Exact self-consistent plane-symmetric solutions of the spinor-field equation with a nonlinear term dependent on the invariant p2

Exact self-consistent plane-symmetric solutions of the spinor-field equation with zero mass parameter and a nonlinear term that is an arbitrary function of the invariant P2 = (iψ̄γ5ψ)2. are obtained in gravitation theory. An equation with power-law nonlinearity in which the nonlinear term in the spinor-field Lagrangian has the form LN = λP2n, where λ is the nonlinearity parameter and n = const, is investigated in detail. It is shown that if λ = -Λ2 < 0, n > 1, the original system of Einstein and nonlinear spinor-field equations has regular solutions with a localized spinor-field energy density. Here the soliton-like configuration of the fields possesses a negative energy. Exact solutions are also obtained for the above spinor-field equation in flat spacetime, and it is demonstrated that there are no soliton-like solutions in that case. Thus it is established that the proper gravitational field plays a decisive, controlling role in the formation of soliton-type solutions of the above nonlinear spinor-field equation. © 1998 Plenum Publishing Corporation.