The stability of particle-like solutions of some non-linear field equations

Abstract

The object of this thesis is to examine the stability of particle-like solutions of the nonlinear field equation ▽²Ψ - 1/c² δ²Ψ/δt² = K²Ψ –μ² ΨΨ*Ψ+λ(ΨΨ*)²Ψ

with the particular form of time-dependence

Ψ = φ (r) e ⁻ ˡʷᵗ Initially our interest is concentrated on the case λ = 0. We begin the analysis by finding spherically symmetric particle-like solutions, and then examining the stability of the lowest-order solution by first- order perturbation theory. Direct perturbation methods are then considered. This solution is found to be highly unstable whether it is time-independent (ω = 0) or not (ω ≠ 0). The more general case λ ≠ 0 is next discussed. Particle-like solutions are found to exist in this case for -∞ < λ (K² - w²/c²) μ⁴ < 3/16 On examining the stability of the lowest-order solutions of this generalised field equation, it is found that for correct choice of the field parameters stable time-dependent solutions can exist, some of which can also have the attractive feature that their energy density is positive definite. We conclude by considering some methods of extending the theory.