Dirac spinor waves and solitons in anisotropic Taub-NUT spaces

We apply a new general method of anholonomic frames with associated nonlinear connection structure to construct new classes of exact solutions of Einstein–Dirac equations in five-dimensional (5D) gravity. Such solutions are parametrized by off-diagonal metrics in coordinate (holonomic) bases or equivalently, by diagonal metrics given with respect to some anholonomic frames (pentads or funfbeins satisfying corresponding constraint relations). We consider two possibilities of generalization of the Taub-NUT metric in order to obtain vacuum solutions of 5D Einstein equations with effective renormalization of constants (by higher dimension anholonomic gravitational interactions) having distinguished anisotropies on an angular parameter or on an extra-dimensional coordinate. The constructions are extended to solutions describing self-consistent propagations of 3D Dirac wave packets in 5D anisotropic Taub-NUT spacetimes. We show that such anisotropic configurations of spinor matter can induce gravitational 3D solitons which are solutions of Kadomtsev–Petviashvili or sine–Gordon equations.