A nonlinear wave equation invariant with respect to unitary representations of the Lorentz group is considered in an attempt to describe extended particles with spin and positive definite energy by means of a self-confined classical field. The wave function has an infinite number of components and, in the specific representations used, the corresponding internal degree of freedom is identified with the spin. A fractional power of the scalar bilinear invariant appears as an appropriate choice for the nonlinearity in order that all the stationary states be localized. Two approximation methods are proposed and both lead to results that bear a resemblance to the results of the MIT bag model.

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In this thesis, we study the "two-point approximation" for highly non-local effective actions, in the particular case of the Chiral Soliton Model of the nucleon. The nucleon in this model is regarded as being made of three valence quarks bound together by a meson field in a soliton form. Mesons are treated in mean field theory and the vacuum energy due to one-quark loops is included. The theory is defined with a finite cut-off in momentum space, consistent with an effective theory for the low-energy description of the strong interactions. We use the two-point approximation to calculate the vacuum correction to the chiral soliton energy for a variety of soliton profile functions, investigating the effect of different regularisation schemes. Results are little influenced by the choice of the cut-off, and are within 20% of exact calculations, done with the full inclusion of the Dirac sea. We then perform a dynamical calculation of the chiral soliton by including sea-quark effects self-consistently in the two-point approximation. We find a typical 20% (or less) deviation in the soliton energy from exact calculations. We apply a further "pole" approximation which leads to a significant algebraic simplification in the self-consistent equations. We show, in particular, that a simple numerical fit of the pole form to the two-point cut-off function yields essentially indistinguishable results from the latter. We finally calculate some static nucleon observables in the two-point approximation and find general agreement with exact calculations. In view of the results obtained, we may hope that the pole form of the twopoint approximation may prove to be a generally useful approach to similar problems involving highly non-local actions.

The work in this thesis is concerned with the numerical study of some stability and scattering properties of two CP¹ models in three dimensional space-time: The non-linear 0(3) model and its modified Skyrme version. Chapter 3 focuses principally on the Skyrme model on compactified plane, the topological sphere. Such model is obtained by supplementing the ordinary 0(3) lagrangian with both a Skyrme term and a potential term which, in the present work, has a rather general form. Under the numerical simulation the skyrmions behave stably and scatter either back-to-back or at 90 to the initial direction of motion, depending on the initial velocity. In the 0(3) limit the solitons are no longer stable and scatter at 90 irrespective of the speed. In the fourth chapter the 0(3) model is studied on a flat torus. Its solitons exhibit the usual instability but can be stabilised by the sole addition of a Skyrme term to the lagrangian. Scattering at right angles is observed in all cases considered, including skyrmions colliding at speeds that would bounce them back were they evolving in compactified plane. The periodic 0(3) model has no analytic solutions of degree one, so when a field configuration that resembles a single soliton is numerically evolved, it shrinks to become infinitely thin. Interestingly, such ansatz may be regarded as a soliton of unit topological charge in the context of the periodic skyrmion model. Chapter 5 closes with a summary and suggestions for future research.

Thesis (M.S. in Applied Science) Naval Postgraduate School, March 1997. / Thesis advisors, Chiu, Ching-Sang ; Smith, Kevin B. ADA331078. Includes bibliographical references (p. 33).

Made available in DSpace on 2014-06-11T19:30:11Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-08-20Bitstream added on 2014-06-13T18:59:53Z : No. of bitstreams: 1 correa_rac_me_guara.pdf: 727192 bytes, checksum: e893f54f230b703f5d40233d4e8e1002 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho, apresentamos uma breve introdução ao estudo dos sólitons. Além disso, introduzimos uma nova classe de configurações oscilantes do tipo lump e do tipo kink. Tais configurações são obtidas de forma analítica e exata, como conseqüência somos capazes de fazer a análise da estabilidade das soluções apresentadas. Por fim, apresentamos no último capítulo soluções de estados ligados de férmions na presença das novas configurações de campo que estamos apresentando neste trabalho / In this work we present a brief introduction to the study of solitons. Furthermore, we introduce a new class of oscillating Lorentz covariant configurations for the evolution of the domain walls which are analytically obtained in diverse dimensions. As a consequence, we are able to perform the analysis of the stability of the presented solutions. Finally, in the last chapter, we present analytical solutions of fermion bound states on the background of kinks of a class of asymmetrical scalar field

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Les solitons dissipatifs (SD) sont des paquets d'onde auto-localisés qui apparaissent dans les systèmes dissipatifs spatialement étendus. En optique, tous les SD observés jusqu'à présent dans des systèmes propagatifs peuvent être classés en deux catégories, suivant la présence ou non d'un forçage externe, i.e. si la symétrie de phase est brisée ou non. Dans les systèmes forcés, les DS sont accrochés en phase au forçage alors que sans forçage, leur phase est libre et peu dériver en fonction du temps. Dans cette thèse nous étudions la formation d'états localisés propagatifs dans deux systèmes expérimentaux qui diffèrent fondamentalement par la présence ou l'absence d'un forçage externe. Le premier système est un laser à semiconducteur à cavité verticale (VCSEL) soumis à une boucle de rétro-action sélective en fréquence, qui accueille des DS se formant dans la dimension transverse à la propagation (2D). Nous analysons comment la synchronisation des fréquences longitudinales du système (verrouillage modal) peut mener à la formation d'un état localisé dans les trois dimensions : balles de lumière. Le deuxième système est un laser en anneau à semi-conducteur fortement multimode le long de la propagation, et forcé par une injection externe. Lorsque le forçage est légèrement désaccordé de la fréquence naturelle du système, il est possible d'observer des états localisés constitués par un tour de phase de 2 pi, immergés dans l'état homogène (synchronisé). Nous reportons ainsi la première observation de SD qui se forment dans la phase de l'onde optique : solitons de phase dissipatifs / Dissipative solitons (DS) are self-localized wave-packets appearing in spatially extended dissipative systems. In optics, all the DS that have been observed in propagative systems can be cast in two categories, depending on the presence or absence of an external forcing, i.e. the phase symmetry is broken or not. In forced systems, DS are locked in phase to forcing whereas without forcing, their phase is free an can wander in the course of time. In this thesis, we study the formation of propagative DS in two different experimental systems that fundamentally differ from the presence or lack of an external forcing. The first one is a Vertical Cavity Surface Emitting Laser (VCSEL) submitted to a frequency selective feedback, in which DS form in the transverse plane of the system (2D). We analyze how the synchronization of the longitudinal frequencies (mode-locking) can give rise to tri-dimensionnal localization of light: light bullets. The second system is a highly multimode semiconductor ring laser with external forcing, whose spatial extension takes place along the propagation dimension. When the forcing frequency is slightly detuned from the natural frequency of the system, we can see the appearance of self-confined 2 pi phase rotations embedded in a homogeneous (synchronized) state. We then report on the first observation of DS that form in the phase of the optical wave : dissipative phase solitons