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. 101
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2
(1 ? u? u? )2u + uµ u? uµ? = 0. (1989)

In the brackets we indicated the years when the conditional symmetry of the
corresponding equation had been investigated.

1. Fushchych W.I., On symmetry and partial solutions of some multi-dimensional equations of ma-
thematical physics, in Theoretical-algebraic methods in problems of mathematical physics, Kiev,
Institute of Mathematics of Ukr. Acad. Sci., 1983, 4–23.
2. Fushchych W.I., How to expand symmetry of differential equations?, in Symmetry and solutions
of nonlinear equations of mathematical physics, Kiev, Institute of Mathematics of Ukr. Acad. Sci.,
1987, 4–16.
3. Fushchych W.I., On symmetry and exact solutions of many-dimensional nonlinear wave equations,
Ukr. Math. J., 1987, 39, 1, 116–123.
4. Fushchych W.I., Tsifra I.M., On a reduction and solutions nonlinear wave equations with broken
symmetry, J. Phys. A, 1987, 20, L45–L48.
5. Fushchych W.I., Nikitin A.G., Symmetries of Maxwell’s equations, Dordrecht, Reidel Publ., 1987,
212 p.
6. Fushchych W.I., Zhdanov R., On some new exact solutions of nonlinear d’Alembert and Hamilton
equations, Preprint N 468, Minneapolis, Inst. for Math. and its Appl., Univ. of Minnesota, 1988.
7. Fushchych W.I., Serov M.I., Chopyk V.I., Conditional invariance and nonlinear heat equations, Dokl.
Ukr. Acad. Sci., Ser. A, 1988, 9, 17–21.
8. Fushchych W.I., Serov M.I., Conditional symmetry and exact solutions of the nonlinear acoustics
equation, Dokl. Ukr. Acad. Sci., Ser. A, 1988, 10, 27–31.
9. Fushchych W.I., Zhdanov R.Z., Symmetry and exact solutions nonlinear spinor equations, Phys.
Rep., 1989, 172, 4, 123–174.
10. Fushchych W.I., Serov M.I., Conditional invariance and solutions of the Boussinesq equation, in
Symmetry and solutions of equations of mathematical physics, Kiev, Institute of Mathematics of
Ukr. Acad. Sci., 1989, 95–102.
11. Bluman G., Cole J., The general similarity solution of the heat equation, J. Math. Mech., 1969, 18,
1025–1042.
12. Olver P., Rosenau Ph., The construction of special solutions to partial differential equations, Phys.
Lett. A, 1986, 112, 3, 107–112.
13. Clarkson P., Kruskal M., New similarity reductions of the Boussinesq equation, J. Math. Phys.,
1989, 30, 10, 2201–2213.
14. Levi D., Winternitz P., Nonclassical symmetry reduction: example of the Boussinesq equation, J.
Phys. A, 1989, 22, 2915–2924.
15. Shul’ga M.W., Symmetry and some partial solutions of the d’Atembert equation with a non-
linear condition, in Group-theoretical studies of equations of mathematical physics, Kiev, Inst. of
Mathematics Ukr. Acad. Sci., 1985, 34–38.
16. Fushchych W.I., Zhdanov R.Z., Revenko I.V., Compatibility and solutions of nonlinear d’Alembert
and Hamilton equations, Preprint N 90.39, Kiev, Inst. of Mathematics Ukr. Acad. Sci., 1990, 67 p.
Conditional symmetry of equations of nonlinear mathematical physics 431

