Variational approximations to nonspherical particle-like solutions of ΔΦ = Φ – Φ3

Abstract

Variational approximations to nonspherical neutral particle-like eigensolutions of ΔΦ = Φ – Φ3 are obtained. Trial functions of the type [Formula: see text] (cos θ) are used for axially symmetrical solutions. The coupled Euler–Lagrange equations, obtained by extremizing the associated energy–momentum tensor component with respect to the Rl's, are solved numerically for N = 1, 2, and 3 when the Rl's are nodeless and for N = 1 and 2 when they have a single node each. Integrals are computed which, for exact solutions, are equal to the extremized energy, but which are not generally simultaneously extremized by the variational approximations. The results are compared with variational calculations of Betts, Schiff, and Strickfadden, who made use of trial functions with adjustable constant parameters rather than functions. Results indicate that no axially symmetric single-particle solution exists which is asymptotically proportional to P1(cos θ).