Konstanz, Univ., Diplomarb., 2002.

Stuttgart, Univ., Diss., 2001.

Ce mémoire de thèse présente une approche semi-analytique des différentes solutions solitons spatio-temporelles de l'équation cubique quintique de Ginzburg-Landau complexe étendue à (3+1)D (GL3D).La méthode semi-analytique choisie est celle des coordonnées collectives qui permet d'approcher le champ exact, dont l'expression analytique est inconnue, par une fonction d'essai, qui comporte un nombre limité de paramètres physiques.En appliquant cette procédure à l'équation GL3D, nous obtenons un système d'équations variationnelles qui gouverne l'évolution des paramètres de la balle de lumière. Nous montrons que cette approche des coordonnées collectives est incomparablement plus rapide que la procédure de résolution directe de l'équation GL3D. cette rapidité permet d'obtenir, en un temps record, une cartographie générale des comportements dynamiques des balles de lumière. Cette cartographie révèle une riche variété d'états dynamiques faite de balles de lumière stationnaires, oscillantes et rotatives.Finalement, les résultats de cette thèse prédisent l'existence de plusieurs familles de balles de lumière, et précisent les domaines respectifs de leurs paramètres physiques. Cette prédiction constitue un pas en avant dans les efforts entrepris ces dernières années en vue d'une démonstration expérimentale de ce type d'impulsions. / This thesis presents a semi-analytical approach for the search of (3+1)D spatio-temporal soliton solutions of the complex cubic-quintic Ginzburg-Landau equation (GL3D).We use a semi-analytical method called collective coordinate approach, to obtain an approximate profile of the unknown pulse field. This ansatz function is chosen to be a function of a finite number of parameters describing the light pulse.By applying this collective corrdinate procedure to the GL3D equation, we obtain a system of variational equations which give the evolution of the light bullet parameters as a function of the propagation distance. We show that the collective coordinate approach is uncomparably faster than the direct numerical simulation of the propagation equation. This permits us to obtain, efficiently, a global mapping of the dynamical behavior of light bullets, which unveils a rich variety of dynamical states comprising stationary, pulsating and rotating light bullets.Finally the existence of several types of light bullets is predicted in specific domains of the equation parameters. Altogether, this theoretical and numerical work may be a useful tool next to the efforts undertaken these last years observing light bullets experimentally.

No description available.

Cette these se compose de quatre chapîtres et une annexe. Le premier chapître est consacre à des idées fondamentales de la theorie du flot de Ricci, qui montre comment nos travaux sont reliés a l'histoire entière. Dans le deuxième chapître, nous construisons une solution du flot de Ricci sur une variete a symétrie de rotation de telle sorte qu'il reste un collecteur complet a l'heure maximale. Nous dérivons également le non-effondrement pour certaines solutions anciennes à proximité de leur temps maximal. Chacun de ces deux resultats sont liés à la régularité des limites des solutions. Dans le troisième chapître, nous montrons qu'une estimation de type Shi d'ordre un est valable pour tenseur de Ricci sur des variétés qui satisfont l'inégalité Bianchi faibles. Le dernier chapître s'interesse aux gradient solitons de Ricci qui sont en expansion. Nous discutons du problème de classification et montrons que chaque cône tangent à l'infini d'un soliton expansion à "fast-than-quadratic-decay" courbure doit être $mathbb{R}^n$. / This thesis consists of four chapters and an appendix. The first chapter is dedicated to the fundamental ideas of the theory of Ricci flow, which shows how our works are connected to the whole story. In the second chapter, we construct a solution of Ricci flow on a rotationally symmetric manifold such that it remains a complete manifold at the maximal time. We also derive a noncollapsing property for certain ancient solutions near their maximal times. Both of these two results are related to the regularity of limits of solutions. In the third chapter, we show that a first order Shi-type estimate holds for Ricci tensor on manifolds which satisfy the weak Bianchi inequality. The last chapter is concerned with expanding gradient Ricci solitons. There we discuss the classification problem and show that every tangent cone at infinity of an expanding soliton with fast-than-quadratic-decay curvature must be $mathbb{R}^n$.

