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Author: Lucie Delcey
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
Dans le cadre de ce projet de thèse nous étudierons la stabilité temporelle des ondes progressives périodiques et localisées pour l'équation de Lugiato-Lefever, récemment redécouverte en photonique dans le contexte de la formation de peignes de fréquences par effet Kerr dans des résonateurs optiques à modes de galerie. L'équation de Lugiato-Lefever peut être vue comme une version de l'équation de Schrödinger non linéaire incluant des termes d'amortissement, d'excitation extérieure, et de décalage fréquentiel de résonance. D'un point de vue mathématique, cette équation a été très peu étudiée et de nombreuses questions, notamment sur la dynamique de ses solutions, sont entièrement ouvertes. Dans un travail en cours de rédaction, Cyril Godey (doctorant au LMB financé par la région Franche-Comté) a effectué une analyse systématique des bifurcations locales des ondes non linéaires de l'équation de Lugiato-Lefever montrant l'existence de nombreuses solutions, dont certaines encore inconnues dans la littérature physique. Une grande partie des solutions observées expérimentalement ont pu être identifiées par cette analyse. L'étape suivante consiste dans l'étude des propriétés de stabilité. Dans ce projet de thèse nous comptons étudier de manière systématique la stabilité des ondes périodiques et localisées, et tout particulièrement -- la stabilité spectrale, qui fournit des conditions nécessaires de stabilité, et -- la stabilité non linéaire, orbitale et asymptotique, qui donne des conditions suffisantes de stabilité. Ces ondes jouent un rôle central dans le problème physique mentionné ci-dessus, et un des buts recherchés, au-delà de l'analyse mathématique, est de permettre une meilleure compréhension du problème physique.
Author: Lucie Delcey
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
Dans le cadre de ce projet de thèse nous étudierons la stabilité temporelle des ondes progressives périodiques et localisées pour l'équation de Lugiato-Lefever, récemment redécouverte en photonique dans le contexte de la formation de peignes de fréquences par effet Kerr dans des résonateurs optiques à modes de galerie. L'équation de Lugiato-Lefever peut être vue comme une version de l'équation de Schrödinger non linéaire incluant des termes d'amortissement, d'excitation extérieure, et de décalage fréquentiel de résonance. D'un point de vue mathématique, cette équation a été très peu étudiée et de nombreuses questions, notamment sur la dynamique de ses solutions, sont entièrement ouvertes. Dans un travail en cours de rédaction, Cyril Godey (doctorant au LMB financé par la région Franche-Comté) a effectué une analyse systématique des bifurcations locales des ondes non linéaires de l'équation de Lugiato-Lefever montrant l'existence de nombreuses solutions, dont certaines encore inconnues dans la littérature physique. Une grande partie des solutions observées expérimentalement ont pu être identifiées par cette analyse. L'étape suivante consiste dans l'étude des propriétés de stabilité. Dans ce projet de thèse nous comptons étudier de manière systématique la stabilité des ondes périodiques et localisées, et tout particulièrement -- la stabilité spectrale, qui fournit des conditions nécessaires de stabilité, et -- la stabilité non linéaire, orbitale et asymptotique, qui donne des conditions suffisantes de stabilité. Ces ondes jouent un rôle central dans le problème physique mentionné ci-dessus, et un des buts recherchés, au-delà de l'analyse mathématique, est de permettre une meilleure compréhension du problème physique.
Author: Cyril Godey
Publisher:
ISBN:
Category :
Languages : en
Pages : 112
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Book Description
In this thesis we present several contributions to qualitative study of solutions of nonlinear partial differential equations in optics and fluid mechanics models. More precisely, we focus on the existence of solutions and their stability properties. In Chapter 1, we study the Lugiato-lefever equation, which is a variant of the nonlinear Schrödinger equation arising in sereval contexts in nonlinear optics. Using tools from bifurcation and normal forms theory, we perfom a systematic analysis of stationary solutions of this equation and prove the existence of periodic and localized solutions. In Chapter 2, we present a simple criterion for linear instability of nonlinear waves. We then apply this result to the Lugiato-Lefever equation, to the Kadomtsev-Petviashvili-I equation and the Davey-Stewartson equations. These last two equations are model equations arising in fluid mechanics. In Chapter 3, we prove a criterion for linear instability of periodic solutions with small amplitude, with respect to certain quasiperiodic perturbations. This result is then applied to the Lugiato-Lefever equation.
