where a · x stands for aµ xµ and ? is an arbitrary real constant.
it is studied in a great detail in .
where a dot over the symbol denotes differentiation with respect to ?.
= gµ? b? + g?? bµ ? 2gµ? b? , hµ?? = ? ?1 (gµ? b? ? gµ? b? ).
able coefficients. That is why there is a little hope to construct their general solutions.
Aµ = e1 kµ exp (??c · x ? ?w2 ) ??2 C 2 + ??2 (C 4 ? ?2 e2 ?2 )1/2 ?
+ e2 ? ??2 C 2 + ??2 (C 4 ? ?2 e2 ?2 )1/2 ?
? sin 2|?|(b · x + w1 ) bµ + kµ w1 .
+ c · x sin w1 + w2 )2 + e2 ? (b · x cos w1 + c · x sin w1 + w2 ) ?
? cµ cos w1 ? bµ sin w1 + 2kµ [(1/4)(?eT + T )(b · x sin w1 ?
Aµ = e1 kµ e?T C1 cosh[e?(b · x cos w1 + c · x sin w1 + w2 )] + C2 sinh[e? ?
+ e2 ? C 2 (b · x cos w1 + c · x sin w1 + w2 )2 + ?2 e2 C ?2 ?
Aµ = e1 kµ Z0 (ie?/2)[(b · x + w1 )2 + (c · x + w2 )2 ] + e2 ? cµ (b · x + w1 ) ?
+ (c · x + w2 )2 ]?e?/2 + e2 ?[(b · x + w1 )2 + (c · x + w2 )2 ]?1 ?
? cµ (b · x + w1 ) ? bµ (c · x + w2 ) ? (1/2)kµ [w1 (c · x + w2 ) ?
? w2 (b · x + w1 )] .
where Js , Ys are the Bessel functions.
Thus, we have obtained broad families of exact non-Abelian solutions of YMEs (1).
are not self-dual, i.e. that they do not satisfy self-dual YMEs.
Let us say a few words about symmetry interpretation of the ans?tze (2), (3), (10).
Q3 = c? ?? ? 2[w1 (kµ b? ? k? bµ ) + w3 (kµ c? ? k? cµ )] Aa? ?Aaµ .
+ 2 w0 (kµ b? ? k? bµ ) + w2 (kµ c? ? k? cµ ) Q1 A? ?
? ? µ (w0 b? + w2 c? )Q1 A? ? k? (w0 Q2 A? + w2 Q3 A? ) ?
+ w2 (w1 b? + w3 c? ) Q1 A? + e (w0 b? + w2 c? )Q1 A? ?
? k? (w0 Q2 A? + w2 Q3 A? ) ? Aµ + 2e(w0 b? A? + w2 c? A? )?Q1 Aµ ?
? 2ek? A? ? (w0 Q2 Aµ + w2 Q3 Aµ ) + eA? ? (w0 bµ + w2 cµ )Q1 A? ?
? ekµ A? ? (w0 Q2 A? + w2 Q3 A? ).
Q3 Aµ ? c? ?? Aµ + 2 w1 (kµ b? ? k? bµ ) + w3 (kµ c? ? k? cµ ) A?
and by the symbol Q2 we denote the second prolongation of the operator Q2 .
theoretical physics equations possessing non-trivial conditional symmetry .
(10b), (10c) and expressions (8h), (8p) are particular cases of the expressions (10d).