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Nonlinear Adiabatic Evolution of Quantum Systems: Geometric Phase and Virtual Magnetic Monopole
by Jie Liu Sheng-Chang Li Li-Bin Fu Di-Fa YeThis book systematically introduces the nonlinear adiabatic evolution theory of quantum many-body systems. The nonlinearity stems from a mean-field treatment of the interactions between particles, and the adiabatic dynamics of the system can be accurately described by the nonlinear Schrödinger equation. The key points in this book include the adiabatic condition and adiabatic invariant for nonlinear system; the adiabatic nonlinear Berry phase; and the exotic virtual magnetic field, which gives the geometric meaning of the nonlinear Berry phase. From the quantum-classical correspondence, the linear and nonlinear comparison, and the single particle and interacting many-body difference perspectives, it shows a distinct picture of adiabatic evolution theory. It also demonstrates the applications of the nonlinear adiabatic evolution theory for various physical systems. Using simple models it illustrates the basic points of the theory, which are further employed for the solution of complex problems of quantum theory for many-particle systems. The results obtained are supplemented by numerical calculations, presented as tables and figures.
Nonlinear Analysis - Theory and Methods (Springer Monographs in Mathematics)
by Nikolaos S. Papageorgiou Vicenţiu D. Rădulescu Dušan D. RepovšThis book emphasizes those basic abstract methods and theories that are useful in the study of nonlinear boundary value problems. The content is developed over six chapters, providing a thorough introduction to the techniques used in the variational and topological analysis of nonlinear boundary value problems described by stationary differential operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations as well as their applications to various processes arising in the applied sciences. They show how these diverse topics are connected to other important parts of mathematics, including topology, functional analysis, mathematical physics, and potential theory. Throughout the book a nice balance is maintained between rigorous mathematics and physical applications. The primary readership includes graduate students and researchers in pure and applied nonlinear analysis.
Nonlinear and Modern Mathematical Physics: NMMP-2022, Tallahassee, Florida, USA (Virtual), June 17–19 (Springer Proceedings in Mathematics & Statistics #459)
by Solomon Manukure Wen-Xiu MaThis book gathers peer-reviewed, selected contributions from participants of the 6th International Workshop on Nonlinear and Modern Mathematical Physics (NMMP-2022), hosted virtually from June 17–19, 2022. Works contained in this volume cover topics like nonlinear differential equations, integrable systems, Hamiltonian systems, inverse scattering transform, Painleve's analysis, nonlinear wave phenomena and applications, numerical methods of nonlinear wave equations, quantum integrable systems, and more. In this book, researchers and graduate students in mathematics and related areas will find new methods and tools that only recently have been developed to solve nonlinear problems. The sixth edition of the NMMP workshop was organized by Florida A&M University in Tallahassee, Florida, USA, with support from the University of South Florida, Florida State University, Embry-Riddle Aeronautical University, Savannah State University, Prairie View A&M University, and Beijing Jiaotong University. The aim was to bring together researchers from around the world to present their findings and foster collaboration for future research.
Nonlinear And Stochastic Climate Dynamics
by Christian L. E. Franzke Terence J. O’kaneIt is now widely recognized that the climate system is governed by nonlinear, multi-scale processes, whereby memory effects and stochastic forcing by fast processes, such as weather and convective systems, can induce regime behavior. Motivated by present difficulties in understanding the climate system and to aid the improvement of numerical weather and climate models, this book gathers contributions from mathematics, physics and climate science to highlight the latest developments and current research questions in nonlinear and stochastic climate dynamics. Leading researchers discuss some of the most challenging and exciting areas of research in the mathematical geosciences, such as the theory of tipping points and of extreme events including spatial extremes, climate networks, data assimilation and dynamical systems. This book provides graduate students and researchers with a broad overview of the physical climate system and introduces powerful data analysis and modeling methods for climate scientists and applied mathematicians.
