Mathematics and its Applications (Dordrecht). 442. Dordrecht: Kluwer Academic Publishers. xix, 590 p. Dfl 450.00; $ 198.00; £ 134.75 (1998).

This is a fundamental comprehensive monograph devoted to analytical and numerical methods for calculation of exponentially small effects in the presence of a small parameter (i.e., “asymptotics beyond all orders”) in both partial and ordinary differential equations (that appear, first of all, in hydrodynamics and nonlinear optics, but also in a number of other physical applications), and to applications of those methods to numerous problems for solitons and other physical objects. In the case of the solitons, central problems are weak (frequently, exponentially small) emission of radiation by slowly decaying solitons, due to their resonant coupling to the model’s linear spectrum, and stationary weakly nonlocal solitons in models of the same type, sitting on top of a quasilinear background in the form of a radiation wave with an (exponentially) small but finite amplitude. In the studies of these effects, an analytical calculation of the core exponentially small factor is frequently straightforward (basically, this is the same exponential smallness as that of an integral taken from $-\infty$ to $+\infty$ of a product of randomly oscillating and slowly varying functions), while calculation of a corresponding pre-exponential factor is very nontrivial (but is usually possible, being based on the technique of matched asymptotic expansions). The numerical approach to searching for the exponentially small effects also has its specific features, being usually based upon spectral methods (Fourier or Galerkin decompositions, etc.). Exponentially small effects also play an important role in the theory of finite-dimensional dynamical systems, an important example being the splitting of separatrices in an integrable system containing a small nonintegrable perturbation.
The book comprises all these and many other topics. It is well rubricated, being split into five nearly independent parts: overview; analytical methods; numerical applications; general applications; and the specific application in the form of the radiative decay of solitons. The parts are further divided into chapters, which consist of precisely defined subchapters. Besides the weakly nonlocal solitary waves in hydrodynamics and nonlinear optics, other important physical applications are considered in detail, e.g., fingering in propagation of solidification fronts, radiative effects in field theory models, and others. The book, and parts thereof, can be recommended as useful reading for graduate students specializing in physics, applied mathematics, and advanced engineering. Simultaneously, the book can be efficiently used for reference by researchers actively working in these areas. The material is presented from the viewpoint of a physicist, rather than from that of a pure mathematician.