Conditional symmetry, Yang-Mills equation

Conditional Symmetry and New Solutions of the Yang-Mills Equations... 

One more important property of the self-dual Yang-Mills equations is that they are equivalent to the compatibility conditions of some overdetermined system of linear partial differential equations [11,12]. In other words, the selfdual Yang-Mills equations admit the Lax representation and, in this sense, are integrable. For this very reason it is possible to reduce Eq. (2) to the widely studied solitonic equations, such as the Euler-Amold, Burgers, and Devy-Stuardson equations [13,14] and Liouville and sine-Gordon equations [15] by use of the symmetry reduction method. 

The present review is based mainly on our publications [33,35-39,49-53]. In Section II we give a detailed description of the general reduction routine for an arbitrary relativistically invariant systems of partial differential equations. The results of Section II are used in Section III to solve the problem of symmetry reduction of Yang-Mills equations (1) by subgroups of the Poincare group P 1,3) and to construct their exact (non-Abelian) solutions. In Section IV we review the techniques for nonclassical reductions of the STJ 2) Yang-Mills equations, which are based on their conditional symmetry. These techniques enable us to obtain the principally new classes of exact solutions of (1), which are not derivable within the framework of the standard symmetry reduction technique. In Section V we give an overview of the known invariant solutions of the Maxwell equations and construct multiparameter families of new ones. 

IV. CONDITIONAL SYMMETRY AND NEW SOLUTIONS OF THE YANG-MILLS EQUATIONS... 

With all the wealth of exact solutions obtainable through Lie symmetries of the Yang-Mills equations, it is possible to construct solutions that cannot be derived by the symmetry reduction method. The source of these solutions is conditional or nonclassical symmetry of the Yang-Mills equations. 

As very cumbersome computations show, the ansatzes determined by formulas 2-4 from (91) also correspond to the conditional symmetry of Yang-Mills equations. Hence it follows, in particular, that Yang-Mills equations should be included in the long list of mathematical and theoretical physics equations possessing nontrivial conditional symmetry [21]. 

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