The Discrete Nonlinear Schrödinger 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 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. TOC:1. General Introduction and Derivation of the DNLS Equation.- 2. The One-Dimensional Case.- 3. The Two-Dimensional Case.- 4. The Three-Dimensional Case.- 5. The Defocusing Case.- 6. Extended Solutions and Modulational Instability.- 7. Multi-Component Solitary Waves.- 8. Experimental Results Related to DNLS Equations.- 9. Numerical Methods for DNLS.- 10. The Dynamics of Unstable Waves.- 11. A Map Approach to Stationary Solutions of the DNLS Equation.- 12. Formation of Localized Modes in DNLS.- 13. Few-lattice-site Systems of Discrete Self-Trapping Equations.- 14. Surface Waves and Boundary Effects in DNLS Equations.- 15. Discrete Nonlinear Schrödinger Equations with Time-Dependent Coefficients.- 16. Exceptional Discretizations of the NLS.- 17. Solitary Wave Collisions.- 18. Related Models.- 19. DNLS with Impurities.- 20. Statistical Mechanics of DNLS.- 21. Traveling Solitary Waves in DNLS Equations.- Decay and Strichartz Estimates for DNLS.