Two areas of affine Toda field theory are explored in this thesis. First the introduction of a boundary into the real coupling affine Toda field theory. It has been shown by other authors that affine Toda field theory stays an integrable theory for certain boundaries. One such theory is the one corresponding to Its integrable boundary condition is described by two continuous parameters. Also, it is continuously connected to the natural Neumann condition, i.e. vanishing space derivative of the fields at the boundary. Classical reflection factors of incoming plane waves in the background of a static soliton solution are calculated for this theory. They fulfil a classical reflection bootstrap equation which is the classical limit of the reflection bootstrap equation for reflection matrices. The second part is concerned with the the αη(^1) affine Toda field theory with imaginary coupling. The behaviour of oscillatory solitonic solutions, breathers is investigated. Explicit construction for breather solution are given. They originate from two soliton solutions. It is found that there are two different types of breathers depending on their constituent solitons. The constituent solitons are either of the same species or are anti-species of each other. Also, the topological charges of breather solutions are calculated and they are either zero or equal to a certain one soliton solution. These topological charges lie in the tensor product representation of the fundamental representations associated with the topological charges of the constituent solitons. The breather masses are, as expected, less than the sum of the masses of the constituent solitons.

Cette thèse est consacrée à l’étude du comportement de propagation de solitons, ou d’impulsions s’apparentant à des solitons, dans des fibres optiques présentant un profil de dispersion variable. La première partie présente les propriétés générales de guidage d’une fibre optique et introduit les principaux processus linéaires et non-linéaires dans lesquels des solitons temporels sont mis en jeu. Dans la partie II, le contrôle spectral, puis spectro-temporel, de solitons est démontré grâce à l’utilisation de fibres optiques topographiques spécialement conçues à cet effet à l’aide d’un algorithme de résolution inverse. La partie III est, elle, dédiée à l’étude de la dynamique d’émission d’ondes dispersives multiples et de cascades d’ondes dispersives dans une fibre dont le zéro de dispersion varie sinusoïdalement. Enfin, la partie IV explore la stabilisation de breathers d’Akhmediev dans une fibre optique comportant un saut de dispersion. / This thesis concerns the study of the propagation of solitons, or assimilated ones, in optical fibers with tailored longitudinal profiles. The first part presents the general guiding properties of an optical fiber and introduces the main linear and non-linear process involved in solitons propagation. In Part II, spectral and then spectrotemporal solitons control is demonstrated through the use of specially designed topographic fibers. Part III explore the emission dynamics of cascaded resonant radiations and multiple resonant radiations in a fiber with an oscillating zero-dispersion wavelength. Finally, Part IV is devoted to the stabilization of Akhmediev breathers in an optical fiber including a dispersion step.

