When the author was a graduate student, introductions to texts on nonlinear evolution equations contained a long description of physical applications, numerous references to the works of others, and sparse details of the justification of analytical results. Times have changed, however, and the main interest in the nonlinear evolution equations has moved from modeling to analysis. It is now more typical for applied mathematics texts to start an introduction with the main equations in the first lines, to give no background information on applications, to reduce the list of references to a few relevant mathematical publications, and to focus discussions on technical aspects of analysis.

Since this book is aimed at young mathematicians, we should reduce the background information to a minimum and focus on useful analytical techniques in the context of the nonlinear Schrödinger equation with a periodic potential. It is only in this introduction that we recall the old times and review the list of nonlinear evolution equations that we are going to work with in this book. The few references will provide a quick glance at physical applications, without distracting attention from equations.