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.

The topic of this thesis is about experimental study of phenomena which are associated with light propagation in nonlinear dielectric media. In the first part of this work, we study experimentally the snake instability of the bright soliton stripe of the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation. The instability is observed, through spectral measurements, on spatially extended femtosecond pulses propagating in a normally dispersive self-defocusing semiconductor planar waveguide. The second part of this thesis is about light propagation in nonlinear periodic media. We experimentally observe a stationary spatial gap (or Bragg) soliton in a periodic semiconductor planar waveguide. Based on the interference pattern of the soliton beam, we measure the power parameter of the soliton which is related to the position of the spatial spectrum in the linear band gap. Cette thèse de doctorat a pour sujet l’étude expérimentale de phénomènes associés à la propagation de la lumière dans les milieux diélectriques non linéaires. La première partie porte sur la démonstration expérimentale de l’instabilité serpentine d’une bande solitonique dans un système décrit par une équation de Schrödinger non linéaire à (2+1)-dimensions. L’instabilité est observée sur base de mesures du spectre spatial ainsi que du profil spatio-fréquentiel d’une impulsion femtoseconde après propagation dans un guide plan semi-conducteur qui présente une dispersion normale et une non-linéarité défocalisante. Le second thème abordé concerne la propagation de la lumière dans les milieux non linéaires périodiques. Les expériences réalisées ont montré l’existence du soliton de Bragg spatial stationnaire sous forme de faisceaux se propageant dans des guides plans semi-conducteurs périodiquement gravés. Sur base du profil de la distribution modale en intensité du faisceau soliton, il a été possible de mesurer le paramètre de puissance du soliton de Bragg qui détermine la position du spectre spatial dans la bande interdite linéaire.

Expérience réalisée a l'aide d'un laser a colorant en anneau a blocage de mode passif, et dont la dispersion de vitesse de groupe peut être contrôlée. On montre que le fonctionnement de ce type de laser est base sur des phénomènes solitons en analysant un régime particulier du laser qui émet alors des impulsions semblables à des solitons d'ordre 3. Description d'une chaine amplificatrice ayant un gain d'environ 10**(6). On présente un nouveau système de compensation de la dispersion de la chaine amplificatrice et une nouvelle méthode de mesure monocoup de la réponse temporelle de l'effet Kerr optique.

