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

Plasma Dynamics

Contains research objectives and summary of research on twenty-one projects split into three sections, with four sub-sections in the second section and reports on twelve research projects.National Science Foundation (Grant ENG75-06242)U.S. Energy Research and Development Administration (Contract E(11-1)-2766)U.S. Energy Research and Development Agency (Contract E(11-1)-3070)U.S. Energy Research and Development Administration (Contract E(11-1)-3070)Research Laboratory of Electronics, M.I.T. Industrial Fellowshi

Analysis and control of rogue waves in fibre lasers and in hydrodynamics: integrable turbulence framework

Understanding mechanisms underlying the formation of extreme events is the problem of primary importance in various domains of study including hydrodynamics, optics, forecasting natural disasters etc. In these domains, extreme events are known as RogueWaves (RWs). RWs are localised coherent structures of unusually large amplitude spontaneously emerging in nonlinear random wave fields, and as such, can have damaging effect on the environment (e.g. offshore engineering structures) or on the medium they propagate through (e.g. optical fibres). Within this PhD project several problems related to the emergence, control and manipulation of RWs in fibre optics and in hydrodynamics have been investigated. The particular emphasis is on the study of RWs emerging in the propagation of the so-called partially coherent waves described by the focusing nonlinear Schr¨odinger equation (fNLSE), the universal model for the propagation of modulationally unstable quasi-monochromatic wavepackets in a broad range of physical media. fNLSE belongs to the class of the completely integrable equations possessing deep mathematical structure and amenable to analytical methods such as Inverse Scattering Transform and Finite-Gap Integration. We use recent mathematical discoveries related to the semi classical, or small-dispersion, limit of fNLSE to investigate analytically, numerically and experimentally the formation of RWs within the framework of integrable turbulence—the emerging theory of random waves in integrable systems. The study of the RW formation in this project has also prompted the investigation of a closely related problem concerned with dynamics of soliton and breather gases as special types of integrable turbulence. The project’s findings fall in five categories: (i) the analytical description of the emergence of the so-called “heavy tails” in the probability distribution for the field intensity at the early stage of the development of integrable turbulence; (ii) the development and experimental realisation in a water tank of nonlinear spectral engineering, the IST-based method of control and manipulation of RWs; (iii) the development of the spectral theory of bidirectional soliton gases; (iv) numerical synthesis of breather gases and the verification of the recently developed spectral kinetic theory for such gases; (v) the investigation of the RWformation in the compression of broad optical pulses in the highly nonlinear propagation regimes, when the higher order effects such as self steepening, third order dispersion and Raman scattering need to be taken into account

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

Abstract book

Welcome at the International Conference on Differential and Difference Equations & Applications 2015. The main aim of this conference is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential & difference equations will be represented with special emphasis on applications. It will be mathematically enriching and socially exciting event. List of registered participants consists of 169 persons from 45 countries. The five-day scientific program runs from May 18 (Monday) till May 22, 2015 (Friday). It consists of invited lectures (plenary lectures and invited lectures in sections) and contributed talks in the following areas: Ordinary differential equations, Partial differential equations, Numerical methods and applications, other topics

Laboratory-directed research and development: FY 1996 progress report

This report summarizes the FY 1996 goals and accomplishments of Laboratory-Directed Research and Development (LDRD) projects. It gives an overview of the LDRD program, summarizes work done on individual research projects, and provides an index to the projects` principal investigators. Projects are grouped by their LDRD component: Individual Projects, Competency Development, and Program Development. Within each component, they are further divided into nine technical disciplines: (1) materials science, (2) engineering and base technologies, (3) plasmas, fluids, and particle beams, (4) chemistry, (5) mathematics and computational sciences, (6) atomic and molecular physics, (7) geoscience, space science, and astrophysics, (8) nuclear and particle physics, and (9) biosciences

Two Phase Flow, Phase Change and Numerical Modeling

The heat transfer and analysis on laser beam, evaporator coils, shell-and-tube condenser, two phase flow, nanofluids, complex fluids, and on phase change are significant issues in a design of wide range of industrial processes and devices. This book includes 25 advanced and revised contributions, and it covers mainly (1) numerical modeling of heat transfer, (2) two phase flow, (3) nanofluids, and (4) phase change. The first section introduces numerical modeling of heat transfer on particles in binary gas-solid fluidization bed, solidification phenomena, thermal approaches to laser damage, and temperature and velocity distribution. The second section covers density wave instability phenomena, gas and spray-water quenching, spray cooling, wettability effect, liquid film thickness, and thermosyphon loop. The third section includes nanofluids for heat transfer, nanofluids in minichannels, potential and engineering strategies on nanofluids, and heat transfer at nanoscale. The forth section presents time-dependent melting and deformation processes of phase change material (PCM), thermal energy storage tanks using PCM, phase change in deep CO2 injector, and thermal storage device of solar hot water system. The advanced idea and information described here will be fruitful for the readers to find a sustainable solution in an industrialized society