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Here C is an arbitrary smooth function.
It is important to note that the formulae (91) under C ? const give the already
known solutions (see, e.g. [8, 11]).

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mathematical physics equations, Kiev, Naukova Dumka, 1989, 336 p.
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Phys. Repts., 1989, 172, № 4, 123–143.
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W.I. Fushchych, Scientific Works 2002, Vol. 4, 557–561.

On the new exact solutions of the nonlinear
Maxwell–Born–Infeld equations
W.I. FUSHCHYCH, R.Z. ZHDANOV, V.F. SMALIJ
Предложены две нелокальные подстановки, сводящие уравнения Максвелла с мате-
риальными уравнениями Борца–Инфельда к скалярным уравнениям. С использова-
нием этих подстановок построены многопараметрические семейства точных решений
нелинейных уравнений Максвелла–Борна–Инфельда.

In the present work we obtain broad classes of exact solutions of the Maxwell
equations
?t B = ?rot E, (1a)
?t D = rot H,

(1a)
div D = 0, div B = 0,

with constitutive equations suggested by Born and Infeld [1] in 1934
H = ? B ? ?1 E,
D = ? E + ?1 B,
(1c)
?1/2
? = 1 + B 2 ? E 2 ? (B E)2 , ?1 = (B E)?.

Here Ea , Ha , Ba , Da are smooth functions on t ? x0 ? R1 , x ? R3 , a = 1, 3.
Symmetry properties of equations (1a–c) are investigated in [2] but we do not
apply symmetry reduction procedure to construct their exact solutions. Our approach
generalizes that of papers [1, 3] and is based on the fact that the general solution of
equations (1a,b) can be represented in the form
E = ??t U , (2)
B = rot U , D = rot W , H = ?t W ,
where U = (U1 , U2 , U3 ), W = (W1 , W2 , W3 ) are arbitrary smooth vector functions.
Substitution of formulas (2) into (1c) gives rise to the first-order system of partial
differential equations (PDE)

rot W = ?? ?t U + ((?t U )(rot U )) rot U ,
(3)
?t W = ? rot U + ((?t U )(rot U ))?t U ,
?1/2
where ? = 1 + (rot U ) ? (?t U ) ? ((?t U )(rot U ))
2 2 2
.
To obtain exact solutions of system (3) we use the ansatz
(4)
U = a?(?0 , ?1 , ?2 ),

where ?0 = t, ?1 = b x, ?2 = c x; ? ? C 2 (R3 , R1 ); a, b, c are arbitrary constant
vectors in the space R3 satisfying conditions
a 2 = b 2 = c 2 = 1, a b = b c = c a = 0.
Доклады АН Украины, 1992, № 10, С. 28–33.
558 W.I. Fushchych, R.Z. Zhdanov, V.F. Smalij

Since rot U = ?c ??1 + b ??2 , the relation (?t U )rot U = 0 holds. Consequently,
?? ??

system (3) is rewritten in the form

?? ?? ??
rot W = ?? a ?c (5)
, ?t W = ? +b ,
??0 ??1 ??2
?1/2
where ? = 1 ? ?2 + ?2 + ?2 , ?µ ? ??/??µ .
0 1 2
Since ?t rot W = rot ?t W , the equality

?? ??
?c
?t (?? a?0 ) = rot ? +b
??1 ??2

holds. The above equation after making some manipulations takes the form

a(1 ? ?µ ?µ )?3/2 [(1 ? ?µ ?µ )2? + ?µ? ?µ ?? ] = 0.
2
Here ?µ? ? ??µ ??? ; µ, ? = 0, 2, the summation over the repeated indices in the
??

Minkowski space R(1, 2) with the metric tensor gµ? = diag (1, ?1, ?1) is implied.
Thus provided the function ?(?) is a solution of the nonlinear PDE

(1 ? ?µ ?µ )2? + ?µ? ?µ ?? = 0, (6)

and what is more 1 ? ?µ ?µ = 0, formulas (2), (4), (5) give particular solutions of
system of nonlinear PDE (1).
We look for a solution of equation (6) in the form

(7)
? = ?(y1 , y2 ),

where y1 = ?0 + ?1 , y2 = ?2 .
Substitution of (7) into (6) gives rise to the following equation for ?:

?2?
(8)
2 = 0,
?y2

whence ? = h1 (y1 )y2 + h2 (y1 ), hi ? C 2 (R1 , R1 ) being arbitrary functions. Substitu-
ting the obtained expressions into (7) we get the class of exact solutions of nonlinear
PDE (6) containing two arbitrary functions on ?0 + ?1 ? t + bx

? = ?2 h1 (?0 + ?1 ) + h2 (?0 + ?1 ).

