?0 /k (??)1/2k ? = 0,
?

where ? = x0 + x3 , ?? = ?/??, a = 1, 2, and B = 1, 2.
Hence it follows that ansatz (3) reduces the initial PDEs to ODEs if the nonlinear
equations hold:

?? ?0 ? 2?a ?a ?0 ? ?a ?a = e?0 /k f1 (?), ?1 ?0 = e?0 /k f2 (?),
?2 ?1 ? ?1 ?2 = e?0 /k f4 (?),
?a ?0 = e?0 /k f3 (?), (4)
?? ? ? 2?a ?a ? = e?0 /k f5 (?), ?1 ? = e?0 /k f6 (?), ?2 ? = e?0 /k f7 (?).

In Eqs. (4) f1 , . . . , f7 are arbitrary smooth real functions.
It is worth noting that as a result of the arbitrariness of the function ?(?) substi-
tution of the expressions

(5)
?(x), ?? (x)

and
? ? (6)
f (?(x)), ?? (x) + f? (?(x)),
??
where f , fa ? C 1 (R1 , R1 ), ? = 0, 2, into formula (3) gives the same ansatz for the
field ?(x). In this sebse solutions of system (4) of the forms (5) and (6) are equivalent.
System (4) contains seven equations for four functions, i.e., it is an overdetermined
system. This fact makes it possible to construct its general solution.
Theorem. The general solution of the nonlinear system of PDEs (4) determined up
to the above equivalence relation is given by one of the following formulas:

?0 = k ln ?1 , ?1 = (2?1 )?1 (?1 x1 + ?2 ),
? ?
?2 = (2?1 )?1 [(2k ? 1)?1 x2 + ?3 ], ? = ?1 x1 + ?2 ;
?
?0 = ?k ln(x1 + ?1 ),
?a = ?3 [(x1 + ?1 )2 + (x2 + ?2 )2 ]k?1 (xa + ?a ) + 1 ?a , a = 1, 2, (7)
2?
? = (x1 + ?1 )(x2 + ?2 )?1 ;
?0 = 0, ?1 = R(x1 + ix2 , x0 + x3 ) + R(x1 ? ix2 , x0 + x3 ) + ?1 x1 ,
?2 = iR(x1 + ix2 , x0 + x3 ) ? iR(x1 ? ix2 , x0 + x3 ) + ?2 x1 , ? = x0 + x3 .

Here ?1 , ?2 and ?3 designate arbitrary real smoothfunctions on x0 + x3 and R
designates an arbitrary analytical function on the first variable.
On the non-Lie reduction of the nonlinear Dirac equation 357

Let us adduce the main steps of the proof. First, an overdetermined system made
up of the second, third, sixth, and seventh equations in (7) is integrated. Upon making
the change of the variable ? = e??0 /k we rewrite this system in the form
?a ? = ??1 Ga (?), Fa , Ga ? C 1 (R1 , R1 ), (8)
?a ? = Fa (?), a = 1, 2.
From the necessary and sufficient compatibility conditions of system (8), ?1 ?2 ? =
?2 ?1 ?, ?1 ?2 ? = ?2 ?1 ?, one has the following relations for Fa (?), Ga (?):
? ? ? ?
G2 G1 ? G1 F2 = G1 G2 ? G2 F1 , (9)
F1 G2 = G1 F2 ,
where the overdot means differentiation with with respect to ?.
The procedure for the integration of the system of ODEs (9) is essentially simpli-
fied by the fact that the equivalence conditions (6) induce the equivalence relation on
the set of solutions of Eqs. (9):
Fa (?) ? Fa (f (?)) ? g(?)Ga (f (?)),
?
(10)
Ga (?) ? (f?(?))?1 Ga (f (?))(g(?))?1 ,
where f, g ? C 1 (R1 , R1 ), f?g = 0.
By integrating the system of PDEs (8) and (9) one establishes that is general
solution up to the equivalence relations (6) and (10) is determined by one of the
following formulas:
?1
F1 = G1 = 1, F2 = G2 = 0, ? = ?1 , ? = ?1 x1 + ?2 ;
F1 = 1, F2 = 0, G1 = ?, G2 = ?? 2 ,
? = x1 + ?1 , ? = (x1 + ?1 )(x2 + ?2 )?1 ;
(11)
F1 = F2 = G1 = G2 = 0, ? = ?, ? = 1;
F1 = F2 = 0, Ga ? C 1 (R1 , R1 ), ? = ?,
? = G1 (?)x1 + G2 (?)x2 + ?3 .
Here ?1 , ?2 , and ?3 are arbitrary real smooth functions on ?.
Substitution on the sxpressions for the functions ?(x), ?0 (x) = ?k ln ?(x) into the
remainder of Eqs. (4) yields four systems of PDEs on ?1 (x) and ?2 (x). By integrating
the first systems of PDEs one arrives at formulas (7). The fourth system of PDEs
proves to be incompatible system.
Substitution of expressions (7) into formula (3) gives three classes of ans?tze for
a
the spinor field:
?(x) = ?1 exp{(2?1 )?1 [(?1 x1 + ? 2 )?1 +
k
? ?
+ ((2k ? 1)?1 x2 + ?3 )?2 ](?0 + ?3 )}?(?1 x1 + ?2 ),
?
?(x) = (x1 + ?1 )?k exp{?3 [(x1 + ?1 )2 + (x2 + ?2 )2 ]k?1 (12)
? ?a (xa + ?a )(?0 + ?3 ) + 1 ?a ?a (?0 + ?3 )}?((x1 + ?1 )(x2 + ?2 )?1 );
2?
?(x) = exp{[(R + R? + ?1 x1 )?1 + (iR ? iR? + ?2 x1 )?2 ](?0 + ?3 )}?(x0 + x3 ),
reducing the nonlinear Dirac equation (1) to the system of ODEs
i?1 ? = ?(??)1/2k ?,
? ?
i(?2 ? ?1 ?)? = ?(??)1/2k ?, (13)
? ?
i(?0 + ?3 )? = ?(??)1/2k ?.

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