Exact self-consistent plane-symmetric solutions of the spinor-field equation with a nonlinear term that depends on the S2 - P2 invariant

Exact plane-symmetric solutions of the spinor-field equation with zero mass parameter and nonlinear term that depends arbitrarily on the S2 -P2 invariant are derived with consideration of an intrinsic gravitational field. The existence of regular solutions with localized energy density among the solutions obtained is investigated. Equations with powerlaw and polynomial nonlinearity types are examined in detail. For the power-law nonlinearity, when the nonlinear term entering into the Lagrangian has the form LN = λIn, where λ is the nonlinearity parameter and n = const, it is shown that the initial system of Einstein and spinor-field equations has regular solutions with localized energy density only under the condition λ = -A2 < 0, n > 1. In this case, the examined field configuration posesses a negative energy. In the case of polynomial nonlinearity, regular solutions with localized energy density T0 0(x), positive energy Ef = ∫+∞ -∞ T0 0 √-3gdx (upon integration over y and z between finite limits), and an everywhere regular metric that transforms into a two-dimensional space-time metric at spatial infinity are obtained. It is shown that the initial nonlinear spinor-field equations in two-dimensional space-time have no solutions with localized energy density. Thus, it is established that the intrinsic gravitational field plays a regularizing role in the formation of regular localized solutions to the examined nonlinear spinor-field equations. © 1999 Kluwer Academic/Plenum Publisher.