Author: Willy Dörfler
Publisher: Springer Nature
ISBN: 3030471748
Category : Mathematics
Languages : en
Pages : 330
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Book Description
Wave phenomena are ubiquitous in nature. Their mathematical modeling, simulation and analysis lead to fascinating and challenging problems in both analysis and numerical mathematics. These challenges and their impact on significant applications have inspired major results and methods about wave-type equations in both fields of mathematics. The Conference on Mathematics of Wave Phenomena 2018 held in Karlsruhe, Germany, was devoted to these topics and attracted internationally renowned experts from a broad range of fields. These conference proceedings present new ideas, results, and techniques from this exciting research area.
Author: Daniele Sanvitto
Publisher: Springer Science & Business Media
ISBN: 3642241867
Category : Science
Languages : en
Pages : 416
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Book Description
In the past decade, there has been a burst of new and fascinating physics associated to the unique properties of two-dimensional exciton polaritons, their recent demonstration of condensation under non-equilibrium conditions and all the related quantum phenomena, which have stimulated extensive research work. This monograph summarizes the current state of the art of research on exciton polaritons in microcavities: their interactions, fast dynamics, spin-dependent phenomena, temporal and spatial coherence, condensation under non-equilibrium conditions, related collective quantum phenomena and most advanced applications. The monograph is written by the most active authors who have strongly contributed to the advances in this area. It is of great interests to both physicists approaching this subject for the first time, as well as a wide audience of experts in other disciplines who want to be updated on this fast moving field.
Author: Joachim Krieger
Publisher: European Mathematical Society
ISBN: 9783037191064
Category : Differential equations, Hyperbolic
Languages : en
Pages : 494
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Book Description
Wave maps are the simplest wave equations taking their values in a Riemannian manifold $(M,g)$. Their Lagrangian is the same as for the scalar equation, the only difference being that lengths are measured with respect to the metric $g$. By Noether's theorem, symmetries of the Lagrangian imply conservation laws for wave maps, such as conservation of energy. In coordinates, wave maps are given by a system of semilinear wave equations. Over the past 20 years important methods have emerged which address the problem of local and global wellposedness of this system. Due to weak dispersive effects, wave maps defined on Minkowski spaces of low dimensions, such as $\mathbb R^{2+1}_{t,x}$, present particular technical difficulties. This class of wave maps has the additional important feature of being energy critical, which refers to the fact that the energy scales exactly like the equation. Around 2000 Daniel Tataru and Terence Tao, building on earlier work of Klainerman-Machedon, proved that smooth data of small energy lead to global smooth solutions for wave maps from 2+1 dimensions into target manifolds satisfying some natural conditions. In contrast, for large data, singularities may occur in finite time for $M =\mathbb S^2$ as target. This monograph establishes that for $\mathbb H$ as target the wave map evolution of any smooth data exists globally as a smooth function. While the authors restrict themselves to the hyperbolic plane as target the implementation of the concentration-compactness method, the most challenging piece of this exposition, yields more detailed information on the solution. This monograph will be of interest to experts in nonlinear dispersive equations, in particular to those working on geometric evolution equations.
