Symmetry reduction exact solutions

V. Symmetry Reduction and Exact Solutions of the Maxwell Equations... 

To the best of our knowledge, the first paper devoted to symmetry reduction of the 57/(2) Yang-Mills equations in Minkowski space has been published by Fushchych and Shtelen [27] (see also Ref. 21). They use two conformally invariant ansatzes in order to perform reduction of Eqs. (1) to systems of ordinary differential equations. Integrating the latter yields several exact solutions of Yang-Mills equations (1). 

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. 

In this section we apply the technique described above in order to perform in-depth analysis of the problems of symmetry reduction and construction of exact invariant solutions of the SU(2) Yang-Mills equations in the (l+3)-dimensional Minkowski space of independent variables. Since the general method to be used relies heavily on symmetry properties of the equations under study, we will briefly review the group-theoretic properties of the SU(2) 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. 

V. SYMMETRY REDUCTION AND EXACT SOLUTIONS OF THE MAXWELL EQUATIONS... 

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