Symmetry reduction 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. 

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 principal aim of the present chapter is twofold. First, we will review the already known ideas, methods, and results centered around the solution techniques that are based on the symmetry reduction method for the Yang-Mills equations (1) in Minkowski space. Second, we will describe the general reduction routine, developed by us in the 1990s, which enables the unified treatment of both the classical and nonclassical symmetry reduction approaches for an arbitrary relativistically invariant system of partial differential equations. As a byproduct, this approach yields exhaustive solution of the problem of... 

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. 

Thanks to Assertion 9, the problem of symmetry reduction of Yang-Mills equations by subalgebras of the algebrap(l, 3) reduces to routine substitution of the corresponding expressions for , > into (61). We give below the final forms of the coefficients (60) of the reduced system of ordinary differential equations (59) for each subalgebras of the algebra p(l, 3) ... 

So, combining symmetry reduction by the number of independent variables and direct reduction by the number of the components of the function to be found, we have reduced the SU(2) Yang-Mills equations (46) to comparatively simple systems of ordinary differential equations (80). 

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. 

We do not consider here the solution techniques based on the symmetry reduction of different versions of the self-dual Yang-Mills equations to integr-able models (we refer the interested reader to several papers [13-15],[22-24,65] for a detailed exposition of the results in this field available to date). 

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