Exact plane-symmetric solutions of the nonlinear equations of a spinor field in the theory of gravitation

Abstract

We obtain exact plane-symmetric solutions of the spinor field equations with a nonlinear term that is an arbitrary functions of the invariant\(S = \bar \psi \psi \) and with the self-gravitational field taken into account. Conditions are formulated for which the initial system of Einstein's equation and the spinor field equations with a power-law nonlinearity have regular solutions with localized (negative) spinor field energy density: so-called soliton-like solutions. Exact solutions of the spinor field equations are also obtained in flat space—time in this case and it is shown that the initial system of equations does not have soliton-like solutions. Hence the self-gravitational field plays a crucial (regularizing) in soliton-like solutions of the nonlinear spinor field equations.