The hierarchy of rogue wave solutions in nonlinear systems

Oceanic freak waves, optical spikes and extreme events in numerous contexts can arguably be modelled by modulationally unstable solutions within nonlinear systems. In particular, the fundamental nonlinear Schroedinger equation (NLSE) hosts a high-amplitude spatiotemporally localised solution on a plane-wave background, called the Peregrine breather, which is generally considered to be the base-case prototype of a rogue wave. Nonetheless, until very recently, little was known about what to expect when observing or engineering entire clusters of extreme events. Accordingly, this thesis aims to elucidate this matter by investigating complicated structures formed from collections of Peregrine breathers. Many novel NLSE solutions are discovered, all systematically classifiable by their geometry. The methodology employed here is based on the well-established concept of Darboux transformations, by which individual component solutions of an integrable system are nonlinearly superimposed to form a compound wavefunction. It is primarily implemented in a numerical manner within this study, operating on periodically modulating NLSE solutions called breathers. Rogue wave structures can only be extracted at the end of this process, when a limit of zero modulation frequency is applied to all components. Consequently, a requirement for breather asymmetry ensures that a multi-rogue wavefunction must be formed from a triangular number of individual Peregrine breathers (e.g. 1, 3, 6, 10, ...), whether fused or separated. Furthermore, the arrangements of these are restricted by a maximum phase-shift allowable along an evolution trajectory through the relevant wave field. Ultimately, all fundamental high-order rogue wave solutions can be constructed via polynomial relations between origin-translating component shifts and squared modulation frequency ratios. They are simultaneously categorisable by both these mathematical existence conditions and the corresponding visual symmetries, appearing spatiotemporally as triangular cascades, pentagrams, heptagrams, and so on. These parametric relations do not conflict with each other, meaning that any arbitrary NLSE rogue wave solution can be considered a hybridisation of this elementary set. Moreover, this hierarchy of structures is significantly general, with complicated arrangements persisting even on a cnoidal background

Hydrodynamics

The phenomena related to the flow of fluids are generally complex, and difficult to quantify. New approaches - considering points of view still not explored - may introduce useful tools in the study of Hydrodynamics and the related transport phenomena. The details of the flows and the properties of the fluids must be considered on a very small scale perspective. Consequently, new concepts and tools are generated to better describe the fluids and their properties. This volume presents conclusions about advanced topics of calculated and observed flows. It contains eighteen chapters, organized in five sections: 1) Mathematical Models in Fluid Mechanics, 2) Biological Applications and Biohydrodynamics, 3) Detailed Experimental Analyses of Fluids and Flows, 4) Radiation-, Electro-, Magnetohydrodynamics, and Magnetorheology, 5) Special Topics on Simulations and Experimental Data. These chapters present new points of view about methods and tools used in Hydrodynamics

Optical Communication

Optical communication is very much useful in telecommunication systems, data processing and networking. It consists of a transmitter that encodes a message into an optical signal, a channel that carries the signal to its desired destination, and a receiver that reproduces the message from the received optical signal. It presents up to date results on communication systems, along with the explanations of their relevance, from leading researchers in this field. The chapters cover general concepts of optical communication, components, systems, networks, signal processing and MIMO systems. In recent years, optical components and other enhanced signal processing functions are also considered in depth for optical communications systems. The researcher has also concentrated on optical devices, networking, signal processing, and MIMO systems and other enhanced functions for optical communication. This book is targeted at research, development and design engineers from the teams in manufacturing industry, academia and telecommunication industries

Geometric Analysis of Nonlinear Partial Differential Equations

This book contains a collection of twelve papers that reflect the state of the art of nonlinear differential equations in modern geometrical theory. It comprises miscellaneous topics of the local and nonlocal geometry of differential equations and the applications of the corresponding methods in hydrodynamics, symplectic geometry, optimal investment theory, etc. The contents will be useful for all the readers whose professional interests are related to nonlinear PDEs and differential geometry, both in theoretical and applied aspects

Conception of a Low-Noise Ultrafast Fiber Laser System Toward nJ Phase Stable Pulses in the Mid-Infrared

