Ablowitz M.J., Prinari B., Trubatch A.D. Discrete and Continuous Nonlinear Schrödinger Systems

Cambridge: Cambridge University Press. – 2004. – 267 p. (London Mathematical Society Lecture Note Series. 302) Nonlinear systems are generic in the mathematical representation of physical phenomena. It is unusual for one to be able to find solutions to most nonlinear equations. However, a certain physically significant subclass of problems admits deep mathematical structure that further allows one to find classes of exact solutions. Solutions are a particularly important subclass of such solutions. Solitons are localized waves that, in an appropriate sense, interact elastically with each other. They have proved to be extremely interesting to physicists and engineers due, in part, to their localized and stable nature. Nevertheless, there are particular problems that, because of their wide applicability, deserve special attention. It is the purpose of this book to investigate one such important set of equations: nonlinear Schrödinger (NLS) systems. The unified description of these physically interesting integrable discrete and continuous NLS systems within the context of the IST methodology does not appear anywhere else. One of our motivations in writing this book was to develop the direct and inverse scattering formalism based on the Riemann–Hilbert approach. From a pedagogical point of view, readers will find that the structure follows the one laid out for the Korteweg–de Vries equation in the monograph of Ablowitz and Clarkson. Here we have also attempted to include many of the mathematical details, to make the book suitable for students as well as researchers who wish to study this topic.Preface page.Nonlinear Schrödinger equation (NLS).
Integrable discrete nonlinear Schrödinger equation (IDNLS).
Matrix nonlinear Schrödinger equation (MNLS).
Integrable discrete matrix NLS equation (IDMNLS).
Summation by parts formula.
Transmission of the Jost function through a localized potential.
Scattering theory for the discrete Schrödinger equation.
Nonlinear Schrödinger systems with a potential term.
Continuous NLS systems with a potential term.
Discrete NLS systems with a potential term.
NLS systems in the limit of large amplitudes.