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

Abstract

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$P^2 = (i\bar \psi \gamma ^5 \psi )^2$, are obtained in gravitation theory. An equation with power-law nonlinearity in which the nonlinear term in the spinor-field Lagrangian has the form L_N=λP²ⁿ, where λ is the nonlinearity parameter and n=const, is investigated in detail. It is shown that 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.