Discrete and Continuous Nonlinear Schrödinger Systems
Solitons (localized and stable waves interacting almost like elastic objects) in nonlinear systems are a fascinating theme. They were initially studied (over a century ago) in relation to water waves, where they appear as solutions of the corresponding nonlinear differential equations (the Korteweg-de Vries equation). These phenomena have been intensively studied over the last thirty years. This book is devoted to the study of solitons for nonlinear Schrödinger systems in nonlinear optics. Such systems describe, among others, wave transmission in optical fibers, which have technological applications of critical importance.
The book follows an earlier monograph by Ablowitz and Clarkson (Solitons, Nonlinear Evolutions Equations and Inverse Scattering, LMS Lecture Notes Series 149, Cambridge University Press, 1991). The central tool used in the study of these systems, which is thoroughly explained in the book, is the inverse scattering transform (IST): a method that can be viewed as a nonlinear version of the Fourier transform and which allows one to linearize certain classes of nonlinear evolution equations. This method is applied to four different types of nonlinear Schrödinger systems in 1+1 and 1+2 dimensions: Nonlinear Schrödinger equations (NLS), integrable discrete NLS, matrix NLS and integrable discrete matrix NLS. To summarize, this valuable book provides a detailed and self-contained presentation of an extremely important tool used in the study of NLS systems.