Este trabalho relata um estudo numÃrico das caracterÃsticas Ãpticas de um interferÃmetro Mach-Zehnder disposto em uma configuraÃÃo com quatro estÃgios e um ressonador Ãptico em anel. O interferÃmetro Mach-Zehnder foi construÃdo numa versÃo com fibra comum e outra com fibra de dispersÃo decrescente considerando um perfil de dispersÃo linear. O ressonador Ãptico em anel foi constuÃdo com um guia de onda cujo o Ãndice de refraÃÃo nÃo linear foi moldado com um perfil linear crescente. A transmissÃo e o fator de compressÃo dos pulsos de saÃda sÃo analisados nos regimes de soliton e quasi-soliton para ambos os dispositivos. Os resultados mostraram que a transmissÃo do interferÃmetro Mach-Zehnder à fortemente dependente da potÃncia de entrada em ambos os regimes. O nÃvel de crosstalk tambÃm à dependente da potÃncia de entrada e da dispersÃo da fibra. Para potÃncias de entrada altas, o nÃvel de crosstalk à mais baixo para o dispositivo com fibra de dispersÃo decrescente no regime de soliton. Para esta configuraÃÃo, o nÃvel de crosstalk mÃnimo encontrado (-22 dB) foi para a potÃncia normalizada de entrada P = 2,7. No regime de quasi-soliton, em geral, o nÃvel de crosstalk mÃnimo à mais baixo para o interferÃmetro Mach-Zehnder com fibra de dispersÃo decrescente em potÃncias de entrada altas, mas com a possibilidade de se obter um nÃvel de crosstalk menor para o dispositivo utilizando fibra comum. A razÃo de extinÃÃo à tambÃm muito dependente da potÃncia de entrada e sofre degeneraÃÃo quando a potÃncia de entrada aumenta. Para potÃncias de entrada baixas, o interferÃmetro Mach-Zehnder apresenta uma razÃo de extinÃÃo melhor em potÃncias de entrada mais altas no regime de soliton. No regime de quasi-soliton, a melhor razÃo de extinÃÃo à obtida para o interferÃmetro Mach-Zehnder com fibra comum em potÃncias de entrada altas. Pode-se dizer que a operaÃÃo do dispositivo como uma chave Ãptica nÃo melhora com o uso de fibra de dispersÃo decrescente. Entretanto, para operaÃÃo de multiplex/demultiplex, o interferÃmetro Mach-Zehnder construÃdo com fibra de dispersÃo decrescente mostrou melhoramentos no nÃvel de crosstalk no regime de soliton e, dependendo da potÃncia de entrada, tambÃm no regime de quasi-soliton. No estudo do ressonador Ãptico em anel foi encontrado que a perda do guia de onda que forma a cavidade anelar diminui a transmissÃo em ambos os regimes, mas mantÃm o mesmo comportamento nÃo linear. O uso de um Ãndice de refraÃÃo nÃo linear com um perfil linear crescente leva à compressÃo ou alargamento temporal, dependendo da potÃncia de entrada, no regime de soliton. No regime de quasi-soliton nÃo foi observado deformaÃÃo temporal dos pulsos de saÃda. Isto indica que, em mÃdia, os pulsos de saÃda estÃo com a mesma duraÃÃo temporal que os pulsos de entrada. Os resultados tambÃm mostraram que hà um aumento da transmissÃo quando um Ãndice de refraÃÃo nÃo linear com um perfil linear crescente à usado. Assim, o decrÃscimo na transmissÃo associado à perda do guia de onda pode ser evitado. As caracterÃsticas de transmissÃo e a forma dos pulsos Ãpticos de saÃda do interferÃmetro Mach-Zehnder e do ressonador Ãptico em anel serÃo de interesse em circuitos Ãpticos e sistemas de comunicaÃÃes totalmente Ãpticos no futuro.

The decay of higher order solitons in optical fiber, initiated by localized channel perturbations such as a step change in dispersion, a localized loss element, or a bandpass filter, is explored theoretically and experimentally as a means of generating pairs of pulses having wavelengths that are up and down-shifted from the input wavelength. The achievable wavelength separation between the two sub pulses increases with increasing the amount of perturbations. Pulse parameter requirements for achieving useful wavelength shifts while avoiding unwanted nonlinear effects are presented. Experimental studies for N=2 solitons having 1 ps initial width are performed to demonstrate tunable wavelength conversion using a step change in dispersion and using a loss element. Wavelength shifts are tunable by varying the magnitude of a dispersion step or loss element that is used to disrupt the soliton cycle. Competing nonlinear effects, such as cubic dispersion, self-steepening, and stimulated Raman scattering, can be minimized by using input pulsewidths of one picosecond or greater. The separated pulses at two wavelengths can in principle be amplified to form separate higher order solitons. The process repeated to produce multiple wavelength replicas of an input data stream, and may thus be of possible use in multi-casting applications in fiber communication systems. The possibility of soliton recovery is also studied. For soliton recovery, conditions are stringent, in that the precise temporal overlap and phase relationship between sub-pulses that occurred at the point of separation is in principle needed at the reverse perturbation location. Experimental studies on soliton recovery for an N=2 soliton is performed by using a dispersion-compensated intermediate link, and reversing the dispersion step. Detrimental effects on soliton recovery are studied.