On The Connection Between Dirac And Ricatti Equations

We analyse the behaviour of the Dirac equation in d = 1 + 1 with Lorentz scalar potential. As the system is known to provide a physical realization of supersymmetric quantum mechanics, we take advantage of the factorization method in order to enlarge the restricted class of solvable problems. To be precise, it suffices to integrate a Ricatti equation to construct one-parameter families of solvable potentials. To illustrate the procedure in a simple but relevant context, we resort to a model which has proved useful in showing the phenomenon of fermion number fractionalization. 1. Introduction. When solving the Dirac equation in d = 1 + 1 with Lorentz scalar potential, the underlying supersymmetric structure is crucial. As the system provides a physical realization of supersymmetric quantum mechanics (susy qm henceforth), the problem reduces itself to a pair of Schrodinger-like hamiltonians related by means of supersymmetry. In doing so, both operators share identical energy spectra u..

Spontaneous Symmetry Breaking Phenomena with Non-Equivalent Vacua

We study the existence of bidimensional bosonic models for which the spontaneous symmetry breaking phenomenon yields, at classical level, non-equivalent vacua. Once we introduce the concept of vacuum manifold V , defined in terms of the vacuum field configurations, the behavior of the models at issue can be analyzed by means of the so-called classical moduli space M c (V). The aforementioned structure allows us to classify the kink-like excitations into loops and links. To be precise, the loops interpolate smoothly between equivalent minima of the classical potential while the links connect vacua located at different points in M c (V). Although exact results are very hard to come by, we resort to models nice enough to provide us with the solitary waves in closed form. 1. Introduction. Since the mid-seventies the search for classical solutions of the corresponding non-linear equations has been one of the most successful tools in both classical and quantum field theories. Unless other..

The problem of area change in tangential longitudinal strain folding

This paper deals with some problems with the concept and properties of the folding mechanism named tangential longitudinal strain. A general two-dimensional mathematical description of this mechanism in terms of displacements and finite strains is presented. In the analysis of this mechanism of folding, two geologically reasonable variants are considered. The first of these, referred to as parallel tangential longitudinal strain folding, involves no finite elongation of lines perpendicular to the layer and produces class 1B (parallel) folds. The second variant is characterized by the conservation of area across the fold profile and is therefore termed equiareal tangential longitudinal strain folding; it produces folds ranging from class 1B to more complex shapes with the development of a bulge in the hinge zone inner arc when amplitude and curvature are high. Using the computer program “FoldModeler” which incorporates the derived equations for displacements and finite strains, the geometrical features of idealized folds produced by these two variants have been studied, together with those arising from their successive or simultaneous combination. The implications of the operation of these two deformation mechanisms in natural folds are then considered and a discussion is presented about the features that can be diagnostic of their operation in nature. It is suggested that the two mechanisms operate together in the formation of natural folds, in a way that deformation probably begins with equiareal tangential longitudinal strain, but subsequently gives way to parallel tangential longitudinal strain when strain concentration in some parts of the folded layer makes area change probable

Some considerations on the kinematics of chevron folds

Geometrical modelling and field analysis of chevron folds suggest that these structures are a result of the combination of several kinematical mechanisms, whose magnitude and order of application vary within certain limits. A possible mechanism operating early in the fold growth is homogeneous layer shortening, whose contribution is restricted by the high competence contrast that the multilayers developing chevron folds usually exhibit. In the early stages of folding, when the curvature is small, equiareal tangential longitudinal strain (ETLS) is an essential mechanism, since the operation of parallel tangential longitudinal strain (PTLS) or flexural flow (FF) would give rise respectively to area changes or strain in the limbs of the final fold that are too high to be geologically realistic. After folding by ETLS, probable mechanisms are PTLS and FF, which can operate in this order or simultaneously. In general, FF is necessary at the last stage of buckling, although the increment of folding due to this mechanism can be very small. High values of slip between layers and area change produced in the later stages of chevron folding can bring an end to buckling, probably at an interlimb angle value of 60–70°, and induce the onset of homogeneous strain (HS). This strain is not coaxial in many cases, with simple shear playing an important role, and gives rise to asymmetrical folds

Kinematic analysis of asymmetric folds in competent layers using mathematical modelling

