The Discrete Nonlinear Schrodinger Equation

This book constitutes the first effort to summarize a large volume of results obtained over the past 20 years in the context of the Discrete Nonlinear Schrodinger equation and the physical settings that it describes. This book constitutes the first effort to summarize a large volume of results obtained over the past 20 years in the context of the Discrete Nonlinear Schrödinger equation and the physical settings that it describes. It contains an introduction to the model, its systematic derivation and its connection to applications, a subsequent analysis of the existence and the stability of fundamental nonlinear structures in 1, 2 and even 3 spatial lattice dimensions. It also covers the case of defocusing nonlinearities, the modulational instabilities of plane wave solutions, and the extension to multi-component lattices. In addition, it features a final chapter on special topics written by a wide array of experts in the field, addressing through short reviews, areas of particular recent interest. 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.