In the present work, we study wave phenomena in strongly nonlinear lattices. Such lattices are characterized by the absence of classical linear waves. We demonstrate that compactons – strongly localized solitary waves with tails decaying faster than exponential – exist and that they play a major role in the dynamics of the system under consideration. We investigate compactons in different physical setups. One part deals with lattices of dispersively coupled limit cycle oscillators which find various applications in natural sciences such as Josephson junction arrays or coupled Ginzburg-Landau equations. Another part deals with Hamiltonian lattices. Here, a prominent example in which compactons can be found is the granular chain. In the third part, we study systems which are related to the discrete nonlinear Schrödinger equation describing, for example, coupled optical wave-guides or the dynamics of Bose-Einstein condensates in optical lattices. Our investigations are based on a numerical method to solve the traveling wave equation. This results in a quasi-exact solution (up to numerical errors) which is the compacton. Another ansatz which is employed throughout this work is the quasi-continuous approximation where the lattice is described by a continuous medium. Here, compactons are found analytically, but they are defined on a truly compact support. Remarkably, both ways give similar qualitative and quantitative results. Additionally, we study the dynamical properties of compactons by means of numerical simulation of the lattice equations. Especially, we concentrate on their emergence from physically realizable initial conditions as well as on their stability due to collisions. We show that the collisions are not exactly elastic but that a small part of the energy remains at the location of the collision. In finite lattices, this remaining part will then trigger a multiple scattering process resulting in a chaotic state. / In der hier vorliegenden Arbeit werden Wellenphänomene in stark nichtlinearen Gittern untersucht. Diese Gitter zeichnen sich vor allem durch die Abwesenheit von klassischen linearen Wellen aus. Es wird gezeigt, dass Kompaktonen – stark lokalisierte solitäre Wellen, mit Ausläufern welche schneller als exponentiell abfallen – existieren, und dass sie eine entscheidende Rolle in der Dynamik dieser Gitter spielen. Kompaktonen treten in verschiedenen diskreten physikalischen Systemen auf. Ein Teil der Arbeit behandelt dabei Gitter von dispersiv gekoppelten Oszillatoren, welche beispielsweise Anwendung in gekoppelten Josephsonkontakten oder gekoppelten Ginzburg-Landau-Gleichungen finden. Ein weiterer Teil beschäftigt sich mit Hamiltongittern, wobei die granulare Kette das bekannteste Beispiel ist, in dem Kompaktonen beobachtet werden können. Im dritten Teil werden Systeme, welche im Zusammenhang mit der Diskreten Nichtlinearen Schrödingergleichung stehen, studiert. Diese Gleichung beschreibt beispielsweise Arrays von optischen Wellenleitern oder die Dynamik von Bose-Einstein-Kondensaten in optischen Gittern. Das Studium der Kompaktonen basiert hier hauptsächlich auf dem numerischen Lösen der dazugehörigen Wellengleichung. Dies mündet in einer quasi-exakten Lösung, dem Kompakton, welches bis auf numerische Fehler genau bestimmt werden kann. Ein anderer Ansatz, der in dieser Arbeit mehrfach verwendet wird, ist die Approximation des Gitters durch ein kontinuierliches Medium. Die daraus resultierenden Kompaktonen besitzen einen im mathematischen Sinne kompakten Definitionsbereich. Beide Methoden liefern qualitativ und quantitativ gut übereinstimmende Ergebnisse. Zusätzlich werden die dynamischen Eigenschaften von Kompaktonen mit Hilfe von direkten numerischen Simulationen der Gittergleichungen untersucht. Dabei wird ein Hauptaugenmerk auf die Entstehung von Kompaktonen unter physikalisch realisierbaren Anfangsbedingungen und ihre Kollisionen gelegt. Es wird gezeigt, dass die Wechselwirkung nicht exakt elastisch ist, sondern dass ein Teil ihrer Energie an der Position der Kollision verharrt. In endlichen Gittern führt dies zu einem multiplen Streuprozess, welcher in einem chaotischen Zustand endet.

Made available in DSpace on 2015-05-14T12:14:16Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 9476998 bytes, checksum: 23f6bc7db98a0b2fa2dfbd3464093f70 (MD5) Previous issue date: 2014-04-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Granular chains are one dimensional systems where elastic grains placed along a line are in contact and interact with neighbors. These systems are excellent nonlinear waveguides allowing the control of some properties of the wave through different disposition of the grains. Among the applications of practical interest, the impact absorption draws attention. In this thesis, we propose to study a tapered granular chain, very short and decorated with grains appropriately positioned to maximize shock absorption. We call this system quasi-one-dimensional chain, because it is a three-dimensional distribution of grains whose dynamic of interest take place along a fixed direction. To study the dynamics of this system, we integrated numerically the equations of motion. Perturbations caused by collision with the grains at the edge of the chain spread quickly throughout the system and do not propagate as solitary waves. Results shows that impact pulses are attenuated almost completely and combinations of materials are important. Futhermore, dissipation, gravity and impact velocities almost do not influence the dynamics of the system. The decorating grains act as an auxiliary chain which traps part of the energy and linear momentum, preventing the formation of pulses and playing a central role in impact mitigation. Therefore, we present a new granular chain where we increase the number of grains per units length and maximize the dispersion of the momentum. / Cadeias granulares são sistemas unidimensionais onde grãos elásticos enfileirados estão em contato e interagem com dois vizinhos. Estes sistemas são excelentes guias de onda não lineares capazes de controlar algumas características das ondas através de diferentes arranjos de grãos. Dentre as aplicações de interesse prático, chama a atenção a absorção de impacto. Nesta tese, o que propomos estudar é uma cadeia granular afilada, bastante curta e decorada com grãos adequadamente posicionados para maximizar a absorção de impactos. Nós chamamos este sistema de cadeia quase unidimensional, a qual é uma distribuição tridimensional dos grãos, cuja dinâmica de interesse se dá ao longo de uma direção fixa. Para estudar a dinâmica desse sistema, integramos numericamente as equações de movimento. Perturbações causadas por colisão com os grãos nas extremidades da cadeia se espalham rapidamente através do sistema inteiro e não se propagam como ondas solitárias. Resultados das simulações mostram que pulsos de impacto são atenuados quase completamente e combinações de materiais são importantes. Além disso, dissipação, gravidade e velocidades de impactos quase não influenciam a dinâmica do sistema. Os grãos decorados atuam como uma cadeia auxiliar que aprisiona parte da energia e momento linear, impedindo assim a formação de pulsos e têm papel central na atenuação de impactos. Assim, apresentamos uma cadeia granular inovadora onde aumentamos a quantidade de grãos por unidade de comprimento e maximizamos a dispersão do momento linear.