Creative artists are well aware that the juxtaposition of contrasting features forms an interesting starting point for a musical composition or a visual painting. I had always wanted to use such a combination of opposites in putting together a description of a scientific theory. I came to realize that blending quantum mechanics and the science of nonlinearity would provide an excellent instance, especially because quantum mechanics is patently linear. Pursuing this program, I found it possible to give expression to my thoughts about the importance to a budding theoretical physicist of acquiring skills exemplified by numerical techniques to supplement analytical study on the one hand, and designing experiments to test the theory on the other. This, in brief, is the genesis of my most recent book,

https:// link.springer.com/book/10.1007/978-3-030-94811-5

Quantum mechanics prides itself on being a linear structure with superposition its overriding principle. It is true that illustrious scientists such as Leggett, Sudarshan and Weinberg have written about possibilities of the violation of linearity in quantum mechanics on a fundamental level. My interest was, however, less ambitious if more practical. It extended to discussing such violation only in an approximate representation arising from a coarse-grained manner of viewing the quantum description. As the subtitle of the book clarifies, Understanding Small-System Dynamics of the Discrete Nonlinear Schrödinger Equation is what I tried to develop within the book.

As everyone knows, the Schrödinger equation is linear in the wave function. In condensed matter physics of strongly interacting systems such as electrons and phonons, clever physical treatments originated by Landau, Pekar, Fröhlich and Holstein led to the conception of composite particles called polarons that have dominated diverse areas of condensed matter physics for decades. One of the varied ways of describing polarons is to present them as obeying the nonlinear Schrödinger equation in which the energy is itself a function of the wave function, the typical nonlinearity being cubic. On a lattice where the basis is perhaps a set of Wannier states centered on the sites, the nonlinear Schrödinger equation becomes discrete. As a consequence of its discrete nature, it has no known analytic solutions for systems of realistic size.

My purpose in writing the book was to show the reader how a suspicion raised by numerical analysis that interesting transitions lie hidden in the system can be pursued by taking the bull by the horns and forcing the system to yield by making it almost ridiculously small. For this purpose, I brought the apparatus of elliptic functions to bear on a system of size so limited that we were looking, initially, at just a dimer (two sites). I was stimulated by the pedagogically valuable possibility of introducing this particular class of special functions that is seldom taught to the beginning physics student. A quick grounding in this material at the very beginning of the book is followed, thus, by the appearance of “pretty bifurcations and nifty transitions” (as my Preface calls them) that starts the storied journey of the book.

The topic of study allowed me to complicate our simple system in several independent directions, three of which were found to be particularly fertile. Translational invariance was broken in one case by making the system energetically nondegenerate. In another, infinitely fast relaxation was generalized to allow for finite rates of vibrational energy transfer. In a third, the system was coupled to a reservoir to allow for thermal interactions. Each direction of study yielded simple yet fascinating insights and an opportunity to help the student of theoretical physics learn hands-on.

The skill of designing experiments even if they are only at the gedanken level is, I believe strongly, essential to acquire if you are a budding theorist. The book is replete with examples of such design activity that starts with neutron scattering and muon spin relaxation in solids. Observations of fluorescence depolarization of stick dimers are also discussed at more than one place, as well as the mobility of carriers. The two large fields where the particular theoretical considerations of the book appear relevant are polaron dynamics as in the transport of excitons in molecular aggregates and behavior of Bose-Einstein condensates in low-temperature optical traps. Particularly in the latter field, the reader is shown how the theoretical concepts expounded in the book allow the delicate manipulation of transitions of different kinds and predict surprising outcomes in experiment.

I also found it appropriate to touch upon a variety of miscellaneous topics lying on the edges of the area of interplay of quantum mechanics with nonlinearity. This was done lightly at the end of the book, with a view to whetting the reader’s appetite. Questions ranging from the effect of external fields to soliton propagation, and from impurity dynamics to excimer formation were analyzed in this fashion. The Davydov Soliton was also discussed as a backdrop for some of the matters described.

I hope that graduate students entering into the research adventure will find the book exposition pedagogically useful and exciting in the sense of the spirit of play in which it has been authored. May I invite them to do so.