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Evolution and scattering of excited topological defects: Interaction between internal modes

D. Miguélez-Caballero

Abstract

This thesis presents an extensive analysis of the behavior of topological solitons when one or more of their internal modes are activated. The first part of this manuscript is devoted to the study of the simplest topological solitons in (1+1) dimensions: kinks. Specifically, we investigate how these solutions emit radiation when one of their internal modes is initially excited, within the framework of the double $φ^4$ model. The simplest kink solution in this theory exhibits a complex internal mode structure that depends on a coupling constant appearing in the potential governing the dynamics. We will show how the amplitude and frequency of the emitted radiation are affected by changes in this coupling constant. We also examine the dynamics of wobbling kink/antikink scattering when the kinks possess more than one internal mode. To this end, we study kink/antikink collisions in the context of the simplest kink solution arising in the MSTB model. This analysis sheds light on the resonant energy exchange mechanism, allowing energy transfer between internal modes and the translational mode. The second part of this thesis focuses on excited vortex solutions in (2+1) dimensions. We begin with a detailed study of the internal mode structure associated with vortex solutions in the Abelian-Higgs model. We demonstrate how the problem can be significantly simplified by choosing an appropriate angular dependence for the eigenfunctions. Furthermore, we investigate the radiation emitted by a vortex when its internal mode is initially activated. To achieve this, we extend the analytical techniques used in (1+1) dimensions to field theories defined in two spatial dimensions. This enables us to compute the radiation amplitude, its frequency, and the decay of the internal mode amplitude due to energy loss via radiation. All analytical results are contrasted with data from numerical simulations.

Evolution and scattering of excited topological defects: Interaction between internal modes

Abstract

This thesis presents an extensive analysis of the behavior of topological solitons when one or more of their internal modes are activated. The first part of this manuscript is devoted to the study of the simplest topological solitons in (1+1) dimensions: kinks. Specifically, we investigate how these solutions emit radiation when one of their internal modes is initially excited, within the framework of the double model. The simplest kink solution in this theory exhibits a complex internal mode structure that depends on a coupling constant appearing in the potential governing the dynamics. We will show how the amplitude and frequency of the emitted radiation are affected by changes in this coupling constant. We also examine the dynamics of wobbling kink/antikink scattering when the kinks possess more than one internal mode. To this end, we study kink/antikink collisions in the context of the simplest kink solution arising in the MSTB model. This analysis sheds light on the resonant energy exchange mechanism, allowing energy transfer between internal modes and the translational mode. The second part of this thesis focuses on excited vortex solutions in (2+1) dimensions. We begin with a detailed study of the internal mode structure associated with vortex solutions in the Abelian-Higgs model. We demonstrate how the problem can be significantly simplified by choosing an appropriate angular dependence for the eigenfunctions. Furthermore, we investigate the radiation emitted by a vortex when its internal mode is initially activated. To achieve this, we extend the analytical techniques used in (1+1) dimensions to field theories defined in two spatial dimensions. This enables us to compute the radiation amplitude, its frequency, and the decay of the internal mode amplitude due to energy loss via radiation. All analytical results are contrasted with data from numerical simulations.
Paper Structure (18 sections, 1 theorem, 67 equations, 13 figures)

This paper contains 18 sections, 1 theorem, 67 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Topological solitons: origin and history
  3. A brief history of kinks
  4. A brief history of vortices
  5. Objectives and structure
  6. Methodology
  7. Topological kinks
  8. Topological kinks: foundations and their dynamics
  9. Stability based on scaling arguments: Derrick's theorem
  10. BPS equations, vacuum structure, kink solutions and topological charge
  11. Internal mode structure
  12. Two simple field theories: the phi4 and phi6 models
  13. The phi4 model
  14. The phi6 model
  15. Wobbling kinks and radiation emission
  16. ...and 3 more sections

Key Result

Theorem 1

Consider the Lagrangian density eqI2:LagDens and suppose a finite energy field configuration $\phi$ that is a solution to the corresponding field equations and is not the vacuum. If $n > 1$, the function $E_\lambda$ has no minima, and therefore, there are no static solutions other than the vacuum.

Figures (13)

  • Figure 1: Kink-antikink velocity diagram for kinks in the $\phi^4$ model. The final velocity $v_f$ of the scattered kinks as a function of the initial velocity $v_0$ has been plotted. The color code shown in the graphs indicates the number of bounces suffered by the kink-antikink pair before moving appart. In initial velocity ranges where no final velocity is shown, a bion is assumed to form. The resonance window where various bounces can be observed has been expanded inside each graphics to better show the fractal pattern. For the sake of comparison the dashed grey line indicates the elastic scenario $v_0 = v_f$.
  • Figure 2: Energy density plots at increasing times during the right-angle scattering of two vortices in a head-on collision in the BPS limit at a velocity $v=0.1c$.
  • Figure 3: Left: kink solutions arising in the $\phi^4$ model and their energy density. Right: kink solutions plotted over the potential \ref{['eqI2:PotPhi4']} (right).
  • Figure 4: Potential well corresponding to the spectral problem \ref{['eqI2:PerProbPhi4']} and the allowed eigenvalues. The shaded area represents the continuum starting at $\omega^2=4$.
  • Figure 5: Kink–antikink pair \ref{['eqI2:KAKPhi4']} with $d=5$, and its corresponding potential energy $U(x)$.
  • ...and 8 more figures

Theorems & Definitions (2)

  • Definition 1: Topological soliton
  • Theorem 1: Derrick's Theorem