Nonlinear Climate Dynamics
by Henk A. DijkstraThis book introduces stochastic dynamical systems theory in order to synthesize our current knowledge of climate variability. Nonlinear processes, such as advection, radiation and turbulent mixing, play a central role in climate variability. These processes can give rise to transition phenomena, associated with tipping or bifurcation points, once external conditions are changed. The theory of dynamical systems provides a systematic way to study these transition phenomena. Its stochastic extension also forms the basis of modern (nonlinear) data analysis techniques, predictability studies and data assimilation methods. Early chapters apply the stochastic dynamical systems framework to a hierarchy of climate models to synthesize current knowledge of climate variability. Later chapters analyse phenomena such as the North Atlantic Oscillation, El Niño/Southern Oscillation, Atlantic Multidecadal Variability, Dansgaard-Oeschger Events, Pleistocene Ice Ages, and climate predictability. This book will prove invaluable for graduate students and researchers in climate dynamics, physical oceanography, meteorology and paleoclimatology.
Nonlinear Differential Equations in Ordered Spaces (Monographs and Surveys in Pure and Applied Mathematics)
by S. Carl Seppo HeikkilaExtremality results proved in this Monograph for an abstract operator equation provide the theoretical framework for developing new methods that allow the treatment of a variety of discontinuous initial and boundary value problems for both ordinary and partial differential equations, in explicit and implicit forms. By means of these extremality res
Nonlinear Dispersive Waves: Based on the 2023 Workshop at University College Cork, Ireland (Advances in Mathematical Fluid Mechanics)
by David HenryThis volume explores a number of exciting developments in the field of nonlinear dispersive waves with a particular focus on waves arising in the ocean. Chapters are based on talks given at the workshop “Nonlinear Dispersive Waves” that was held at University College Cork, Ireland, on April 24-25, 2023. Specific topics covered include: The recovery of steady rotational wave surface profiles; Hamiltonian models for the propagation of long gravity waves; Waves propagating at the surface of a fluid covered by floating ice plates; The use of spherical coordinates to describe arctic ocean waves; Boundary value problems related to the Muskat Problem. Nonlinear Dispersive Waves will appeal to researchers as well as graduate students interested in this active area of research.
Nonlinear Dynamics: Exploration Through Normal Forms (Dover Books On Physics Series #Vol. 5)
by Prof. Yair Zarmi Peter B. KahnGeared toward advanced undergraduates and graduate students, this exposition covers the method of normal forms and its application to ordinary differential equations through perturbation analysis. In addition to its emphasis on the freedom inherent in the normal form expansion, the text features numerous examples of equations, the kind of which are encountered in many areas of science and engineering. The treatment begins with an introduction to the basic concepts underlying the normal forms. Coverage then shifts to an investigation of systems with one degree of freedom that model oscillations, in which the force has a dominant linear term and a small nonlinear one. The text considers a variety of nonautonomous systems that arise during the study of forced oscillatory motion. Topics include boundary value problems, connections to the method of the center manifold, linear and nonlinear Mathieu equations, pendula, Nuclear Magnetic Resonance, coupled oscillator systems, and other subjects. 1998 edition.
Nonlinear Dynamics of Structures Under Extreme Transient Loads
by Adnan Ibrahimbegovic Naida AdemovićThe effect of combined extreme transient loadings on a structure is not well understood—whether the source is man-made, such as an explosion and fire, or natural, such as an earthquake or extreme wind loading. A critical assessment of current knowledge is timely (with Fukushima-like disasters or terrorist threats). The central issue in all these problems is structural integrity, along with their transient nature, their unexpectedness, and often the uncertainty behind their cause. No single traditional scientific discipline provides complete answers, rather, a number of tools need to be brought together: nonlinear dynamics, probability theory, some understanding of the physical nature of the problem, as well as modeling and computational techniques for representing inelastic behavior mechanisms. Nonlinear Dynamics of Structures Under Extreme Transient Loads covers model building for different engineering structures and provides detailed presentations of extreme loading conditions. A number of illustrations are given quantifying; a plane crash or explosion induced impact loading, the effects of strong earthquake motion, and the impact and long-duration effects of strong stormy winds—along with a relevant framework for using modern computational tools. The book considers the levels of reserve in existing structures, and ways of reducing the negative impact of high-risk situations by employing sounder design procedures.
