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This new edition has been thoroughly revised, expanded and contain some updates function of the novel results and shift of scientific interest in the topics. The book has a Foreword by Jerry L. Bona and Hongqiu Chen. The book is an introduction to nonlinear waves and soliton theory in the special environment of compact spaces such a closed curves and surfaces and other domain contours. It assumes familiarity with basic soliton theory and nonlinear dynamical systems.The first part of the book introduces the mathematical concept required for treating the manifolds considered, providing relevant notions from topology and differential geometry. An introduction to the theory of motion of curves and surfaces - as part of the emerging field of contour dynamics - is given.The second and third parts discuss the modeling of various physical solitons on compact systems, such as filaments, loops and drops made of almost incompressible materials thereby intersecting with a large number of physical disciplines from hydrodynamics to compact object astrophysics.This book is intended for graduate students and researchers in mathematics, physics and engineering.

The 2nd edition of this book provides novel topics and studyies in boundaries of networks and Big Data Systems.The central theme of this book is the extent to which the structure of the free dynamical boundaries of a system controls the evolution of the system as a whole. Applying three orthogonal types of thinking - mathematical, constructivist and morphological, it illustrates these concepts using applications to selected problems from the social and life sciences, as well as economics. In a broader context, it introduces and reviews some modern mathematical approaches to the science of complex systems. Standard modeling approaches (based on non-linear differential equations, dynamic systems, graph theory, cellular automata, stochastic processes, or information theory) are suitable for studying local problems. However they cannot simultaneously take into account all the different facets and phenomena of a complex system, and new approaches are required to solve the challenging problem of correlations between phenomena at different levels and hierarchies, their self-organization and memory-evolutive aspects, the growth of additional structures and are ultimately required to explain why and how such complex systems can display both robustness and flexibility. This graduate-level text addresses a broader interdisciplinary audience, keeping the mathematical level essentially uniform throughout the book, and involving only basic elements from calculus, algebra, geometry and systems theory.  

Spontaneous pattern formation in nonlinear dissipative systems far from equilibrium is a paradigmatic case of emergent behaviour associated with complex systems. It is encountered in a great variety of settings, both in nature and technology, and has numerous applications ranging from nonlinear optics through solid and fluid mechanics, physical chemistry and chemical engineering to biology.This book is a first-hand account by one of the leading players in this field, which gives in-depth descriptions of analytical methods elucidating the complex evolution of nonlinear dissipative systems, and brings the reader to the forefront of current research.Since the publication of the first edition, applications of the theory of nonlinear dynamics have been substantially extended to the novel area of active systems, largely motivated by problems of biophysics and biomorphic technology. These problems typically involve media with internal orientation. This new edition incorporates a chapter discussing dynamics of liquids and soft solids with internal orientation, including special features of their instabilities and motion of topological defects, which form the background for various applications to the motion of cells, tissues, and activated soft materials. The contents of the first edition have also been substantially reworked, improving graphics, emphasizing more complex secondary instabilities, and dropping some material pertaining to dynamical systems.This book caters for graduate students and young researchers from many pertinent areas including applied mathematics, physical chemistry, chemical engineering and biophysics, as well as the seasoned scientist in search of a modern source of reference.

This book provides the dynamics of non-equilibrium dissipative systems with asymmetric interactions (Asymmetric Dissipative System; ADS). Asymmetric interaction breaks "the law of action and reaction" in mechanics, and results in non-conservation of the total momentum and energy. In such many-particle systems, the inflow of energy is provided and the energy flows out as dissipation. The emergences of non-trivial macroscopic phenomena occur in the non-equilibrium energy balance owing to the effect of collective motions as phase transitions and bifurcations. ADS are applied to the systems of self-driven interacting particles such as traffic and granular flows, pedestrians and evacuations, and collective movement of living systems. The fundamental aspects of dynamics in ADS are completely presented by a minimal mathematical model, the Optimal Velocity (OV) Model. Using that model, the basics of mathematical and physical mechanisms of ADS are described analytically with exact results.The application of 1-dimensional motions is presented for traffic jam formation. The mathematical theory is compared with empirical data of experiments and observations on highways. In 2-dimensional motion pattern formations of granular media, pedestrians, and group formations of organisms are described. The common characteristics of emerged moving objects are a variety of patterns, flexible deformations, and rapid response against stimulus. Self-organization and adaptation in group formations and control of group motions are shown in examples. Another OV Model formulated by a delay differential equation is provided with exact solutions using elliptic functions. The relations to soliton systems are described. Moreover, several topics in ADS are presented such as the similarity between the spatiotemporal patterns, violation of fluctuation dissipation relation, and a thermodynamic function for governing the phase transition in non-equilibrium stationary states.

This book contains a systematic analysis of the formalisms of quantum electro- dynamics in the presence of an intense external field able to create pairs from the vacuum, and thereby violate the stability of the latter. The approach developed is not specific to quantum electrodynamics, and can equally well be applied to any quantum field theory with an unstable vacuum. It should be noted that only macroscopic external fields are considered, whereas problems associated with the superstrong Coulomb (micro) field are not treated. As a rule, the discussion is confined to those details of the formalism and calculations that are specific to the instability property. For instance, renormalization is not discussed here since, in practical calculations, it is carried out according to standard methods. The presentation is based mainly on original research undertaken by the authors. Chapter 1 contains a general introduction to the problem. It also presents some standard information on quantum electrodynamics, which will be used later in the text. In addition, an interpretation of the concept of an external field is given, and the problems that arise when one tries to keep the interaction with the external field exactly are discussed. In Chapter 2, the perturbation expansion in powers of the radiative interac- tion is developed for the matrix elements of transition processes, taking the arbitrary external field into account exactly.