BEZERRA, Victor Hugo Ferreira. Estudo de colisões de paredes de Bloch e localização de modos ferminônicos. 2005. 90 f. Dissertação (Mestrado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2005. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2015-06-23T20:24:45Z No. of bitstreams: 1 2005_dis_vhfbezerra.pdf: 2731698 bytes, checksum: d87aed2d86c122faae1ddc27ad887419 (MD5) / Approved for entry into archive by Edvander Pires(edvanderpires@gmail.com) on 2015-06-23T21:03:25Z (GMT) No. of bitstreams: 1 2005_dis_vhfbezerra.pdf: 2731698 bytes, checksum: d87aed2d86c122faae1ddc27ad887419 (MD5) / Made available in DSpace on 2015-06-23T21:03:25Z (GMT). No. of bitstreams: 1 2005_dis_vhfbezerra.pdf: 2731698 bytes, checksum: d87aed2d86c122faae1ddc27ad887419 (MD5) Previous issue date: 2005 / In this work we numerically study the collision process of domain-walls in a system with two coupled scalar fields, which generates a wall with internal structures, the so called Bloch Walls. If the initial velocity is less than 0, 57... occurs inelastic scattering and part of the energy is lost as radiation. We realize that the objects resulting from the collision belong to a topological sector which is different from those chosen in the beginning of the process. In other words, the system transit from a sector AA-like to a sector AB-like one. The reflected solitary waves have approximately half of the initial energy. Besides of the decay of the sector AA to AB one, we note that part of the energy is converted into a coherent structure which oscillates in the center of the net, with approximately half of the initial energy of the system. We have studied also bound states of the system in the case of two walls which initially are close and with null velocity. The resulting structure lose energy as radiation. However, the energy loss is gradual, like in the case of the breathers of the Á4 model. On the other hand, we have studied the coupling of fermions to the Bloch domainwall. The coupling between the fermionic field and two different scalar fields named Á and  it is a Yukawa-type coupling. We have showed that the fermionic zero mode is localized at x = 0 in the scalar field Â, which means that the ferminonic field is localized inside wall. The fermionic mass in the vacuum sector is the same obtained for the Á4 model, since that the mass of the scalar field  in the vacuum sector is zero. / Neste trabalho estudamos numericamente o processo de colisão de paredes de domínio em um modelo de dois campos escalares acoplados que geram paredes com estrutura interna, as chamadas paredes de Bloch. Percebemos que os objetos resultantes da colisão pertencem a um setor topológico diferente dos setores escolhidos no início do processo, ou seja, o sistema passou de um setor do tipo AA para um setor do tipo AB. Para a velocidade inicial acima de 0, 57..., percebemos a aniquilação dos objetos. As ondas solitárias refletidas possuem aproximadamente metade da energia inicial. Além do decaimento do setor AA para um tipo AB, verificou-se que parte da energia é convertida numa estrutura coerente oscilante no centro da rede, com aproximadamente metade da energia inicial do sistema. Estudamos ainda estados ligados a partir de duas paredes inicialmente próximas com velocidade nula. A estrutura formada perde energia em forma de radiação. Entretanto, a perda de energia se dá lentamente, assim como os breathers do modelo Á4. Estudou-se ainda a presença de férmions acoplados nas paredes de domínio tipo Bloch. O acoplamento entre o campo fermiônico e os campos Á e  é do tipo Yukawa. Mostrou-se que o modo zero fermiônico é localizado em x = 0 no campo x, ou seja, o campo fermiônico é localizado dentro da parede. A massa fermiônica no setor de vácuo é a mesma para o modelo Á4, uma vez que a massa do campo  no setor de vácuo é nula.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho estudamos numericamente o processo de colisÃo de paredes de domÃnio em um modelo de dois campos escalares acoplados que geram paredes com estrutura interna, as chamadas paredes de Bloch. Percebemos que os objetos resultantes da colisÃo pertencem a um setor topolÃgico diferente dos setores escolhidos no inÃcio do processo, ou seja, o sistema passou de um setor do tipo AA para um setor do tipo AB. Para a velocidade inicial acima de 0, 57..., percebemos a aniquilaÃÃo dos objetos. As ondas solitÃrias refletidas possuem aproximadamente metade da energia inicial. AlÃm do decaimento do setor AA para um tipo AB, verificou-se que parte da energia à convertida numa estrutura coerente oscilante no centro da rede, com aproximadamente metade da energia inicial do sistema. Estudamos ainda estados ligados a partir de duas paredes inicialmente prÃximas com velocidade nula. A estrutura formada perde energia em forma de radiaÃÃo. Entretanto, a perda de energia se dà lentamente, assim como os breathers do modelo Ã4. Estudou-se ainda a presenÃa de fÃrmions acoplados nas paredes de domÃnio tipo Bloch. O acoplamento entre o campo fermiÃnico e os campos à e à à do tipo Yukawa. Mostrou-se que o modo zero fermiÃnico à localizado em x = 0 no campo x, ou seja, o campo fermiÃnico à localizado dentro da parede. A massa fermiÃnica no setor de vÃcuo à a mesma para o modelo Ã4, uma vez que a massa do campo à no setor de vÃcuo à nula. / In this work we numerically study the collision process of domain-walls in a system with two coupled scalar fields, which generates a wall with internal structures, the so called Bloch Walls. If the initial velocity is less than 0, 57... occurs inelastic scattering and part of the energy is lost as radiation. We realize that the objects resulting from the collision belong to a topological sector which is different from those chosen in the beginning of the process. In other words, the system transit from a sector AA-like to a sector AB-like one. The reflected solitary waves have approximately half of the initial energy. Besides of the decay of the sector AA to AB one, we note that part of the energy is converted into a coherent structure which oscillates in the center of the net, with approximately half of the initial energy of the system. We have studied also bound states of the system in the case of two walls which initially are close and with null velocity. The resulting structure lose energy as radiation. However, the energy loss is gradual, like in the case of the breathers of the Ã4 model. On the other hand, we have studied the coupling of fermions to the Bloch domainwall. The coupling between the fermionic field and two different scalar fields named à and à it is a Yukawa-type coupling. We have showed that the fermionic zero mode is localized at x = 0 in the scalar field Ã, which means that the ferminonic field is localized inside wall. The fermionic mass in the vacuum sector is the same obtained for the Ã4 model, since that the mass of the scalar field à in the vacuum sector is zero.

Dans cette thèse, on analyse par des méthodes de variétés invariantes deux problèmes distincts: le phénomène des roll-waves en hydraulique et l'existence de breathers discrets dans des réseaux non linéaires discrets. Les roll-waves sont des ondes progressives périodiques et discontinues solutions entropiques des équations de Saint Venant. Grace aux théorèmes de Fenichel, on montre l'existence de roll-waves continues "visqueuses" proches des roll-waves discontinues lorsqu'on ajouté aux équations un petit terme de viscosité. On étudie ensuite la stabilité linéaire de ces roll-waves discontinues. Enfin, on montre l'existence de roll-waves de petite amplitude dans des canaux à fond périodiques.\\ Les breathers discrets sont des oscillations périodiques, localisées en espace dans des réseaux non linéaires discrets. On analyse d'abord le modèle Fermi-Pasta-Ulam (FPU) diatomique. En formulant le problème sous la forme d'un mapping en dimension infinie, on montre, via une réduction à une variété centrale, l'existence de breathers discrets de petite amplitude pour des rapports de masses arbitraires. On utilise aussi cette approche pour montrer l'existence de breathers discrets dans des chaines de spins ferromagnétiques.

