The stability of particle-like solutions of some non-linear field equations
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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.
Thesis, PhD Doctor of Philosophy
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