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convince oneself that the following equalities hold:
(n) (n)
n ? Z,
pµ pµ A? = 4(m2 + ?2 )A? ,
n
(5.10)
(n)
p? A? pµ pµ ? (n) = m2 ? (n) .
= 0,
Symmetry and exact solutions of nonlinear spinor equations 619

The above relations seem to admit the following interpretation: the interaction of
a spinor and a massless electromagnetic field according to the nonlinear eqs. (5.1)
(n)
generates massive electromagnetic fields Aµ (x) with masses Mn = 2(m2 + ?2 )1/2 n
(in other words, the nonlinear interaction of the fields Aµ (x) and ?(x) generates the
mass spectrum). If one puts n = 0 then M0 = 2m, m being the mass of the electron.
As solutions (5.8), (5.9) have an analytical dependence on m, then the solutions of
the massless Maxwell–Dirac equations can be obtained by putting m = 0. The case
m = 0 deserves special consideration because the massless Maxwell–Dirac equations
are conformally invariant (see, e.g., ref. [53]).
It is not difficult to obtain the general solution of the two-dimensional massless
Dirac equation

? = (? · b + i? · c)?1 (z, ?0 ) + (? · b ? i? · c)?2 (z ? , ?0 ), (5.11)

where ?1 , ?2 are arbitrary spinors depending analytically on z, z ? ; z = b · x + ic · x.
Substituting (5.11) into (5.4) one obtains the following expression for ?(?):

?(?) = F (z, ?0 ) + F (z ? , ?0 ) +
z?
z
?
f2 (z ? , ?0 )dz ? (5.12)
+e z f1 (z, ?0 )dz + z ,
0 0

f1 = ?1 (? · ?)[1 ? i(? · b)(? · c)]?2 , f2 = ?2 (? · ?)[1 + i(? · b)(? · c)]?1 .
? ?

Substitution of the above formulae into (5.2) gives rise to a multi-parameter family
of exact solutions including three arbitrary complex functions,

?(x) = (? · a + ? · d)[(? · b + i? · c)?1 (z, a · x + d · x) +
+ (? · b ? i? · c)?2 (z ? , a · x + d · x)],
Aµ (x) = (aµ + dµ ) F (z, a · x + d · x) + F (z ? , a · x + d · x) +
(5.13)
z?
z
+ e z? f2 (z ? , a · x + d · x)dz ?
f1 (z, a · x + d · x)dz + z ,
0 0
z = b · x + ic · x.

Using the solution generating formula with the group of special conformal transfor-
mations [24, 47]
?II (x) = ? ?2 (x)[1 ? (? · x)(? · ?)]?I (x ),
AII (x) = ? ?2 (x)[gµ? ?(x) + 2(?µ x? ? ?? xµ +
µ
+ 2? · xxµ ?? ? x · x?µ ?? ? ? · ?xµ x? )]A? (x ),
I
xµ = (xµ ? ?µ x · x)? ?1 (x), ?(x) = 1 ? 2? · x + (? · ?)(x · x),

it is possible to obtain a larger family of solutions of the system of equations (5.1).
We omit the corresponding formulae because of their cumbersome character.
6. Conclusions
In this review we described Poincar?-invariant nonlinear systems of first-order
e
differential equations for spinor fields which are nonlinear generalizations of the clas-
sical Dirac equation without using variational principles. The large class of nonlinear
620 W.I. Fushchych, R.Z. Zhdanov

?
spinor equations invariant under the extended Poincar? group P (1, 3) and the confor-
e
mal group is constructed. It contains, in particular, the well-known nonlinear Dirac–
lvanenko, Dirac–Heisenberg and Dirac–G?rsey equations. Besides there are many
u
equations which so far have not been considered in the literature.
The main aim of this review is to suggest a constructive method of solution of
nonlinear Dirac-type spinor equations, that is, to construct in explicit form families of
exact solutions of these equations without applying methods of perturbation theory.
The key idea of our method is a symmetry reduction of the many-dimensional spinor
equation to systems of ordinary differential equations. Many of them can be integrated
in quadratures. Such a reduction is carried out with the help of special ansatze
constructed using the symmetry properties of the equation in question.
To our mind the important result of the present paper is that we have obtained
nongenerable families of exact solutions of nonlinear spinor equations. These solutions
possess the same symmetry as the equation of motion. So nongenerable families of
solutions can be quantized in a standard way without losing the invariance under the
Poincar? group.
e
It is worth noting that some solutions depend on the coupling constant ? in a
singular way.
It is shown how to construct the simplest fields with spin s = 0 using solutions of
the fundamental spinor equation. Such bosonic fields satisfy the nonlinear d’Alembert
equations.
A new approach to the problem of the mass spectrum is suggested (section 4).
It is established that exact solutions of the system of nonlinear equations for spinor
and scalar fields make it possible to calculate the ratio of the masses of spinor and
scalar fields. It occurs that this ratio is determined by the non-linearity degrees of
the spinor and scalar fields.
We hope that the results presented in our paper will make it possible to understand
more deeply the role played by nonlinear spinor equations in the unified theory of
bosonic and fermionic fields with spins s = 0, 1/2, 1, 3/2, 2, . . ..
Suggested methods can be applied to equations of motion in R(1, n) [54, 55]. The
?
problem of subgroup classification of generalized Poincar? groups P (1, n), P (1, n),
e
P (2, n) and Galilei groups G(1, n) was solved in refs. [55–60].