Dans le contexte du développement technologique d’un nouveau laboratoire en photonique quantique ultrarapide, un système laser tout-fibre à maintien de polarisation est présenté. En exploitant les propriétés de stabilité exceptionnelle des lasers à fibres dopées à l’erbium, l’objectif de ce projet est de générer un train d’impulsions fs dont le niveau de bruit se rapproche de la limite quantique. Dans le but de faire de la recherche novatrice en spectroscopie à résolution temporelle, l’intensité des impulsions produites par le système laser fibré est maximisée pour entrainer efficacement un processus non linéaire de génération de fréquences dans l’infra-rouge moyen. Dans le présent travail, un laser à fibre dopée à l’erbium atteignant passivement la synchronisation de modes est construit, et celui-ci produit de manière fiable et robuste un train d’impulsions ultrabrèves à un taux de répétition de 75.4 MHz. Le bruit d’intensité de l’oscillateur maître est profondément caractérisé. Les résultats indiquent un bruit d’intensité relative de 0.023 % (0.048 %) intégré sur une plage de fréquence allant de 10 Hz à 100 kHz (1 MHz). L’analyse de la densité spectrale de puissance du bruit d’intensité relative de l’oscillateur révèle une domination du bruit technique à basses fréquences induit par le laser de pompe. Afin d’augmenter significativement l’intensité des impulsions produites par l’oscillateur maître, un amplificateur à fibre dopée à l’erbium est conçu et caractérisé. Une distribution longitudinale ajustée de la dispersion à travers le système fibré permet d’obtenir une énergie d’impulsion de 4.2 nJ tout en contrôlant l’impact des effets optiques non linéaires dans l’amplificateur. De même, la composition spectrale des impulsions est élargie jusqu’à 55 nm de largeur à mi-hauteur afin de permettre une durée temporelle d’impulsion théoriquement minimale de 53 fs. Un modèle numérique de la propagation des impulsions dans le système d’amplification laser fibré est développé afin de faciliter le pro-cessus d’optimisation du spectre en sortie de l’amplificateur. Les simulations révèlent qu’une compression temporelle excessive dans la fibre de sortie de l’amplificateur pourrait causer une dégradation de la qualité des impulsions. Pour contrer les impacts indésirables de la variation de phase spectrale irrégulière observée dans le présent système, le modèle numérique est utilisé pour proposer une nouvelle configuration de dispersion. Les prochaines étapes pour l’optimisation du système sont claires, et l’efficacité d’un processus de compression temporelle des impulsions en sortie de l’amplificateur s’en trouvera améliorée, tout comme la forme des impulsions elle-même.----------Abstract An ultrafast all-PM Er:fiber laser system is presented in the context of a larger technological scheme for research in quantum optics. The objective is to obtain an ultralow-noise high peak power pulse train for eÿcient driving of nonlinear frequency mixing processes and generation of phase-stable pulses in the MIR. A self-starting passively mode-locked Er:fiber laser operating at the repetition rate of 75.4 MHz has been constructed and optimized for noise reduction. The intensity noise of the master oscillator has been extensively studied, and shows an integrated rms RIN of 0.023 % (0.048 %) on a frequency range of 10 Hz to 100 kHz (1 MHz). Through analysis of the RIN PSD, we find that our laser free-running intensity noise is dominated by pump technical noise at low frequency. A broadband EDFA running in the self-similar pulse amplification regime has been constructed for pulse energy amplification to 4.2 nJ and its dispersion map permits controlled SPM spectral broadening to 55 nm (FWHM). For further optimization of the EDFA output spectrum and pulse compressibily, a numerical model of the EDFA pulse propagation has been developed solving the NLSE, with gain implementation. From analysis of the EDFA output spectrum together with the simulations results, it is found that excessive temporal compression of the pulse in the EDFA output fiber may compromise the pulse shape quality. The numerical simulations are utilized to propose a strategic change to the amplification stage setup that could yield a clean output spectrum and eÿcient GVD compensation of the residual chirp for a compressed pulse duration approaching the theoretical transform-limited value of 53 fs (FWHM) and peak powers above 60 kW

The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series

Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41670/1/10440_2004_Article_193995.pd

Nonlinear Dynamics

This volume covers a diverse collection of topics dealing with some of the fundamental concepts and applications embodied in the study of nonlinear dynamics. Each of the 15 chapters contained in this compendium generally fit into one of five topical areas: physics applications, nonlinear oscillators, electrical and mechanical systems, biological and behavioral applications or random processes. The authors of these chapters have contributed a stimulating cross section of new results, which provide a fertile spectrum of ideas that will inspire both seasoned researches and students