Mathematical 2D modelling of asymmetric folds is carried out by applying a combination of different kinematic folding mechanisms: tangential longitudinal strain, flexural flow and homogeneous deformation. The main source of fold asymmetry is discovered to be due to the superimposition of a general homogeneous deformation on buckle folds that typically produces a migration of the hinge point. Forward modelling is performed mathematically using the software ‘FoldModeler’, by the superimposition of simple shear or a combination of simple shear and irrotational strain on initial buckle folds. The resulting folds are Ramsay class 1C folds, comparable to those formed by symmetric flattening, but with different length of limbs and layer thickness asymmetry. Inverse modelling is made by fitting the natural fold to a computer-simulated fold. A problem of this modelling is the search for the most appropriate homogeneous deformation to be superimposed on the initial fold. A comparative analysis of the irrotational and rotational deformations is made in order to find the deformation which best simulates the shapes and attitudes of natural folds. Modelling of recumbent folds suggests that optimal conditions for their development are: a) buckling in a simple shear regime with a sub-horizontal shear direction and layering gently dipping towards this direction; b) kinematic amplification due to superimposition of a combination of simple shear and irrotational strain with a sub-vertical maximum shortening direction for the latter component. The modelling shows that the amount of homogeneous strain necessary for the development of recumbent folds is much less when an irrotational strain component is superimposed at this stage that when the superimposed strain is only simple shear. In nature, the amount of the irrotational strain component probably increases during the development of the fold as a consequence of the increasing influence of the gravity due to the tectonic superimposition of rocks

La cinemática del plegamiento: algunas claves geométricas para su interpretación

Computer modeIling of fold profiles formed by a specific folding mechanism is possible when the equations to find transformation relationships from points of the initial configuration to points of the folded configuration of the layer are known. From these equations, folds formed by the successive or simultaneous superposition of tangential longitudinal strain, flexural flow and several types of homogeneous strain (layer shortening, compaction, and fold flattening parallel or perpendicular to the axial trace) have been modelled. The geometry and strain pattern of the theoretical folds allow predictions about the characteristics of natural folds formed by the above mechanisms. The strain state in the folded layer is a typical feature of every mechanism or mechanism superposition and it can be described by two types of curves, which show the variation of the major axis plunge of the strain ellipse and the variation of the aspect ratio of this ellipse with the layer dip. Tangential longitudinal strain is the only mechanism analysed that produces different curves for the two boundaries of the folded layer. Ramsay's classification or classifications based on parameters derived from it also give results specific for every folding mechanism or mechanism superposition. Analysis of folding mechanisms that operated in a specific natural fold can be made by trial and error modelling a fold with the same geometrical characteristics as the natural fold. The geometrical features to be analysed in the latter are those involved in the modelling. In natural folds with cleavage, an approach to the curve of the majar axis plunge of the strain ellipse against the layer dip can be obtained by measuring the dip variation of the cleavage as a function of the layer dip. The greatest problem in the kinematical analysis of folding is posed by the difficulty of obtaining strain measurements in folded rocks.La modelización por ordenador de perfiles de pliegues formados por un determinado mecanismo de plegamiento es posible cuando se conocen las ecuaciones que permiten encontrar las relaciones de transformación de puntos de la configuración inicial de la capa en puntos de la configuración plegada. De este modo, se han modelizado pliegues formados mediante la superposición sucesiva o simultánea de deformación longitudinal tangencial, "flexural-flow" y diversos tipos de deformación homogénea (acortamiento de la capa, compactación, y aplastamiento y achatamiento de pliegues). La geometría y distribución de la deformación interna en los pliegues teóricos así obtenidos permiten predecir las características de pliegues naturales formados por los citados mecanismos de plegamiento. El estado de deformación de la capa plegada es típico de cada mecanismo o superposición de mecanismos y puede describirse mediante dos tipos de curvas, que muestran la variación de la inclinación del eje mayor de la elipse de deformación y la variación del cociente entre las longitudes de los ejes de dicha elipse en función del buzamiento de la capa plegada. La deformación longitudinal tangencial es el único de los mecanismos analizados que, cuando interviene en el plegamiento, da lugar a curvas distintas para los dos límites de la capa plegada. La clasificación de Ramsay o las clasificaciones basadas en parámetros derivados de ella dan lugar también a resultados específicos para cada mecanismo o superposición de mecanismos de plegamiento. El análisis de los mecanismos que formaron un pliegue natural dado puede hacerse por tanteo mediante la modelización de un pliegue con las mismas características geométricas que el pliegue natural. Los rasgos geométricos a analizar en este último son los implicados en la modelización. En pliegues naturales con clivaje, puede obtenerse una aproximación a la curva de la inclinación del eje mayor de la elipse de deformación en función del buzamiento, midiendo la variación del buzamiento del clivaje en función del buzamiento de la capa. El mayor problema en el análisis cinemático de pliegues estriba en la dificultad de realizar medidas de deformación interna en las rocas plegadas