Nonlinear Elastic and Inelastic Models for Shock Compression of Crystalline Solids (Shock Wave and High Pressure Phenomena)
by John D. ClaytonThis book describes thermoelastic and inelastic deformation processes in crystalline solids undergoing loading by shock compression. Constitutive models with a basis in geometrically nonlinear continuum mechanics supply these descriptions. Large deformations such as finite strains and rotations, are addressed. The book covers dominant mechanisms of nonlinear thermoelasticity, dislocation plasticity, deformation twinning, fracture, flow, and other structure changes. Rigorous derivations of theoretical results are provided, with approximately 1300 numbered equations and an extensive bibliography of over 500 historical and modern references spanning from the 1920s to the present day. Case studies contain property data, as well as analytical, and numerical solutions to shock compression problems for different materials. Such materials are metals, ceramics, and minerals, single crystalline and polycrystalline.The intended audience of this book is practicing scientists (physicists, engineers, materials scientists, and applied mathematicians) involved in advanced research on shock compression of solid materials.
Nonlinear Evolution Equations
by Songmu ZhengNonlinear evolution equations arise in many fields of sciences including physics, mechanics, and material science. This book introduces some important methods for dealing with these equations and explains clearly and concisely a wide range of relevant theories and techniques. These include the semigroup method, the compactness and monotone operator
Nonlinear Filtering and Optimal Phase Tracking
by Zeev SchussThis book offers an analytical rather than measure-theoretical approach to the derivation of the partial differential equations of nonlinear filtering theory. The basis for this approach is the discrete numerical scheme used in Monte-Carlo simulations of stochastic differential equations and Wiener's associated path integral representation of the transition probability density. Furthermore, it presents analytical methods for constructing asymptotic approximations to their solution and for synthesizing asymptotically optimal filters. It also offers a new approach to the phase tracking problem, based on optimizing the mean time to loss of lock. The book is based on lecture notes from a one-semester special topics course on stochastic processes and their applications that the author taught many times to graduate students of mathematics, applied mathematics, physics, chemistry, computer science, electrical engineering, and other disciplines. The book contains exercises and worked-out examples aimed at illustrating the methods of mathematical modeling and performance analysis of phase trackers.
Nonlinear Fokker-Planck Flows and their Probabilistic Counterparts (Lecture Notes in Mathematics #2353)
by Viorel Barbu Michael RöcknerThis book delves into a rigorous mathematical exploration of the well-posedness and long-time behavior of weak solutions to nonlinear Fokker-Planck equations, along with their implications in the theory of probabilistically weak solutions to McKean-Vlasov stochastic differential equations and the corresponding nonlinear Markov processes. These are widely acknowledged as essential tools for describing the dynamics of complex systems in disordered media, as well as mean-field models. The resulting stochastic processes elucidate the microscopic dynamics underlying the nonlinear Fokker-Planck equations, whereas the solutions of the latter describe the evolving macroscopic probability distributions. The intended audience for this book primarily comprises specialists in mathematical physics, probability theory and PDEs. It can also be utilized as a one-semester graduate course for mathematicians. Prerequisites for the readers include a solid foundation in functional analysis and probability theory.
