This book explores the impact of nonlinearity on a broad range of areas, including time-honored fields such as biology, geometry, and topology, but also modern ones such as quantum mechanics, networks, metamaterials and artificial intelligence. The concept of nonlinearity is a universal feature in mathematics, physics, chemistry and biology, and is used to characterize systems whose behavior does not amount to a superposition of simple building blocks, but rather features complex and often chaotic patterns and phenomena. Each chapter of the book features a synopsis that not only recaps the recent progress in each field but also charts the challenges that lie ahead. This interdisciplinary book presents contributions from a diverse group of experts from various fields to provide an overview of each field's past, present and future. It will appeal to both beginners and seasoned researchers in nonlinear science, numerous areas of physics (optics, quantum physics, biophysics), and applied mathematics (ODEs, PDEs, dynamical systems, machine learning) as well as engineering.

This book, written by experts in the fields of atomic physics and nonlinear science, covers the important developments in a special aspect of Bose-Einstein condensation, namely nonlinear phenomena in condensates. Topics covered include bright, dark, gap and multidimensional solitons; vortices; vortex lattices; optical lattices; multicomponent condensates; mathematical methods/rigorous results; and the beyond-the-mean-field approach.

The sine-Gordon model is a ubiquitous model of Mathematical Physics with a wide range of applications extending from coupled torsion pendula and Josephson junction arrays to gravitational and high-energy physics models. The purpose of this book is to present a summary of recent developments in this field, incorporating both introductory background material, but also with a strong view towards modern applications, recent experiments, developments regarding the existence, stability, dynamics and asymptotics of nonlinear waves that arise in the model. This book is of particular interest to a wide range of researchers in this field, but serves as an introductory text for young researchers and students interested in the topic. The book consists of well-selected thematic chapters on diverse mathematical and physical aspects of the equation carefully chosen and assigned.

Adventures of nonlinear science were perhaps most notably seeded at the Los Alamos National Laboratory (LANL) over half a century ago with the fundamental questionsofenergyequipartitioninnonlinearsystems, astheywereposedbyFermi, Pasta, and Ulam. At the time, probably little could be imagined of the far-reaching implications that the studies of nonlinear phenomena would have, continuing to expandtothisday.TheGinzburg-Landautheoryofsuperconductivityandtheord- parameter descriptions of super uidity, the "soliton revolution" through the works of Zabusky and Kruskal on the KdV equation and the subsequent widespread - plicationsof the nonlinear Schrodi .. ngerequation in optical bers and Bose-Einstein condensates,the developmentsof bifurcationtheory and chaotic dynamicsand their widespread applicationsfrom climate and geophysics,to biological phenomenaand chemical kineticsare only a few of the multiple arenas in which nonlineardynamics have emerged as the appropriate description of important physical systems. I well remember my own early days of nonlinear science appreciation, rst at Cornell University in the early 1970s and then at Los Alamos where we began the Center for Nonlinear Studies (CNLS) in 1980. These were years marked by interdisciplinary discovery and by the recognition that many nonlinear equations have an inherent ability to exhibit both coherence and chaos - the beginnings of our appreciation today of spatio-temporal complexity and the functional role that this plays in multiple branches of science, technology, and engineering.

This book presents a careful selection of the most important developments of the \phi^4 model, offering a judicious summary of this model with a view to future prospects and the challenges ahead. Over the past four decades, the \phi^4 model has been the basis for a broad array of developments in the physics and mathematics of nonlinear waves. From kinks to breathers, from continuum media to discrete lattices, from collisions of solitary waves to spectral properties, and from deterministic to stochastic models of \phi^4 (and \phi^6, \phi^8, \phi^12 variants more recently), this dynamical model has served as an excellent test bed for formulating and testing the ideas of nonlinear science and solitary waves.

This book summarizes a number of fundamental developments at the interface of granular crystals and the mathematical and computational analysis of some of their key localized nonlinear wave solutions. The subject presents a blend of the appeal of granular crystals as a prototypical engineering tested for a variety of diverse applications, the novelty in the nonlinear physics of its coherent structures, and the tractability of a series of mathematical and computational techniques to analyse them. While the focus is on principal one-dimensional solutions such as shock waves, traveling waves, and discrete breathers, numerous extensions of the discussed patterns, e.g., in two dimensions, chains with defects, heterogeneous settings, and other recent developments are discussed. The emphasis on the subject was motivated by models in condensed matter physics, ferroelectrics, high energy physics, and statistical mechanics, leading to developments in mathematical analysis, numerical computation and insights on the physical aspects of the model. The book appeals to researchers in the field, as well as for graduate and advanced undergraduate students. It will be of interest to mathematicians, physicists and engineers alike.

The sine-Gordon model is a ubiquitous model of Mathematical Physics with a wide range of applications extending from coupled torsion pendula and Josephson junction arrays to gravitational and high-energy physics models. The purpose of this book is to present a summary of recent developments in this field, incorporating both introductory background material, but also with a strong view towards modern applications, recent experiments, developments regarding the existence, stability, dynamics and asymptotics of nonlinear waves that arise in the model. This book is of particular interest to a wide range of researchers in this field, but serves as an introductory text for young researchers and students interested in the topic. The book consists of well-selected thematic chapters on diverse mathematical and physical aspects of the equation carefully chosen and assigned.

Adventures of nonlinear science were perhaps most notably seeded at the Los Alamos National Laboratory (LANL) over half a century ago with the fundamental questionsofenergyequipartitioninnonlinearsystems, astheywereposedbyFermi, Pasta, and Ulam. At the time, probably little could be imagined of the far-reaching implications that the studies of nonlinear phenomena would have, continuing to expandtothisday.TheGinzburg-Landautheoryofsuperconductivityandtheord- parameter descriptions of super uidity, the "soliton revolution" through the works of Zabusky and Kruskal on the KdV equation and the subsequent widespread - plicationsof the nonlinear Schrodi .. ngerequation in optical bers and Bose-Einstein condensates,the developmentsof bifurcationtheory and chaotic dynamicsand their widespread applicationsfrom climate and geophysics,to biological phenomenaand chemical kineticsare only a few of the multiple arenas in which nonlineardynamics have emerged as the appropriate description of important physical systems. I well remember my own early days of nonlinear science appreciation, rst at Cornell University in the early 1970s and then at Los Alamos where we began the Center for Nonlinear Studies (CNLS) in 1980. These were years marked by interdisciplinary discovery and by the recognition that many nonlinear equations have an inherent ability to exhibit both coherence and chaos - the beginnings of our appreciation today of spatio-temporal complexity and the functional role that this plays in multiple branches of science, technology, and engineering.