Hence by using formulas (2), (4), (5) we obtain a family of exact solutions of the
nonlinear Maxwell–Born–Infeld equations

H = (1 + h2 )?1/2 [bh1 ? c(h1 c x + h2 )],
? ? ? ?
E = ?a(h1 c x + h2 ), 1
(9)
D = ?a(1 + h2 )?1/2 [h1 c x + h2 ], B = bh1 ? c(h1 c x + h2 ).
? ? ? ?
1

Similarly, using exact solutions of equation (6) constructed in [4, 5], that satisfy
the condition 1 ? ?µ ?µ = 0 we shall write down corresponding particular solutions
On the new exact solutions of the nonlinear Maxwell–Born–Infeld equations 559

of system (1) (as earlier, we shall use the designations ?0 = t, ?1 = bx, ?2 = cx,
?2 = ?0 ? ?1 ? ?2 ).
2 2 2

v
?2 1/2
c2
dy 1
1. ?(?0 , ?1 , ?2 ) = c1 , ?= 1+ ,
1 + c2 ? 4 ? c2
1 + c2 y 4
0 1

c1 ta c1 [?b(c x) + c(b x)]
E=? (10)
, B= ,
? 2 (1 + c2 ? 4 ) ? 2 (1 + c2 ? 4 )
c1 ta c1 [?b(c x) + c(b x)]
D=? , H= .
? 2 (1 + c2 ? 4 ? c2 ) ? 2 (1 + c2 ? 4 ? c2 )
1 1

1/2
?0 ? ?1
?(?0 , ?1 , ?2 ) = ±
2. th(c1 (?0 + ?1 ) + c2 ) + c3 ,
c1
? = ch(c1 (?0 + ?1 ) + c2 ),
1/2
a cth(c1 (t + b x) + c2 )
E=? ?
c1 (t ? b x)
4

2c1 (t ? b x) + sh 2(c1 (t + b x) + c2 )
? ,
ch2 (c1 (t + cx) + c2 )
1/2
(11)
c cth(c1 (t + b x) + c2 )
B=? ?
c1 (t ? b x)
4

2c1 (t ? b x) ? sh 2(c1 (t + b x) + c2 )
? ,
ch2 (c1 (t + bx) + c2 )
1/2
a 2{2c1 (t ? b x) + sh 2(c1 (t + b x) + c2 )}2
D=? ,
c1 (t ? b x) sh 2(c1 (t + b x) + c2 )
4
1/2
c 2{2c1 (t ? b x) ? sh 2(c1 (t + b x) + c2 )}2
H=? ,
c1 (t ? b x) sh 2(c1 (t + b x) + c2 )
4

1/2
2
c3 (?0 ??1 )
?(?0 , ?1 , ?2 ) = ± c2 e
3. + (?0 + ?1 ) + c1 ,
c3
1/2
2(?0 + ?1 ) + c2 c1 ec3 (?0 ??1 )
?= ,
2(?0 + ?1 ) ? c2 c1 ec3 (?0 ??1 )
2
+ c2 c3 ec3 (t?b x) (12)
a c3
E=? ,
1/2
2 2
c2 ec3 (t?c x) + c3 (t + b x)

? c2 c3 ec3 (t?b x)
2
c c3
B=? ,
1/2
2 2
ec3 (t?b x)
c2 + c3 (t + b x)
560 W.I. Fushchych, R.Z. Zhdanov, V.F. Smalij

? c2 c3 ec3 (t?b x)
2
a c3
H=? ,
1/2
2
?c2 2
ec3 (t?b x) + c3 (t + b x)
2
+ c2 c3 ec3 (t?b x)
a c3
D=? .
1/2
2
?c2 ec2 (t?b x) + 2
c3 (t + b x)

By direct check one can become convinced of the fact that solutions (9)–(12) are
such that the vectors E and D as well as B and H are parallel. Besides, the conditions

E B = DB = E H = DH = 0

hold.
Some other classes of exact solutions of system (3) are obtained by putting in (3)

(13)
rot U = 0, Utt = 0.

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