||The precise project for this book took shape after the writing of the lecture notes for a course on the 'Physical and mathematical models of nonlinear waves in solids' delivered in 1993 at the International Centre for Mechanical Sciences, Udine, Italy. Although the style and contents of the book follow these Notes, the size of the book has grown considerably.
It was neither to be a book on crystallography per se nor a treatise of mathematics. Rather, it was intended to be devoted to applied mathematics exploited in a specific physical field of basic interest to many scientists and engineers. This book would decidedly be different from the numerous books already present on our shelves. Whereas I did not want to duplicate the existing elaborate books on the fascinating subject ofsolitons, I also wanted to show that the physical reality practically always escapes the simplifications with which one derives such exceptional equations as the exactly integrable ones. In addition, as I have been involved, with my co-workers, simultaneously in elaborating the physical models and in studying the mathematical formulations and carrying out the numerical simulations, I wanted this work to reflect fully the spirit of constant cooperation and cross-fertilization between three levels of understanding: physics, applied mathematics and numerics. I am not an expert in numerical computations, and I must acknowledge in this respect my immense debt for some of my co-workers, in particular, J. Pouget, B. Collet, H. Hadouaj, A. Salupere and C. I. Christov. For the friendly help, advice and cooperation on the theoretical developments, in addition to these scientists, I am grateful to A. N. Abd-Alla, W. Ani, A. Askar, S. Cadet, N. Daher, J. Engelbrecht, M. Epstein, A. C. Eringen, A. Fomethe, A. F. Ghaleb, I. Z. Hefni, B. A. Malomed, A. Miled, S. Motogi, W. Muschik, D. F. Parker and C. Trimarco, some as my Ph.D. students and others as my colleagues or elders. All errors, however, are mine and only mine.
Although this book addresses 'nonlinear waves', it may come as a surprise to many readers that I have not discussed hyperbolic systems in detail. There are two reasons for this. First, as stated already, I wanted to avoid duplication with other books—some of which contributed to the history of applied mathematics— which are replete with this subject. Second, my focus was primarily on the dispersive systems. The phenomenon of dispersion has two origins: discreteness of the substratum of propagation, e.g. the crystal lattice, and the critical appearance of a length scale when some relatively long-range interactions are accounted for. The analysis of dispersion leads to a wealth of interesting information on the subject. This is indeed the realm of elastic crystals that a standard continuum approach may have missed. Simultaneously, this has strengthened my belief in the validity of the microcontinuum theory, where one escapes from the underlying dogmatism created by some school of continuum mechanics. The continuous