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Dynamics and interaction of solitons in the BPS limit and their internal modes

S. Navarro-Obregón

Abstract

The main objective of this thesis has been to analyse soliton dynamics in detail, with special attention paid to the role of the internal modes associated with these configurations. Specifically, the thesis has focused on the study of one- and two-dimensional models, with the aim of developing a solid basis that can then be extended to the study in three-dimensional theories. This thesis concentrates on the study of kinks, oscillons, vortices, and sphalerons. Nevertheless, field theories constitute systems with an infinite number of degrees of freedom, which poses challenges both for obtaining analytical results and for predictive modeling. To address these challenges, this thesis employs the construction of effective models that retain the essential degrees of freedom required to capture the phenomenology observed in numerical simulations of the full theory, using the well-known collective coordinate method. In addition, other complementary mathematical tools, such as perturbative techniques, have also been employed. Among the main achievements of this doctoral thesis, it is worth highlighting the introduction, for the first time, of genuine radiation modes within the collective coordinate framework. Furthermore, a generalisation of Samols' moduli space metric for local vortices in the Abelian-Higgs model has been developed through the incorporation of vibrational degrees of freedom. Additionally, a new class of sphalerons that we have coined semi-BPS sphalerons has been identified and analysed. Finally, the role of oscillatory internal modes in the decay process of sphalerons has been studied in detail, leading to the proposal of a dynamic stabilisation mechanism. This mechanism has been further explained and extended to more general models, demonstrating the robustness and potential applicability of this phenomenon to physically relevant theories.

Dynamics and interaction of solitons in the BPS limit and their internal modes

Abstract

The main objective of this thesis has been to analyse soliton dynamics in detail, with special attention paid to the role of the internal modes associated with these configurations. Specifically, the thesis has focused on the study of one- and two-dimensional models, with the aim of developing a solid basis that can then be extended to the study in three-dimensional theories. This thesis concentrates on the study of kinks, oscillons, vortices, and sphalerons. Nevertheless, field theories constitute systems with an infinite number of degrees of freedom, which poses challenges both for obtaining analytical results and for predictive modeling. To address these challenges, this thesis employs the construction of effective models that retain the essential degrees of freedom required to capture the phenomenology observed in numerical simulations of the full theory, using the well-known collective coordinate method. In addition, other complementary mathematical tools, such as perturbative techniques, have also been employed. Among the main achievements of this doctoral thesis, it is worth highlighting the introduction, for the first time, of genuine radiation modes within the collective coordinate framework. Furthermore, a generalisation of Samols' moduli space metric for local vortices in the Abelian-Higgs model has been developed through the incorporation of vibrational degrees of freedom. Additionally, a new class of sphalerons that we have coined semi-BPS sphalerons has been identified and analysed. Finally, the role of oscillatory internal modes in the decay process of sphalerons has been studied in detail, leading to the proposal of a dynamic stabilisation mechanism. This mechanism has been further explained and extended to more general models, demonstrating the robustness and potential applicability of this phenomenon to physically relevant theories.
Paper Structure (71 sections, 5 theorems, 383 equations, 97 figures, 4 tables)

This paper contains 71 sections, 5 theorems, 383 equations, 97 figures, 4 tables.

Table of Contents

  1. Introduction
  2. From solitary waves to solitons
  3. Solitons in field theories
  4. Objectives of the thesis
  5. Structure of the thesis and methodology employed
  6. On the Role of Internal Modes in the Dynamics of Topological Solitons
  7. Introduction to topological solitons
  8. Lagrangian field theory
  9. Topological protection
  10. Linear stability
  11. Bogomolny equations
  12. Derrick's theorem
  13. Topological solitons in one spatial dimension
  14. The phi4 model
  15. The phi6 model
  16. ...and 56 more sections

Key Result

Theorem 2.1

Let $\phi^a \rightarrow \phi^a + \epsilon\,\delta\phi^a$ be a continuous symmetry of a Lagrangian density, that is, a continuous transformation that changes the Lagrangian density by a divergence To this continuous symmetry corresponds a conserved current $J^{\mu} = (J^0,\vec{J})$ such that and a conserved charge given by

Figures (97)

  • Figure 1.1: A scheme of a solitary wave in a shallow channel of water.
  • Figure 1.2: Dynamics of a two-soliton solution of the KdV equation.
  • Figure 2.1: The scheme illustrates two homotopic functions, $f$ and $g$, with $f(A) = g(A)$ and $f(B) = g(B)$, which map the segment $AB$ onto the green upper curve and the blue lower curve, respectively. The dashed curve represents the action of the map $F:X \times[0,1] \rightarrow Y$ for an intermediate value $c \in [0,1]$.
  • Figure 2.2: Example of a function $f$ mapping between two one-spheres $\mathbbm{S}^1$. As the domain variable ranges over $\theta \in [0,2\pi)$ once, the image of $f$ winds twice around the target one-sphere. In this case, deg $f$ = 2.
  • Figure 2.3: Potential function of the $\phi^4$ model (\ref{['e:potential_phi4']}) for $m = 1$ and $\lambda = 1/2$.
  • ...and 92 more figures

Theorems & Definitions (16)

  • Definition 1.1: Solitary wave: Ablowitz and Clarkson Ablowitz:1991
  • Definition 1.2: Soliton: Drazin and Johnson Drazin:1996
  • Definition 1.3: Soliton
  • Definition 1.4: Stable and unstable soliton
  • Theorem 2.1: Noether's theorem Noether:1918
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Homotopy
  • Definition 2.4: $n$-th homotopy group
  • Definition 2.5: Topological degree Manton:2002
  • ...and 6 more