From Nonlinearity To Coherence

Nonlinear physics has been growing at an astounding rate over the past two decades and has changed its character from a collection of exotic examples of nonstandard behaviour to an all-embracing scientific methodology. This practical hands-on guide provides an overview of the features ofcondensed matter systems. This book provides self-contained background material, however the centrepiece of the text is the chapter dealing with a systematic development of nonlinear field equations for many-body systems. In order to equip the reader with concrete skills in tackling nonlinearproblems in physics, the authors analyse in great detail several important applications such as metamagnetism, superconductivity, the Hubbard Hamiltonian and the multi-electron atom to name a few. A separate mathematical chapter shows in an easy-to-follow manner how the various integrable nonlineardifferential equations that arise in physics can be solved analytically. This book will serve as a compendium of facts and references related to the subject area of nonlinear condensed matter physics. In addition, it can be used as practical introduction into currently developed nonlinear researchmethods in theoretical physics in general.

This book is based on the lectures delivered at the 19th Canberra International Physics Summer School held at the Australian National University in Canberra (Australia) in January 2006.The problem of turbulence and coherent structures is of key importance in many fields of science and engineering. It is an area which is vigorously researched across a diverse range of disciplines such as theoretical physics, oceanography, atmospheric science, magnetically confined plasma, nonlinear optics, etc. Modern studies in turbulence and coherent structures are based on a variety of theoretical concepts, numerical simulation techniques and experimental methods, which cannot be reviewed effectively by a single expert.The main goal of these lecture notes is to introduce state-of-the-art turbulence research in a variety of approaches (theoretical, numerical simulations and experiments) and applications (fluids, plasmas, geophysics, nonlinear optical media) by several experts. A smooth introduction is presented to readers who are not familiar with the field, while reviewing the most recent advances in the area. This collection of lectures will provide a useful review for both postgraduate students and researchers new to the advancements in this field, as well as specialists seeking to expand their knowledge across different areas of turbulence research.

The centrepiece of the text is the chapter dealing with a systematic development of nonlinear field equations for many-body systems. The book provides an overview of condensed matter systems in a practical 'hands-on guide' format.

This book offers an essential introduction to the notions of sound wave topology, duality, coherence and wave-mixing, which constitute the emerging new science of sound. It includes general principles and specific examples that illuminate new non-conventional forms of sound (sound topology), unconventional quantum-like behavior of phonons (duality), radical linear and nonlinear phenomena associated with loss and its control (coherence), and exquisite effects that emerge from the interaction of sound with other physical and biological waves (wave mixing). The book provides the reader with the foundations needed to master these complex notions through simple yet meaningful examples. General principles for unraveling and describing the topology of acoustic wave functions in the space of their Eigen values are presented. These principles are then applied to uncover intrinsic and extrinsic approaches to achieving non-conventional topologies by breaking the time reversal symmetry of acoustic waves. Symmetry breaking can impart topological immunity to wave degradation from imperfection scattering and catalyze controlled coherence. In the intrinsic case and the phonon representation of acoustic waves, the self-interaction/interference of a wave through its supporting medium exposes the notion of duality in the quantum statistics (i.e. boson vs. fermion characterized by the symmetry of multiple particle states) and how the quantum analogue behaviors of sound can be exploited in the form of novel sound-based information transfer and processing devices. By considering media that mix different types of waves, the book addresses the interaction of sound with other physical and biological waves but also brings to light examples of extrinsic processes that can lead to symmetry breaking. The coherent conversion of sound into other types of waves as well as the sound-induced non-conventional topology of elastic, electronic, spin and biological waves are presented in the case of media exhibiting elasto-electronic, photo-elastic, magneto-elastic effects and biological mechano-transduction.

This volume contains the Proceedings of the NATO Advanced Research Workshop (ARW) and Emil-Warburg-Symposium (EWS) "Nonlinear Coherent Structures in Phy sics and Biology" held at the University of Bayreuth from June 1 -4, 1993. Director of the ARW was K. H. Spatschek, while F.G. Mertens acted as the co-director, host, and organizer of the EWS. The other members of the scientific organizing committee were A.R. Bishop (Los Alamos), J.C. Eilbeck (Edinburgh), and M. Remoissenet (Dijon). This was the eighth meeting in a series of interdisciplinary workshops founded by our French colleagues who had organized all the previous workshops, e.g. 1989 in Montpel lier and 1991 in Dijon. We were asked to organize the meeting this time in Germany. Of course, we wanted to keep the character defined by the previous meetings, which were always characterized by an open and friendly atmosphere, being not too large in quantity, but high in quality. This time altogether 103 participants attended the workshop. During the past years most of the participants met several times and discussed problems connected with the generation of nonlinear coherent structures in physics and biology.

The second edition of this book consists of three parts. The first one is dedicated to the WKB methods and the semi-classical limit before the formation of caustics. The second part treats the semi-classical limit in the presence of caustics, in the special geometric case where the caustic is reduced to a point (or to several isolated points). The third part is new in this edition, and addresses the nonlinear propagation of coherent states. The three parts are essentially independent.Compared with the first edition, the first part is enriched by a new section on multiphase expansions in the case of weakly nonlinear geometric optics, and an application related to this study, concerning instability results for nonlinear Schrödinger equations in negative order Sobolev spaces.The third part is an overview of results concerning nonlinear effects in the propagation of coherent states, in the case of a power nonlinearity, and in the richer case of Hartree-like nonlinearities. It includes explicit formulas of an independent interest, such as generalized Mehler's formula, generalized lens transform.

A rich variety of real-life physical problems which are still poorly understood are of a nonlinear nature. Examples include turbulence, granular flows, detonations and flame propagation, fracture dynamics, and a wealth of new biological and chemical phenomena which are being discovered. Particularly interesting among the manifestations of nonlinearity are coherent structures. This book contains reviews and contributions reporting on the state of the art regarding the role of coherent structures and patterns in nonlinear science.