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Nonlinear waves, solitons and chaos. (English) Zbl 0726.76018

Cambridge etc.: Cambridge University Press. xi, 423 p. £45.00/hbk; $ 85.00/hbk; £17.50/pbk; $ 29.95/pbk (1990).
To a large extent, we find here a progress report for the areas covered in the title for more than the two last decades, though without any ambition for completeness. With emphasis on discussion and evaluation, the authors try to explain arguments for novel approaches, under search for interrelations and limitation. Accordingly, this book may appeal to specialists from various branches of physics, though the examples and all phenomenological evidence are taken from fluid dynamics and plasma physics, where both authors can refer to an impressive stoch of original contributions to the first two topics, investigations on stability in particular.
On the other hand, we could classify this work as a series of exercises in developing new methods of applied mathematics (in the broad anglo- saxon sense), partial d.e.’s in particular, such as the Klein-Gordon equation, the nonlinear Schrödinger equation and the Korteweg-de Vries \((=KdV)\) equation (victim of a misprint when introduced in (1.2.8)!). For these 3 exact methods of solution are presented: by Bäcklund transformation and by a direct method in addition to inverse scattering. For the KdV-equation, we thus obtain a one-crested soliton \((=\) solitary wave as first detected by Scott Russel advancing without change of shape along a canal in England (1834)), or even systems of such uni-crested waves, regaining their original shape after penetrating each other.
For approximate treatment of the dynamics (and stability) of nonlinear plane waves, four methods are presented: Two of them due to Whitham (through averaging conservation laws or of Lagrangians) [e.g. G. B. Whitham, Linear and nonlinear waves (1974; Zbl 0373.76001); J. Jimenez and G. B. Whitham, Proc. R. Soc. Lond., Ser. A 349, 277- 287 (1976; Zbl 0326.70019)], one by Hayes (averaging the Hamiltonian) [e.g. M. Hayes, Arch. Ration. Mech. Anal. 85, 41-80 (1984); 97, 221-270 (1987; Zbl 0613.73106)] and finally the authors’ approach of expansion in terms of the wave number ratio. These methods are tested simultaneously on the sine-Gordon d. e. - It is stated that the last method ranks first in applicability (no existence of a Lagrangian or a Hamiltonian is required!), though worst in aspects of formal elegance. Further topics treated can be quoted from the chapter headings: (3) Convective and non-convective instabilities; (5) Model equations for small amplitude waves and solitons (7) Cartesian solitons in one and two space dimensions; (8) Evolution of stability of initially one dimensional waves and solitons; (9 \[ ylindrical\text{ and } spherical\quad solitons\text{ in } plasmas\text{ and } other\quad media. \] It should be mentioned that in chapter 10 deterministic chaos only is dealt with, in order to demonstrate that pseudo-random nonperiodic time dependence can arise from very simple nonlinear equations, both discrete and continuous.
The book is illustrated by more than 100 sketches and diagrams together with some photos; there are 21 pages of references! Each chapter concludes with a set of problems.
Reviewer: K.Eggers (Hamburg)

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
35Q51 Soliton equations
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