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Contents
А.Ф. Баранник, В.И. Фущич, О непрерывных подгруппах
псевдоортогональных и псевдоунитарных групп . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Л.Ф. Баранник, А.Ф. Баранник, В.И. Фущич, О непрерывных подгруппах
обобщенной группы Пуанкаре P (1, n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
Л.Ф. Баранник, В.И. Фущич, Операторы Казимира для обобщенных
групп Пуанкаре и группы Галилея . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Л.Ф. Баранник, В.И. Фущич, Инварианты подгрупп обобщенной группы
Пуанкаре P (1, n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
В.И. Фущич, О симметрии и точных решениях некоторых многомерных
уравнений математической физики . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
В.И. Фущич, А.Ф. Баранник, Л.Ф. Баранник, Непрерывные подгруппы
обобщенной группы Евклида . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
В.И. Фущич, Р.М. Чернига, О точных решениях двух многомерных
нелинейных уравнений шредингеровского типа . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
В.И. Фущич, В.М. Федорчук, И.М. Федорчук, Подгрупповая структура
обобщенной группы Пуанкаре и точные решения некоторых
нелинейных волновых уравнений . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116
В.И. Фущич, В.М. Штелень, Р.З. Жданов, Конформно-инвариантное
обобщение уравнения Дирака–Гейзенберга и его точные решения . . . . . . . . . 134
В.И. Фущич, С.Л. Славуцкий, О нелинейном галилей-инвариантном
обобщении уравнений Ламе . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
В.И. Фущич, С.Л. Славуцкий, О симметрии некоторых уравнений
идеальной жидкости . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
В.И. Фущич, И.М. Цифра, Конформно-инвариантные нелинейные
уравнения для электромагнитного поля . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
L.F. Barannik, W.I. Fushchych, On subalgebras of the Lie algebra
?
of the extended Poincar? group P (1, n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
e
W.I. Fushchych, The symmetry and exact solutions of some multidimensional
nonlinear equations of mathematical physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
В.И. Фущич, Как расширить симетрию дифференциальных уравнений? . . . . . . .180
В.И. Фущич, О симметрии и точных решениях многомерных нелинейных
волновых уравнений . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
В.И. Фущич, И.Ю. Кривский, В.М. Симулик, О векторных лагранжианах
для электромагнитного и спинорного полей . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
W.I. Fushchych, A.G. Nikitin, On the new invariance algebras
and superalgebras of relativistic wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
W.I. Fushchych, N.I. Serov, On some exact solutions of the three-dimensional
non-linear Schr?dinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .236
o
В.И. Фущич, В.М. Штелень, О редукции и точных решениях
нелинейного уравнения Дирака . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
W.I. Fushchych, I.M. Tsyfra, On a reduction and solutions of non-linear
wave equations with broken symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
В.И. Фущич, Р.З. Жданов, Об одном обобщении метода разделения
переменных для линейных систем дифференциальных уравнений . . . . . . . . . 256
W.I. Fushchych, R.Z. Zhdanov, On some exact solutions of a system
of non-linear differential equations for spinor and vector fields . . . . . . . . . . . . . . . 261
Л.Ф. Баранник, В.И. Фущич, О непрерывных подгруппах конформной
группы пространства Минковского R1,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
W.I. Fushchych, Exact solutions of multidimensional nonlinear Dirac’s
and Schr?dinger’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
o
Л.Ф. Баранник, В.И. Лагно, В.И. Фущич, Подалгебры алгебры Пуанкаре
AP (2, 3) и симметрийная редукция нелинейного ультрагиперболического
уравнения Даламбера. I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
В.I. Фущич, А.С. Галiцин, А.С. Полубинський, Нова математична модель
дифузiйних процесiв зi скiнченною швидкiстю . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
W.I. Fushchych, I. Krivsky, V. Simulik, On vector and pseudovector
Lagrangians for electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
В.И. Фущич, Н.И. Серов, Условная инвариантность и точные решения
нелинейного уравнения акустики . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
В.И. Фущич, Н.И. Серов, Об условной инвариантности нелинейных
уравнений Даламбера, Лиувилля, Борна–Инфельда, Монжа–Ампера
относительно конформной алгебры . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
В.I. Фущич, М.I. Сєров, В.I. Чопик, Умовна iнварiантнiсть та нелiнiйнi
рiвняння теплопровiдностi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
В.И. Фущич, И.А. Егорченко, О симметрийных свойствах
комплексно-значных нелинейных волновых уравнений . . . . . . . . . . . . . . . . . . . . 350
W.I. Fushchych, R.Z. Zhdanov, On the reduction and some new exact
solutions of the non-linear Dirac and Dirac–Klein–Gordon equations . . . . . . . . . 356
W.I. Fushchych, R.Z. Zhdanov, Non-local ans?tze for the Dirac equation . . . . . . . 361
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