Nonlinear Hyperbolic Waves in Multidimensions (Monographs and Surveys in Pure and Applied Mathematics)
by Phoolan PrasadThe propagation of curved, nonlinear wavefronts and shock fronts are very complex phenomena. Since the 1993 publication of his work Propagation of a Curved Shock and Nonlinear Ray Theory, author Phoolan Prasad and his research group have made significant advances in the underlying theory of these phenomena. This volume presents their results and pr
Nonlinear Internal Waves in Lakes
by Kolumban HutterInternal wave dynamics in lakes (and oceans) is an important physical component of geophysical fluid mechanics of 'quiescent' water bodies of the Globe. The formation of internal waves requires seasonal stratification of the water bodies and generation by (primarily) wind forces. Because they propagate in basins of variable depth, a generated wave field often experiences transformation from large basin-wide scales to smaller scales. As long as this fission is hydrodynamically stable, nothing dramatic will happen. However, if vertical density gradients and shearing of the horizontal currents in the metalimnion combine to a Richardson number sufficiently small (< ¼), the light epilimnion water mixes with the water of the hypolimnion, giving rise to vertical diffusion of substances into lower depths. This meromixis is chiefly responsible for the ventilation of the deeper waters and the homogenization of the water through the lake depth. These processes are mainly formed as a result of the physical conditions, but they play biologically an important role in the trophicational state of the lake.
Nonlinear Least Squares for Inverse Problems
by Guy ChaventThis book provides an introduction into the least squares resolution of nonlinear inverse problems. The first goal is to develop a geometrical theory to analyze nonlinear least square (NLS) problems with respect to their quadratic wellposedness, i.e. both wellposedness and optimizability. Using the results, the applicability of various regularization techniques can be checked. The second objective of the book is to present frequent practical issues when solving NLS problems. Application oriented readers will find a detailed analysis of problems on the reduction to finite dimensions, the algebraic determination of derivatives (sensitivity functions versus adjoint method), the determination of the number of retrievable parameters, the choice of parametrization (multiscale, adaptive) and the optimization step, and the general organization of the inversion code. Special attention is paid to parasitic local minima, which can stop the optimizer far from the global minimum: multiscale parametrization is shown to be an efficient remedy in many cases, and a new condition is given to check both wellposedness and the absence of parasitic local minima. For readers that are interested in projection on non-convex sets, Part II of this book presents the geometric theory of quasi-convex and strictly quasi-convex (s.q.c.) sets. S.q.c. sets can be recognized by their finite curvature and limited deflection and possess a neighborhood where the projection is well-behaved. Throughout the book, each chapter starts with an overview of the presented concepts and results.
Nonlinear Mathematical Physics and Natural Hazards
by Boyka Aneva Mihaela Kouteva-GuentchevaThis book is devoted to current advances in the field of nonlinear mathematical physics and modeling of critical phenomena that can lead to catastrophic events. Pursuing a multidisciplinary approach, it gathers the work of scientists who are developing mathematical and computational methods for the study and analysis of nonlinear phenomena and who are working actively to apply these tools and create conditions to mitigate and reduce the negative consequences of natural and socio-economic disaster risk. This book summarizes the contributions of the International School and Workshop on Nonlinear Mathematical Physics and Natural Hazards, organized within the framework of the South East Europe Network in Mathematical and Theoretical Physics (SEENET MTP) and supported by UNESCO. It was held at the Bulgarian Academy of Sciences from November 28 to December 2, 2013. The contributions are divided into two major parts in keeping with the scientific program of the meeting. Among the topics covered in Part I (Nonlinear Mathematical Physics towards Critical Phenomena) are predictions and correlations in self organized criticality, space-time structure of extreme current and activity events in exclusion processes, quantum spin chains and integrability of many-body systems, applications of discriminantly separable polynomials, MKdV-type equations, and chaotic behavior in Yang-Mills theories. Part II (Seismic Hazard and Risk) is devoted to probabilistic seismic hazard assessment, seismic risk mapping, seismic monitoring, networking and data processing in Europe, mainly in South-East Europe. The book aims to promote collaboration at the regional and European level to better understand and model phenomena that can cause natural and socio-economic disasters, and to contribute to the joint efforts to mitigate the negative consequence of natural disasters. This collection of papers reflects contemporary efforts on capacity building through developing skills, exchanging knowledge and practicing mathematical methods for modeling nonlinear phenomena, disaster risk preparedness and natural hazards mitigation. The target audience includes students and researchers in mathematical and theoretical physics, earth physics, applied physics, geophysics, seismology and earthquake danger and risk mitigation.
