ALBERTO ALONSO IZQUIERDO

The flow of shape eigenmodes of the small fluctuation operator around BPS 2-vortex solutions is calculated, as a function of the intervortex separation 2d. For the rotationally-invariant 2-vortex, with d= 0, there are three discrete modes; the lowest is non-degenerate and the upper two are degenerate. As d increases, the degeneracy splits, with one eigenvalue increasing and entering the continuous spectrum, and the other decreasing and asymptotically coalescing with the lowest eigenvalue, where they jointly become the eigenvalue of the 1-vortex radial shape mode. The behaviour of the eigenvalues near d= 0 is clarified using a perturbative analysis, and also in light of the 2-vortex moduli space geometry.

The dynamics of a wobbling kink in a two-component coupled ϕ 4 scalar field theory (with an excited orthogonal shape mode) is addressed. For this purpose, the vibration spectrum of the second order small kink fluctuation is studied in order to find the corresponding vibration modes associated to the first (longitudinal) and second (orthogonal) field components. By means of this analysis, it was found that the number of possible shape modes depends on the value of the coupling constant. It is notable that when one of the orthogonal field shape modes is initially triggered, the unique shape mode of the longitudinal field is also activated. This coupling causes the kink to emit radiation with twice the frequency of excited mode in the first field component. Meanwhile, in the orthogonal channel we find radiation with two different frequencies: one is three times the frequency of the orthogonal wobbling mode and another is …

We construct a simple field theory in which a sphaleron, i.e., a saddle-point particle-like solution, forms a semi-BPS state with a background defect that is an impurity. This means that there is no static force between the sphaleron and the impurity. Therefore such a sphaleron-impurity system is very much like usual BPS multi-solitons, however, still possessing an unstable direction allowing for its decay. We study dynamics of the sphaleron in such a system.

In this paper the interaction between the shape modes of the wobbling kinks arising in the family of two-component MSTB scalar field theory models is studied. The spectrum of the second order small kink fluctuation in this model has two localized vibrational modes associated to longitudinal and orthogonal fluctuations with respect to the kink orbit. It has been found that the excitation of the orthogonal shape mode immediately triggers the longitudinal one. In the first component channel the kink emits radiation with twice the orthogonal wobbling frequency (not the longitudinal one as happens in the ϕ 4-model). The radiation emitted in the second component has two dominant frequencies: one is three times the frequency of the orthogonal wobbling mode and the other is the sum of the frequencies of the longitudinal and orthogonal vibration modes. This feature is explained analytically using perturbation expansion …

We construct a simple field theory in which a sphaleron, ie, a saddle-point particle-like solution, forms a semi-BPS state with a background defect that is an impurity. This means that there is no static force between the sphaleron and the impurity. Therefore, such a sphaleron-impurity system is very much like usual BPS multisolitons, however, still possessing an unstable direction allowing for its decay. We study dynamics of the sphaleron in such a system.

In this paper we construct a family of Hamilton–Jacobi separable non-linear S 1× S 1 Sigma models for which the kink variety can be analytically identified and for which the linear stability of the emerging kinks is ensured. Furthermore, a model with only one vacuum point is found, where all kinks are forced to be non-topological. The non-simply connectedness of the torus guarantees the global stability of all the non-topological kinks in these models.

The asymmetric scattering between wobblers and kinks in the standard ϕ 4 model is numerically investigated in two different scenarios. First, the collision between wobblers with opposite phases is analyzed. Here, a destructive interference between the shape modes of the colliding wobblers takes place at the impact time. The second scenario involves the scattering between a wobbler and an unexcited kink. In this case the energy transfer from the wobbler to the kink can be examined. The dependence of the final velocities and wobbling amplitudes of the scattered wobblers on the collision velocity and on the initial wobbling amplitude is discussed. Both situations lead to very different fractal structures in the velocity diagrams.

In this paper the whole kink varieties arising in several massive non-linear Sigma models whose target space is the torus S 1× S 1 are analytically calculated. This possibility underlies the construction of first-order differential equations by adapting the Bogomol’nyi procedure to non-Euclidean spaces. Among the families of solutions non-topological kinks connecting the same vacuum are found. This class of solutions are usually considered to be not globally stable. However, in this context the topological constraints obtained by the non-simply connectedness of the target space turn these non-topological kinks into globally stable solutions. The analytical resolution of the equations allows the complete study of the linear stability for some basic kinks.

