Symmetry groups, Yang-Mills equations

Note that the maximal symmetry groups admitted by the self-dual Yang-Mills equations (2) coincide with the symmetry groups of the corresponding 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. 

Let us also discuss briefly the discrete symmetries of equations (46). It is straightforward to check that the Yang-Mills equations admit the following groups of discrete transformations ... 

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