17. Fushchych W.I., Serov M.I., On some exact solutions of the three-dimensional Schr?dinger equation,
o
J. Phys. A, 1987, 20, L929–L933.
18. Fushchych W.I., Chopyk V.I., Conditional invariance of the nonlinear Schr?dinger equations, Dokl.
o
Ukr. Acad. Sci., Ser. A, 1990, 4, 30–33.
19. Fushchych W.I., Serov M.I., Conditional invariance and reduction of the nonlinear heat equation,
Dokl. Ukr. Acad. Sci., Ser. A, 1990, 7, 24–28.
20. Fushchych W.I., Serov M.I., Amerov T.K., Conditional invariance of the heat equation, Dokl. Ukr.
Acad. Sci., Ser. A, 1990, 11, 16–21.
21. Fushchych W.I., On one generalization of S. Lie method, in Theoretical-algebraic analysis of equati-
ons of mathematical physics, Kiev, Institute of Mathematics of Ukr. Acad. Sci., 4–9.
22. Fushchych W.I., Shtelen W.M., Serov M.I., Symmetry analysis and exact solutions of the equations
of mathematical physics, Kiev, Naukova Dumka, 1989, 336 p. (to be published in Kluwer Publishers
in 1993).
23. Fushchych W.I., Serov M.I., Amerov T.K., On conditional symmetry of the generalized Korteweg-de
Vries equation, Dokl. Ukr. Acad. Sci., Ser. A, 1991, 12, 28–30.
24. Ames W.F., Lohner R.I., Adams E., Group properties of utt = (f (u)ux )x , Intern. J. Nonlinear
Mech., 1981, 16, 5/6, 439–447.
25. Fushchych W.I., Serov M.I., Repeta V.K., Conditional symmetry, reduction and exact solutions of
nonlinear wave equation, Dokl. Ukr. Acad. Sci., Ser. A, 1991, 5, 29–34.
26. Fushchych W.I., Chopyk V.I., Myronyuk P.I., Conditional invariance and exact solutions of the
three-dimensional nonlinear acoustics, Dokl. Ukr. Acad. Sci., Ser. A, 1990, 9, 25–28.
27. Fushchych W., Serov M., Amerov T., Conditional invariance and exact solutions of gas dynamics
equations, Dokl. Ukr. Acad. Sci., Ser. A, 1992, 5, 35–40.
28. Fushchych W., Serov M., Vorobyeva A., Conditional symmetry and exact solutions of equations of
nonstationary filtration, Dokl. Ukr. Acad. Sci., Ser. A, 1992, 6, 20–24.
29. Fushchych W., Myronyuk P., Conditional symmetry and exact solutions of the nonlinear acoustics
equations, Dokl. Ukr. Acad. Sci., Ser. A, 1991, 6, 23–29.
W.I. Fushchych, Scientific Works 2002, Vol. 4, 432–435.

New nonlinear equations for electromagnetic
field having velocity different from c
W.I. FUSHCHYCH
,
, c.
, .

1. The Maxwell equations
?D ?B
= c rot H ? j, = ?c rot E, div D = ?, div H = 0. (1)
dt ?t
play a basic role in modern electromagnetic theory. When considered in vacuum,
Eqs. (1) take the form
?E ?H
= ?c rot E, (2)
= c rot H, div E = 0, div H = 0.
?t ?t
Provided D = ?H, J = ? E, B = µH, ?, ?, µ being constants, from (1) it follows
that the wave equations hold
?2E ?2E ?2H
1 ?H
? c2 ?E + ?µ = ? ??, ?µ 2 ? c2 ?H + ?µ (3)
?µ = 0.
?t2 ?t ? ?t ?t
When considered in vacuum (? = µ = 1, ? = 0) Eqs. (3) read
?2E ?2H
? c ?E = 0, ? c2 ?H = 0.
2
(4)
2
?t ?t
It is a generally accepted axiom of the modern theory of elementary interactions
(classical and quantum) that the quantity in (1)–(4) is identified with the velocity of
light. That is the fundamental constant.
There are few works devoted to study of nonlinear generalizations of equations
(1)–(3) (see, e.g., lists of references in [1–3]).
In the present paper we suggest new nonlinear generalization of Eqs. (1)–(4) based
on the following idea: the velocity of light may not coincide with the constant c.
2. Let us admit following Poyting (1884) the standard definition of the energy
density and of the electromagnetic flow
12
(E + H 2 ), ?vk = c?kln El Hn , k, l, n = 1, 3, (5)
?=
2
v = (v1 , v2 , v3 ) is the velocity of the electromagnetic flow.
It is easy to see of that the formula
1
v 2 = c2 1 ? ??2 (E 2 ? H 2 )2 ? ??2 (E H)2 (6)
4
holds.
, 1992, 4, . 24–27.
New nonlinear equations for electromagnetic field 433

From (6) it follows that v 2 ? c2 and what is more v 2 = c2 ? E 2 ? H 2 = 0,
E H = 0.
Let us make in Eqs. (1)–(4) the change

c > v, c2 > v 2 .