We prove the existence and linear stability of quasi-periodic breather solutions in a 1d Hamiltonian lattice of identical, weakly-coupled, anharmonic oscillators with general on-site potentials and under the effect of long-ranged interaction, via de KAM technique. We prove the persistence of finite-dimensional tori which correspond in the uncoupled limit to N arbitrary lattice sites initially excited. The frequencies of the invariant tori of the perturbed system are only slightly deformed from the frequencies of the unperturbed tori.

This thesis is concerned with analysis of spectral and orbital stability of solitary wave solutions to discrete and continuous PT-symmetric nonlinear Schroedinger equations. The main tools of this analysis are inspired by Hamiltonian systems, where conserved quantities can be used for proving orbital stability and Krein signature can be computed for prediction of instabilities in the spectrum of linearization. The main results are obtained for the chain of coupled pendula represented by a discrete NLS model, and for the trapped atomic gas represented by a continuous NLS model. Analytical results are illustrated with various numerical examples. / Thesis / Doctor of Philosophy (PhD)

Made available in DSpace on 2014-06-11T19:30:54Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-02-18Bitstream added on 2014-06-13T19:00:57Z : No. of bitstreams: 1 cortezgutierrez_ho_dr_sjrp.pdf: 1171634 bytes, checksum: dab1a18fa9cc0c38b49fe73954595b2d (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / O objetivo deste trabalho é investigar a localização de energia e o aprisionamento do breather no modelo de Peyrard Bishop com o potencial original de Morse, potencial simétrico de Morse e potenciais quárticos em cadeias homogêneas e não homogêneas usando o Teorema do limite anticontinuum. No caso não homogêneo, a impureza é introduzida pela profundidade do potencial. Foi observado que o modelo SPB apresenta pequenas amplitudes e menor densidade de energia comparada com o modelo PB. Para potenciais quárticos simétricos e assimétricos foi verificado numericamente que não temos transição de fase usando o conceito de amplitude média das vibrações. Entretanto, um potencial híbrido formado por um quártico e Morse pode fornecer uma grande amplitude de vibração das fitas de DNA . Finalmente, no sistema não homogêneo, se verificou a hipótese de aprisionamento por meio de uma simulação de interação do breather móvel com a região de TATA box para uma cadeia longa de DNA, correspondente à seqüencia de nucleotídeos que codifica a insulina. Este resultado deve ajudar a estender a aplicação dos breathers discretos a sistemas biológicos que levam em conta reações bioquímicas localizadas (como, por exemplo, reações enzimáticas) em macromoléculas biológicas. O modelo PB pode ser modificado para explicar a existência de movimentos localizados de grande amplitude. O decaimento da função potencial para valores grandes mostraram uma vibração localizada no equilíbrio de grande amplitude. Isso é feito alterando o potencial on site. / The objective of this work is to investigate the energy localization and the breather trapping in the Peyrard-Bishop model with the original Morse potential, symmetric Morse potential and quartics potentials in homogeneous and inhomogeneous chains. In the inhomogeneous case, the impurity is introduced by the depth of the potential. It was observed that the SPB model shows small amplitude and lower density of energy compared with the PB model. Quartics potentials, for symmetric and asymmetric cases, do not show phase transition. It is verified by numerical calculation using the concept of mean amplitude of the vibrations. Meanwhile, a potential hybrid formed by a quartic and Morse is ideal for DNA applications. Finally, it was verified the trapping hypotheses through the simulation of interaction of mobile breather with the region of TATA box for a long chain of DNA corresponding to the nucleotide sequence that encodes the insulin. This result should help to extend the application of discrete breathers the biological systems that take into account located biochemical reactions (such as enzymatic reactions) in biological macromolecules. In addition, the PB model can be modified to explain the existence of highly localized large amplitudes motions of the base pairs in DNA. We do this change the on site potential.

We address the excitation of quantum breathers in small nonlinear networks of two and three degrees of freedom, in order to study their properties. The invariance under permutation of two sites of these networks substitutes the translation invariance that is present in nonlinear lattices, where (classical) discrete breathers are time periodic space localized solutions of the underlying classical equations of motion. We do a systematic analysis of the spectrum and eigenstates of such small systems, characterizing quantum breather states by their tunnelling rate (energy splitting), site correlations, fluctuations of the number of quanta, and entanglement. We observe how these properties are reflected in the time evolution of initially localized excitations. Quantum breathers manifest as pairs of nearly degenerate eigenstates that show strong site correlation of quanta, and are characterized by a strong excitation of quanta on one site of the network which perform slow coherent tunnelling motion from one site to another. They enhance the fluctuations of quanta, and are the least entangled states among the group of eigenstates in the same range of the energy spectrum. We use our analysis methods to consider the excitation of quantum breathers in a cell of two coupled Josephson junctions, and study their properties as compared with those in the previous cases. We describe how quantum breathers could be experimentally observed by employing the already developed techniques for quantum information processing with Josephson junctions.

In this work the interaction between a spatial localized and time periodic state (discrete breather) and small amplitude plane waves is studied.