We study topological vortex solutions in a generalized Abelian Higgs model with nonpolynomial dielectric and potential functions. These quantities are chosen by requiring integrability of the self-dual limit of the theory for all values of the magnetic flux. All the vortex profiles are described by exact analytical expressions that solve the self-dual vortex equations. There is only a symmetry-breaking superconducting phase and the model sustains regular phenomenology.

We study topological vortex solutions in a generalized Abelian Higgs model with non-polynomial dielectric and potential functions. These quantities are chosen by requiring integrability of the self-dual limit of the theory for all values of the magnetic flux. All the vortex profiles are described by exact analytical expressions that solve the self-dual vortex equations. There is only a symmetry-breaking superconducting phase and the model sustains regular phenomenology.

In this paper, kink scattering in the dimensional reduction of the bosonic sector of a one-parameter family of generalized Wess-Zumino models with three vacuum points is discussed. The value of the model parameter determines the specific location of the vacua. The influence of the vacuum arrangements (evolving from three collinear vacua to three vacua placed at the vertices of an equilateral triangle) on the kink scattering is investigated. Two different regimes can be distinguished: in the first one, two symmetric BPS kinks/antikinks arise whereas in the second one a new different BPS kink/antikink emerges, with the exception of a three-fold rotational symmetry case, where the three topological defects are identical. The scattering between the two symmetric kinks is thoroughly analyzed. Two different scattering channels have been found: kink-kink reflection and kink-kink hybridization. In the last case, the collision …

We compute the vacuum fermion current in the (2+ 1) dimensional Jackiw-Rossi model by using the 1/m expansion. The current is expressed through a weighted η-function with a matrix weight. In the presence of such a weight, the usual proof of topological nature of η (0) is no longer applicable. Direct computations confirm the following surprising result; the fermion number induced by vortices in the Jackiw-Rossi model is not topological.

In this paper the scattering between a wobbling kink and a wobbling antikink in the standard ϕ 4 model is numerically investigated. The dependence of the final velocities, wobbling amplitudes and frequencies of the scattered kinks on the collision velocity and on the initial wobbling amplitude is discussed. The fractal structure becomes more intricate due to the emergence of new resonance windows and the splitting of those arising in the nonexcited kink scattering. Outside this phase the final wobbling amplitude exhibits a linear dependence of the collision velocity, which is almost independent of the initial wobbling amplitude.

Two-dimensional scalar field theories with spontaneous symmetry breaking subject to the action of Jackiw-Teitelboim gravity are studied. Solutions for the ϕ 4 and sine-Gordon self-gravitating kinks are presented, both for general gravitational coupling and in the perturbative regime. The analysis is extended to deal with a hierarchy of kinks related to transparent Pöschl-Teller potentials.

This chapter contains some programs and specific software that could help lecturers to solve some of the proposed problems. Algorithms and mathematical contents are included in corresponding chapters in the book.

This chapter presents some applications of first-order differential equations and systems of first-order differential equations. Fundamentally, the chapter is divided in three sections: the first section includes a quick theoretical journey through the basic notions associated with the topic of the chapter, with special emphasis on what is known as qualitative theory of differential equations; the second section makes a quick tour of those problems that are modeled through first-order differential equations and systems of differential equations that show the power of this tool to solve many different problems; the third section deals with two real problems with more depth, which are particularly interesting and in general less well known.

We study a semilocal Abelian Higgs sigma model with CP 2-valued scalar fields coupled to gravity. We establish the conditions needed for self-duality in this system and obtain cosmic strings, both of self-dual character and out of the self-dual point, in the reference chart of CP 2. In any of the other two charts of a minimum atlas, the transition functions break the global S U (2) symmetry to the Abelian subgroup e i σ 3 α. In this context, using the transition functions, a new self-dual structure can be constructed, which defines a second species of cosmic strings. Because of the presence of a potential energy for the scalar fields, each species takes values in a different bounded region of the target space and is most naturally described using a particular chart of CP 2.

In this paper the scattering between the non-topological kinks arising in a family of two-component scalar field theory models is analyzed. A winding charge is carried by these defects. As a consequence, two different classes of kink scattering processes emerge: (1) collisions between kinks that carry the same winding number and (2) scattering events between kinks with opposite winding number. The variety of scattering channels is very rich and it strongly depends on the collision velocity and the model parameter. For the first type of events, four distinct scattering channels are found: kink reflection (kinks collide and bounce back), one-kink (partial) annihilation (the two non-topological kinks collide causing the annihilation of one half of each kink and the subsequent recombination of the other two halves, giving rise to a new non-topological kink with the opposite winding charge), winding flip kink reflection (kinks …