Interplay of quantum mechanics and nonlinearity : understanding small system dynamics of the discrete nonlinear Schrödinger equation Chapter 1. The Discrete Nonlinear Schrödinger Equation and the Two State System (Dimer) Chapter 2. Dimer Solutions Mobility Reduction and Neutron Scattering Chapter 3. Initial Delocalization Phase Nonlinearity Interplay and Fluorescence Depolarization Chapter 4. What Polarons Owe to their Harmonic Origins Chapter 5. Static Energy Mismatch in the Nonlinear Dimer: Nondegeneracy Chapter 6. Extended Systems with Global Interactions and Nonlinear Trapping Chapter 7. Slow Relaxation: the Nonadiabatic Nonlinear Dimer Chapter 8. Thermal Effects: Phase Space and Langevin Formulations Chapter 9. Microscopic Origin Issues about the DNLSE for Polarons Chapter 10. Bose Einstein Condensate Tunneling: the Gross Pitaevskii Equation Chapter 11. Miscellaneous Topics and Summary of the Book. This book presents an in depth study of the discrete nonlinear Schrödinger equation (DNLSE) with particular emphasis on spatially small systems that permit analytic solutions. In many quantum systems of contemporary interest the DNLSE arises as a result of approximate descriptions despite the fundamental linearity of quantum mechanics. Such scenarios exemplified by polaron physics and Bose Einstein condensation provide application areas for the theoretical tools developed in this text. The book begins with an introduction of the DNLSE illustrated with the dimer development of fundamental analytic tools such as elliptic functions and the resulting insights into experiment that they allow. Subsequently the interplay of the initial quantum phase with nonlinearity is studied leading to novel phenomena with observable implications in fields such as fluorescence depolarization of stick dimers followed by analysis of more complex and or larger systems. Specific examples analyzed in the book include the nondegenerate nonlinear dimer nonlinear trapping rotational polarons and the nonadiabatic nonlinear dimer. Phenomena treated include strong carrier phonon interactions and Bose Einstein condensation. This book is aimed at researchers and advanced graduate students with chapter summaries and problems to test the readers understanding along with an extensive bibliography. The book will be essential reading for researchers in condensed matter and low temperature atomic physics as well as any scientist who wants fascinating insights into the role of nonlinearity in quantum physics