Métodos espectrais aplicados à dinâmica de defeitos topológicos e óscilons

Abstract

In this work, we solve the Klein-Gordon equation numerically using spectral methods to describe the dynamics of kinks (1 + 1) and oscillons (3 + 1). We remark that kinks and oscillons appear in distinct areas such as particle physics, condensed matter, cosmology, optics, etc. We implemented efficient codes based on spectral methods taking advantage of multiprocessing resources through matrices. To be more specific, we have studied the interaction of the so-called two-kinks described by the potential Vp(φ) = 1/2 φ² (φ⁻ᵖ⁻¹−φᵖ⁻¹)², with p = 1, 3, 5, .... Also, we have studied the interaction of kinks belonging to the potential Vn(φ) = 1/2 (1 − φ²ⁿ)², n = 2, 3, 4, ... These topological defects are of compactlike kinks. In the last part of the thesis, we have considered the dynamics of oscillons; in other words, we have integrated the Klein-Gordon equation numerically in 3+1 dimensions. We first recovered the known results of the case of the φ⁴ model, and in the sequence, we explore oscillons described by the potentials cited previously.