Nonlinear Modeling of Solar Radiation and Wind Speed Time Series
by Luigi Fortuna Giuseppe Nunnari Silvia NunnariThis brief is a clear, concise description of the main techniques of time series analysis --stationary, autocorrelation, mutual information, fractal and multifractal analysis, chaos analysis, etc. -- as they are applied to the influence of wind speed and solar radiation on the production of electrical energy from these renewable sources. The problem of implementing prediction models is addressed by using the embedding-phase-space approach: a powerful technique for the modeling of complex systems. Readers are also guided in applying the main machine learning techniques for classification of the patterns hidden in their time series and so will be able to perform statistical analyses that are not possible by using conventional techniques. The conceptual exposition avoids unnecessary mathematical details and focuses on concrete examples in order to ensure a better understanding of the proposed techniques. Results are well-illustrated by figures and tables.
Nonlinear, Nonlocal and Fractional Turbulence: Alternative Recipes for the Modeling of Turbulence
by Kolumban Hutter Peter William EgolfExperts of fluid dynamics agree that turbulence is nonlinear and nonlocal. Because of a direct correspondence, nonlocality also implies fractionality. Fractional dynamics is the physics related to fractal (geometrical) systems and is described by fractional calculus. Up-to-present, numerous criticisms of linear and local theories of turbulence have been published. Nonlinearity has established itself quite well, but so far only a very small number of general nonlocal concepts and no concrete nonlocal turbulent flow solutions were available. This book presents the first analytical and numerical solutions of elementary turbulent flow problems, mainly based on a nonlocal closure. Considerations involve anomalous diffusion (Lévy flights), fractal geometry (fractal-β, bi-fractal and multi-fractal model) and fractional dynamics. Examples include a new ‘law of the wall’ and a generalization of Kraichnan’s energy-enstrophy spectrum that is in harmony with non-extensive and non-equilibrium thermodynamics (Tsallis thermodynamics) and experiments. Furthermore, the presented theories of turbulence reveal critical and cooperative phenomena in analogy with phase transitions in other physical systems, e.g., binary fluids, para-ferromagnetic materials, etc.; the two phases of turbulence identifying the laminar streaks and coherent vorticity-rich structures. This book is intended, apart from fluids specialists, for researchers in physics, as well as applied and numerical mathematics, who would like to acquire knowledge about alternative approaches involved in the analytical and numerical treatment of turbulence.
Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems (Monographs And Surveys In Pure And Applied Math #76)
by Songmu ZhengThis monograph is devoted to the global existence, uniqueness and asymptotic behaviour of smooth solutions to both initial value problems and initial boundary value problems for nonlinear parabolic equations and hyperbolic parabolic coupled systems. Most of the material is based on recent research carried out by the author and his collaborators.The book can be divided into two parts. In the first part, the results on decay of solutions to nonlinear parabolic equations and hyperbolic parabolic coupled systems are obtained, and a chapter is devoted to the global existence of small smooth solutions to fully nonlinear parabolic equations and quasilinear hyperbolic parabolic coupled systems. Applications of the results to nonlinear thermoelasticity and fluid dynamics are also shown.Some nonlinear parabolic equations and coupled systems arising from the study of phase transitions are investigated in the second part of the book. The global existence, uniqueness and asymptotic behaviour of smooth solutions with arbitrary initial data are obtained. The final chapter is further devoted to related topics: multiplicity of equilibria and the existence of a global attractor, inertial manifold and inertial set.A knowledge of partial differential equations and Sobolev spaces is assumed. As an aid to the reader, the related concepts and results are collected and the relevant references given in the firstchapter. The work will be of interest to researchers and graduate students in pure and applied mathematics, mathematical physics and applied sciences.
