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?

? = [cos(g(?)) ? i(??) sin(g(?))]?, (4.6)

??µ ? = ?µ cos(2g(?)) + cµ sin(2g(?)), (4.7)
?

i
?µ = ?? µ ?, cµ = ?[(??), ? µ ]?. (4.8)
? ?
2
Clearly, ?? = 0. Equation (4.3) together with (4.6), (4.7), (4.8), can be written as
1 1
g = ?1 z 4 + ?2 (??) 3 ,
? ?
?? f?µ? = e(?µ cos(2g) + cµ sin(2g)), (4.9)
µ?
?
?
?? f = 0.

We now seek solutions of (4.9) of the form
g(?) = ??,
(4.10)
f µ? = ?[(?µ ? ? ? ?? ? µ ) sin(2??) ? (cµ ? ? ? c? ? µ ) cos(2??)],

where ?, ? are constants. Without loss of generality, we assume ?2 = c2 = ?1, since
we have ? 2 = 1, ?? = ?c = ?c = 0. With these conventions, (4.9) and (4.10) give
v 1
(4.11)
? = ?1 ? + ?2 (??) 3 , e = 2??.
?

Let us now consider solutions of (4.11). The first case is when ?1 = 0, ?2 = 0. Then
1
2
e?2 3
3
e 1
(4.12)
?= , ?= .
2?1 2
The second case is ?1 = 0, ?2 = 0, which gives
e
1
(4.13)
? = ?2 (??) 3 ,
? ?= .
1
2?2 (??) 3
?
Finally, when ?1 = 0, ?2 = 0 equation (4.12) becomes the cubic equation
e
y 3 + py + q = 0, (4.14)
?= ,
2?
98 W.I. Fushchych, W. Shtelen, P. Basarab-Horwath

where
v e?2 ?2 (??)
q q ?
?+ ?? ?2
1
3 3
y= ?= Q+ Q, Q= .
2 2 8 27
In this way we obtain exact solutions of the system (4.1), (4.2) in the following form
(4.15)
?(x) = exp(?i?(??)?)?, ? = ?x,
e
[(?µ ? ? ? ?? ? µ ) sin(2??) ? (cµ ? ? ? c? ? µ ) cos(2??)],
F µ? = (4.16)
2?
i
?µ = ?? µ ?, cµ = ?[(??), ? µ ]?,
? ?
2
?2 = c2 = ?1,
? 2 = 1, ?? = ?c = ?c = 0.

For conformally invariant solutions of (4.1) we exploit the ansatzes [6, 7]
?x ?x
?(?), ? = 2 , ? 2 = 1,
?(x) =
(x2 )2 x
(4.17)
f (?) 2x? [x f (?) ? x? f ?? (?)]
µ? µ ??
?
µ?
F = .
x2 (x2 )3
Combining (4.1) and (4.17) yields the system of ordinary differential equations
1 1
?i(??)? = ?1 z 4 + ?2 (??) 3 ,
? ?
?? f?µ? = ?e?? µ ?,
? (4.18)
µ?
?
?
?? f =0

with z = ? 1 fµ? f µ? , which is formally similar to (4.3). Using this fact, we can write
2
down the following solutions of (4.1), (4.17):
?x ?x
(4.19)
?(x) = exp(i?(??)?)?, ?= ,
(x2 )2 x2
e
? µ ?? ? ? µ ?? ) + 2(?µ x? ? ?? xµ )? +
F µ? = 2 )2
2?(x
?x
+ 2 2 (xµ ? ? ? x? ? µ ) sin(2??) + (cµ ? ? ? c? ? µ ) + (4.20)
x
cx
+ 2?(xµ c? ? x? cµ ) ? 2 2 (xµ ? ? ? x? ? µ ) cos(2??) ,
x
where
i
?µ = ?? µ ?, cµ = ?[(??), ? µ ]?,
? ?
2
?2 = c2 = ?1, ?? = ?c = ?c = 0.
? 2 = 1,
The solutions found show that the system (1.1), (1.2) is consistent, at least in
certain cases of the mass function. Furthermore, we can calculate the mass corres-
ponding to these solutions:
?
1
?1
m = ?1 u 4 + ?2 (??) 3 = 2 .
x
A new conformal-invariant non-linear spinor equation 99

5. Conclusion
We have shown that there exists a consistent non-linear dynamical model for
a classical spinor particle, in which the mass is generated by an electromagnetic
field and a spinor field, which the particle itself creates. The proposed model (3.1)
is conformally-invariant, as is the class of solutions we obtain, For these solutions,
we have also found an explicit form for the Lorentz-invariant mass. The question of
quantizing the model (3.1) will be taken up in future papers.

1. Yukawa H. (ed.), Nonlinear field theory, Progress of Theoretical Physics, Supplements, 1959, V.9,
1–128.
2. Heisenberg W., Introduction to the unified theory of elementary particles. Collected works, Series B,
Berlin, Springer-Verlag, 1984.
3. Bedrij O., Fushchych W., On the electromagnetic structure of elementary particle masses, Procee-
dings of the Ukrainian Academy of Sciences, 1991, № 2, 38–40.
4. Fushchych W., Shtelen W., Serov M., Symmetry and exact solutions of equationsof nonlinear
mathematical physics, Dordrecht, Kluwer, 1993.
5. Fushchych W., Nikitin A., Symmetries of Maxwell’s equations, Dordrecht, D. Reidel, 1987.
6. Fushchych W., Shtelen W., On some exact solutions of the nonlinear Dirac equation, J. Phys. A,
1983, 16, 271–277.
7. Fushchych W., Shtelen W., On some exact solutions of the nonlinear equations of quantum electro-
dynamics, Phys. Lett. B, 1983, V.128, 215–217.
8. Olver P., Applications of Lie groups to differential equations, New York, Springer, 1986.
9. Ovsjannikov L.V., Group analysis of differential equations, New York, Academic Press, 1982.
10. Fushchych W., Zhdanov R., Symmetry and exact solutions of non-linear spinor equations, Physics
Reports, 1989, V.172, 123–174.
11. Fushchych W., Zhdanov R., Non-linear spinor equations: symmetry and exact solutions, Kiev,
Naukova Dumka, 1992 (in Russian).
W.I. Fushchych, Scientific Works 2003, Vol. 5, 100–104.