This change yields nonlinear equations for electromagnetic field. For example, Eqs. (2)
take the form
?E ?H
= ?v rot H, (7)
= v rot H, div E = 0, div H = 0.
?t ?t
The above equations can be generalized in the following way:

?E ?H
= rot (H ? v), = rot (v ? E). (8)
?t ?t
Eqs. (7), (8) can be interpreted as equations of motion for an electromagnetic field
which spreads with velocity v. Provided v is determined by (5), (6), the velocity of
electromagnetic field is smaller than c.
One can impose on v = v(t, x) equations of hydrodynamics type
? ?
?? (9)
??v = 0, + ?vk
?t ?vk
or

??2 v = 0, (10)

whence
?E ?H
= rot (H ? v), = rot (v ? E),
?t ?t (11)
??v = 0, div E = 0, div H = 0.

Thus system (11) describes woth the electromagnetic field and its velocity.
Note 1. Eq. (9) possesses unique symmetry properties. Is is invariant under the
Poincar? and Galilei groups [3]. That is Eq. (9) satisfies. both the Lorentz–Poincar?–
e e
Einstein and Galilei relativity principles.
In addition, we adduce another nonlinear equation

?E ?E
+ ?1 (E 2 ? H 2 , E H)Hk + ?2 (E 2 ? H 2 , E H) rot H = 0,
?t ?xk
?E ?H
+ ?3 (E 2 ? H 2 , E H)Ek + ?4 (E 2 ? H 2 , E H) rot E = 0,
?t ?xk
where ?1 , ?2 , ?3 , ?4 are some smooth functions.
Eqs. (1), (3), (4) are generalized in an analogous way. For example, Eq. (4) is
generalized in a way

?2E ?2H
? v 2 ?E = 0, ? v 2 ?H = 0. (12)
2 2
?t ?t
434 W.I. Fushchych

or

?2E ?2H
? ?E ? ?E
? ?
cln (v 2 ) cln (v 2 ) (13)
= 0, = 0,
?t2 ?t2
?xl ?xn ?xl ?xn

or
? ?
cµ? (v 2 ) (14)
F?? = 0, µ, ?, ?, ? = 0, 3,
?xµ ?x?

where clm (v 2 ), cµ? (v 2 ) are smooth functions on v 2 . One can impose on the scalar
function v the eikonal equation
2 2
?v ?v
? ? = 0, ±1. (15)
= ?,
?t ?xk

Note 2. In Eqs. (12)–(14) the vector v can be defined according to the formula (6).
3. Let us turn to the generalization of the linear d’Alembert–Klein–Gordon–Fock
equation
2
2? u
? = (? 2 c2 ? + m2 c4 )u. (16)
?t2
After the change c > v(t, x) it takes the form
2
2? u
? = (? 2 v 2 ? + m2 c4 )u, (17)
?t2
where v is determined by (6), functions E, H satisfying Eqs. (1)–(4) or Eqs. (7)–(8).
Note 3. The vector of velocity of spread of the scalar field u can be defined in the
following way:

?u?
?u
vk = ?(|u|) u? (18)
+u
?xk ?xk
or
?u?
?u
vµ = ?(|u|) u? (19)
+u ,
?xµ ?xµ

where ?(|u|) is an arbitrary smooth function.
4. The Dirac equation for the spinor field

(20)
(?i ?µ ?µ + mc)? = 0

is rewritten in the form of nonlinear system
1/2
2 2 2
(21)
(?i ?µ ?µ + mv)? = 0, v = v 1 + v2 + v3 ,

?vµ
or va (22)
??vk = 0 = 0.
?xa
New nonlinear equations for electromagnetic field 435

Note 4. The vector of velocity of spread of the spinor field can be defined by the
formula (6). In this case, E, H are vectors characterizing elecromagnetic field which
is generated by the spinor field
?
Fµ? = ?1 ?(?µ ?? ? ?? ?µ )?, (23)

?1 is some small parameter.
Note 5. The vector of velocity of the spinor field can be defined as follows
?
vk = ?2 ??k ?

or
?
vµ = ?3 ??µ ?.

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. 101
( 135 .)



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