Nonlinear Partial Differential Equations
by Kenneth H. Karlsen Helge HoldenThe topic of the 2010 Abel Symposium, hosted at the Norwegian Academy of Science and Letters, Oslo, was Nonlinear Partial Differential Equations, the study of which is of fundamental importance in mathematics and in almost all of natural sciences, economics, and engineering. This area of mathematics is currently in the midst of an unprecedented development worldwide. Differential equations are used to model phenomena of increasing complexity, and in areas that have traditionally been outside the realm of mathematics. New analytical tools and numerical methods are dramatically improving our understanding of nonlinear models. Nonlinearity gives rise to novel effects reflected in the appearance of shock waves, turbulence, material defects, etc., and offers challenging mathematical problems. On the other hand, new mathematical developments provide new insight in many applications. These proceedings present a selection of the latest exciting results by world leading researchers.
Nonlinear Partial Differential Equations for Future Applications: Sendai, Japan, July 10–28 and October 2–6, 2017 (Springer Proceedings in Mathematics & Statistics #346)
by Shigeaki Koike Hideo Kozono Takayoshi Ogawa Shigeru SakaguchiThis volume features selected, original, and peer-reviewed papers on topics from a series of workshops on Nonlinear Partial Differential Equations for Future Applications that were held in 2017 at Tohoku University in Japan. The contributions address an abstract maximal regularity with applications to parabolic equations, stability, and bifurcation for viscous compressible Navier–Stokes equations, new estimates for a compressible Gross–Pitaevskii–Navier–Stokes system, singular limits for the Keller–Segel system in critical spaces, the dynamic programming principle for stochastic optimal control, two kinds of regularity machineries for elliptic obstacle problems, and new insight on topology of nodal sets of high-energy eigenfunctions of the Laplacian. This book aims to exhibit various theories and methods that appear in the study of nonlinear partial differential equations.
Nonlinear Partial Differential Equations for Scientists and Engineers
by Lokenath DebnathThe revised and enlarged third edition of this successful book presents a comprehensive and systematic treatment of linear and nonlinear partial differential equations and their varied and updated applications. In an effort to make the book more useful for a diverse readership, updated modern examples of applications are chosen from areas of fluid dynamics, gas dynamics, plasma physics, nonlinear dynamics, quantum mechanics, nonlinear optics, acoustics, and wave propagation. Nonlinear Partial Differential Equations for Scientists and Engineers, Third Edition, improves on an already highly complete and accessible resource for graduate students and professionals in mathematics, physics, science, and engineering. It may be used to great effect as a course textbook, research reference, or self-study guide.
Nonlinear Physics of Ecosystems
by Ehud MeronNonlinear Physics of Ecosystems introduces the concepts and tools of pattern formation theory and demonstrates their utility in ecological research using problems from spatial ecology. Written in language understandable to both physicists and ecologists in most parts, the book reveals the mechanisms of pattern formation and pattern dynamics. It als
Nonlinear Resonances
by Miguel A.F. Sanjuan Shanmuganathan RajasekarThis introductory text presents the basic aspects and most important features of various types of resonances and anti-resonances in dynamical systems. In particular, for each resonance, it covers the theoretical concepts, illustrates them with case studies, and reviews the available information on mechanisms, characterization, numerical simulations, experimental realizations, possible quantum analogues, applications and significant advances made over the years. Resonances are one of the most fundamental phenomena exhibited by nonlinear systems and refer to specific realizations of maximum response of a system due to the ability of that system to store and transfer energy received from an external forcing source. Resonances are of particular importance in physical, engineering and biological systems - they can prove to be advantageous in many applications, while leading to instability and even disasters in others. The book is self-contained, providing the details of mathematical derivations and techniques involved in numerical simulations. Though primarily intended for graduate students, it can also be considered a reference book for any researcher interested in the dynamics of resonant phenomena.