Author: Yuri S. Kivshar
Publisher: Academic Press
ISBN: 0080538096
Category : Technology & Engineering
Languages : en
Pages : 557
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Book Description
The current research into solitons and their use in fiber optic communications is very important to the future of communications. Since the advent of computer networking and high speed data transmission technology people have been striving to develop faster and more reliable communications media. Optical pulses tend to broaden over relatively short distances due to dispersion, but solitons on the other hand are not as susceptible to the effects of dispersion, and although they are subject to losses due to attenuation they can be amplified without being received and re-transmitted.This book is the first to provide a thorough overview of optical solitons. The main purpose of this book is to present the rapidly developing field of Spatial Optical Solitons starting from the basic concepts of light self-focusing and self-trapping. It will introduce the fundamental concepts of the theory of nonlinear waves and solitons in non-integrated but physically realistic models of nonlinear optics including their stability and dynamics. Also, it will summarize a number of important experimental verification of the basic theoretical predictions and concepts covering the observation of self-focusing in the earlier days of nonlinear optics and the most recent experimental results on spatial solitons, vortex solitons, and soliton interaction & spiraling.* Introduces the fundamental concepts of the theory of nonlinear waves and solitons through realistic models * Material is based on authors' years of experience actively working in and researching the field* Summarizes the most important experimental verification of the basic theories, predictions and concepts of this ever evolving field from the earliest studies to the most recent
Author: John Scott Russell
Publisher:
ISBN:
Category : Waves
Languages : en
Pages : 124
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Author: George Papanicolaou
Publisher: Springer Science & Business Media
ISBN: 1461216788
Category : Mathematics
Languages : en
Pages : 301
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Book Description
This IMA Volume in Mathematics and its Applications WAVE PROPAGATION IN COMPLEX MEDIA is based on the proceedings of two workshops: • Wavelets, multigrid and other fast algorithms (multipole, FFT) and their use in wave propagation and • Waves in random and other complex media. Both workshops were integral parts of the 1994-1995 IMA program on "Waves and Scattering." We would like to thank Gregory Beylkin, Robert Burridge, Ingrid Daubechies, Leonid Pastur, and George Papanicolaou for their excellent work as organizers of these meetings. We also take this opportunity to thank the National Science Foun dation (NSF), the Army Research Office (ARO, and the Office of Naval Research (ONR), whose financial support made these workshops possible. A vner Friedman Robert Gulliver v PREFACE During the last few years the numerical techniques for the solution of elliptic problems, in potential theory for example, have been drastically improved. Several so-called fast methods have been developed which re duce the required computing time many orders of magnitude over that of classical algorithms. The new methods include multigrid, fast Fourier transforms, multi pole methods and wavelet techniques. Wavelets have re cently been developed into a very useful tool in signal processing, the solu tion of integral equation, etc. Wavelet techniques should be quite useful in many wave propagation problems, especially in inhomogeneous and nonlin ear media where special features of the solution such as singularities might be tracked efficiently.
Author: Lubomir Banas
Publisher: Walter de Gruyter
ISBN: 3110307103
Category : Mathematics
Languages : en
Pages : 248
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Book Description
This monograph examines magnetization dynamics at elevated temperatures which can be described by the stochastic Landau-Lifshitz-Gilbert equation (SLLG). The first part of the book studies the role of noise in finite ensembles of nanomagnetic particles: we show geometric ergodicity of a unique invariant measure of Gibbs type and study related properties of approximations of the SLLG, including time discretization and Ginzburg-Landau type penalization. In the second part we propose an implementable space-time discretization using random walks to construct a weak martingale solution of the corresponding stochastic partial differential equation which describes the magnetization process of infinite spin ensembles. The last part of the book is concerned with a macroscopic deterministic equation which describes temperature effects on macro-spins, i.e. expectations of the solutions to the SLLG. Furthermore, comparative computational studies with the stochastic model are included. We use constructive tools such as e.g. finite element methods to derive the theoretical results, which are then used for computational studies. The numerical experiments motivate an interesting interplay between inherent geometric and stochastic effects of the SLLG which still lack a rigorous analytical understanding: the role of space-time white noise, possible finite time blow-up behavior of solutions, long-time asymptotics, and effective dynamics.
Author: Mark Ainsworth
Publisher: Springer
ISBN: 9783642554841
Category : Mathematics
Languages : en
Pages : 410
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Book Description
These ten detailed and authoritative survey articles on numerical methods for direct and inverse wave propagation problems are written by leading experts. Researchers and practitioners in computational wave propagation, from postgraduate level onwards, will find the breadth and depth of coverage of recent developments a valuable resource. The articles describe a wide range of topics on the application and analysis of methods for time and frequency domain PDE and boundary integral formulations of wave propagation problems. Electromagnetic, seismic and acoustic equations are considered. Recent developments in methods and analysis ranging from finite differences to hp-adaptive finite elements, including high-accuracy and fast methods are described with extensive references.