Generation of solutions for nonlinear
?
equations via the Euler–Ampere
transformation
W.I. FUSHCHYCH, V.A. TYCHYNIN

За допомогою контактного перетворення Ейлера–Ампера одержано формули роз-
множення розв’язкiв. Побудовано класи ДРЧП другого порядку, якi iнварiантнi вiд-
носно цього перетворення.

The invariance of DE under a nonlocal transformation of variables allows us to
generate its solutions from the known ones. The reducing of a nonlinear PDE to
a linear equation makes it possible to construct for it the formula of a nonlinear
superposition of solutions. In the present paper the solutions generating formulae
are obtained via the Euler–Amper? contact transformation. Classes of Euler–Amper?
e e
invariant PDEs are constructed. The efficiency of the obtained formulae is illustrated
in several of examples.
1. Nonlocal invariance and the solutions generating formula. Let us consider the
Euler–Amper? transformation in the space R(1, n?1) of n independent variables [1, 2]:
e

u = ya va ? v, x0 = y0 , xa = va ,
(1)
?v
, a, b = 1, n ? 1, ? ? det(vab ) = 0, µ, ? = 1, n ? 1.
vµ = ? µ v =
?yµ

The first and second order derivatives are changing as

u0 = ?v0 , ua = ya ,
v00 = ?det?1 (vab ) det(vµ? ), u0a = ?det?1 (vab )v0b aba (vcd ), (2)
?1
uab = ?det (vcd )aab (vcd ) (a, b, c, d = 1, n ? 1).

Hereafter the summation over repeated Greek indices is understood in the space
R(1, n ? 1) with the metric gµ? = diag (1, ?1, . . . , ?1) and over repeated Latin indices
it is understood in the space R(0, n ? 1) with the metric gab = diag (1, 1, . . . , 1),
?2u
uµ? = ?µ? u = ?xµ ?x? , det(uab ) = a00 (uab ). a?? (uµ? ), aab (ucd ) are the cofactors to
the elements u?? and uab respectively, ?, ? = 0, n ? 1.
Following expressions are absolute differential invariants of order ? 2 with respect
to (1) due to its involutivity:

f 2 (x0 , ?u0 ), f 3 (u, xa ua ? u),
f 0 (x0 ), f 1 (xa , ua ),
f 4 (u00 , ?det?1 (uab ) det(uµ? )), f 5 (u0a , ?det?1 (uab )u0b aba (ucd )), (3)
?1
f 6 (uab , ?det (ucd )aab (ucd )).

Доповiдi АН України, 1993, № 7, С. 40–45.
Solutions for nonlinear equations via the Euler–Amper? transformation
e 101

Here f 0 is an arbitrary smooth function, and f k , k = 1, 6 are arbitrary smooth and
symmetric on arguments functions:
f k (x, z) = f k (z, x).
Let us construct by means of the expressions (3) the absolutely invariant under
transformation (1) second order PDE
F ({f ? }) = 0 (4)
(? = 0, 6).
F (·) is an arbitrary smooth function. Such equations are contained in the class (4):
u0 ? ua ua + x2 = 0, x2 = xa xa ; (5.1)

?u0 ? ?u ? det?1 (ucd ) Slid (ucd ) = 0; (5.2)

u00 ? det?1 (ucd ) det(uµ? ) = 0; (5.3)

?u0 ? detm (ucd ) + (?1)m·n det?m·n (ucd ) det[aab (ucd )] = 0; (5.4)

?u2h + ?(xc )ua ua + ?(uc )x2 = 0; (5.5)
0

?u2h + ?(uc )ua ua + ?(xc )x2 = 0; (5.6)
0

?u0 ? ?(xc , u)? ? det?1 (ucd )?(uc , xa ua ? u) · Slid (ucd ) = 0. (5.7)

One can continue this list of equations (5) in the obvious manner. ? is the Laplacian,
def
Slid (ucd ) = gab aab (ucd ),
?(x, z) is an arbitrary smooth function, ? is an arbitrary parameter, m, h are real
numbers.
(1)
Let u (x0 , x) be a known partial solution of Eq. (4). For constructing new solution
(2)
u (x0 , x) of this Eq. (4) we rewrite the formula (1) in parametric form, replacing xa
(1) (1)
for parameters ? a , a = 1, n ? 1. Substitute u (x0 , ? ), u a (x0 , ? ) to (1). So, as a result,
we obtain the formula
(2) (1) (1) (1)
u (x0 , x) = ? a u a (x0 , ? ) ? u (x0 , ? ) = ? a xa ? u (x0 , ? ),
(6)
(1)
a = 1, n ? 1.
xa = u a (x0 , ? ),

Here x = (x1 , x2 , . . . , xn?1 ), ? = (? 1 , ? 2 , . . . , ? n?1 ). The formula (6) allows us to
construct efficiently the new solutions of nonlinear equations (5) by resolving the last
system (6) with respect to parameters ? .
Example 1. Let us consider the equation
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