7R10. Strain Solitons in Solids and How to Construct Them. Monographs and Surveys in Pure and Applied Mathematics, Vol 117. - AM Samsonov (Theor Dept, Ioffe Physico-Tech Inst, Russian Acad of Sci, St Petersburg, Russia). Chapman and Hall/CRC, Boca Raton FL. 2001. 230 pp. ISBN 0-8493-0684-1. $69.95.

Reviewed by GA Maugin (Lab de Modelisation en Mec, Univ Pierre et Marie Curie, Tour 66, 4 Place Jussieu, Case 162, Paris Cedex 05, 75252, France).

Within an interval of three years, five books have been published that refer to the propagation of localized nonlinear waves in deformable solids. These are the books by J Engelbrecht (Nonlinear Wave Dynamics, Complexity and Simplicity, Kluwer, Dordrecht, 1997), LA Ostrovsky and AI Potapov (Modulated Waves, Johns Hopkins University Press, Washington DC, 1999), GA Maugin (Nonlinear Waves in Elastic Crystals, Oxford University Press, Oxford, UK, 1999), AM Kosevich (Solitons, VCH-J Wiley, 1999), and the present book by Professor AM Samsonov from the celebrated Ioffe Physico-Technical Institute of the Russian Academy of Sciences in St Petersburg. What is remarkable in this instance is that all these authors are interrelated via cooperative European research programs. In particular, the undersigned shares publications with members of the other four groups. In spite of what could have been considered a friendly competition at a time when a domain of knowledge reaches an evident maturity, each of the authors has succeeded in producing a book with his own interests, style, and idiosyncrasies. This means that while Engelbrecht’s book gave a general background of the field of nonlinearity, the Ostrovsky-Potapov book is more an advanced textbook borrowing examples from both fluid and solid mechanics. Kosevich’s book gives the view point of a condensed-matter physicist in the Landau-Lifshitz tradition, and Maugin’s book is more focused on the case of ordered solid media such as crystals and the natural output of physical modeling, that is, nonexactly integrable systems of equations, in contradistinction with traditionally examined classical models. The present book by Samsonov offers still another and complementary vision and focus of interest in that, in spite of its general title, is devoted to the mathematical study, experimental observation, and numerical simulation of so-called solitons in quasi-one-dimensional models of propagation issued from the modeling of elastic rods. This is a very specific subject matter of which the author, a long-time researcher in the field, expands all facets in a rather concisely written book.

In effect, basic definitions such as those pertaining to wave terminology and nonlinear elasticity are first reminded in a brief Chapter 1. Chapter 2 presents a first derivation for the basic coupled equations that govern elastic longwave propagation in a rod. Depending on the approximation made regarding the lateral behavior and the type of embedding in the external environment (Winkler, Pasternak, or Kerr “foundations”), a series of more or less complex one-dimensional quasi-hyperbolic equations is deduced in this chapter. The basic characteristic instrument for further studies is a so-called doubly-dispersive equation (for short DDE) which contains two wave operators with different characteristic speeds. This is the main ingredient combined with the nonlinearity which is mostly of physical origin. At this stage, one may envisage two direct analytical methods to obtain solutions of the various DDE. One method has become traditional and consists of effecting a standard reduction to a nonlinear evolution equation by means of asymptotics (method of reductive perturbation). The final reasoning for integration is then made on a one-directional equation. The author prefers to consider a direct integration of the DDE by considering general traveling wave solutions in terms of the Weierstrass elliptic complex-valued function (Ch 3). This is somewhat technical, but the author shows well (basing on his own original research) the fruitfullness of the method even for dissipative equations such as those obtained in biophysics (FitzHugh-Nagumo equation). The solutions obtained are not exactly solitons in the mathematical sense. They are solitary waves of different shapes that do not accept the “superposition principle” of solitons’ mathematics, as the systems considered are not exactly integrable. The book’s title, therefore, contains a slight abuse of language. Then come two of the most interesting and original chapters of the book, those expanding the original experimental works of Samsonov’s team. At this point, it should be noted that while solitons in fluid mechanics (shallow channel flow), in physical systems such as ferromagnets and Josephson junctions, and in nonlinear optics are supported by direct physical evidence and even industrial applications, the objective existence of soliton-like signals in solids is a more touchy matter. Thanks to the exploitation of sophisticated methods such as schlieren photography and laser shadowgraphy, interferograms are produced which allow one to exhibit transients, longwave solitons, their reflection at boundaries, and to measure their essential characteristic parameters in transparent materials. In nonuniform rods of varying cross section (eg, tapered rod) or varying elasticity coefficients, the experiments exhibit the phenomenon of focusing of longitudinal strain solitons, and effect similar to the self-focusing of modulated solitons in nonlinear optical fibers. Some of these properties can also be shown in elastic plates, and the effect of the environment by accounting for the direct interaction of the lateral faces with a substratum is also demonstrated (Ch 5). For a layer sitting on an elastic half space, the quasione-dimensional equation obtained is of the nonlinear integro-differential type (Benjamin-Ohno equation including Hilberttransforms of the longitudinal signal). Some numerical simulations accompany the presented results. The final Chapter 6 deals more properly with this, exhibiting some interactions, formations of trains of solitons from a point-like source, head-on collision, passing over, attenuation, and focusing in various specific cases. The whole is completed by a rich bibliography (214 items) which nonetheless misses some of the important related works done in the West. The subject index is rather poor.

Samsonov’s book, Strain Solitons in Solids and How to Construct Them, is an original one. It clearly deserves to be on the shelves of all researchers interested in the nonlinear dynamical elasticity of structures. It could have received more attention in so far as editing is concerned, but it does transmit a message in a clear and efficient way.