On Conditional Symmetries of Multidimensional Nonlinear Equations of Quantum Field Theory

Below we give a brief account of results of studying conditional symmetries of multidimensional nonlinear wave, Dirac and Yang-Mills equations obtained in collaboration with W.I. Fushchych in 1989–1995. It should be noted that till our papers on exact solutions of the nonlinear Dirac equation [1]–[4], where both symmetry and conditional symmetry reductions were used to obtain its exact solutions, only two-dimensional (scalar) partial differential equations (PDEs) were studied (for more detail, see, [5]). The principal reason for this is the well-known fact that the determining equations for conditional symmetries are nonlinear (we recall that determining equations for obtaining Lie symmetries are linear). Thus, to find a conditional symmetry of a multidimensional PDE, one has to find a solution of the nonlinear system of partial differential equations whose dimension is higher that the dimension of the equation under study! In paper [3], we have suggested a powerful method enabling one to obtain wide classes of conditional symmetries of multidimensional Poincaré-invariant PDEs. Later on it was extended in order to be applicable to Galilei-invariant equations [6] which yields a number of conditionally-invariant exact solutions of the nonlinear Levi-Leblond spinor equations [7]. The modern exposition of the above-mentioned results can be found in monograph [8]. Historically, the first physically relevant example of conditional symmetry for a multidimensional PDE was obtained for the nonlinear Dirac equation. However, in this paper, we will concentrate on the nonlinear wave equation which is easier for understanding the basic techniques used to construct its conditional symmetries. As is well known, the maximal invariance group of the nonlinear wave equation