Springer Series in Nonlinear Dynamics. Berlin: Springer-Verlag. xii, 337 p. DM 138.00; öS 1076.40; sFr. 138.00 (1994).

The link between certain differential equations in mathematical physics and the theory of abelian functions can be traced back to the last decades of the 19th century, when K. Neumann, S. Kowalevskaya, and others discovered that some phenomena in classical mechanics could be described by nonlinear systems integrable in terms of Jacobi-Riemann theta functions. In the sequel, this remarkable link remained rather disregarded. Many years later, between 1922 and 1928, Burchnall, Chaundy, and Baker published a series of (three) papers, in which they discussed an apparently deep connection between commuting ordinary differential operators and the function theory on algebraic curves associated with them. However, also this discovery had not been taken up. Almost half a century later, around 1974, the Burchnall-Chaundy-Baker theory was rediscovered, basically in connection with the spectacular success in constructing solutions of the famous Korteweg-de Vries equation (KdV) and P. Lax’s method of interpreting certain nonlinear PDEs by commutator relations for differential operators. In the following years, the theory of the so-called finite-gap solutions of soliton equations underwent a rapid development. The actually old link between algebras of commuting differential operators and data on algebraic curves (or compact Riemann surfaces, respectively) has been extended to a systematic and deep-going theory, on the one hand, and has been effectively used for constructing explicit solutions of the inverse spectral problems for nonlinear operators with (quasi-) periodic coefficients as well as solutions of several KdV-like equations in terms of higher-order theta functions on Jacobians of algebraic curves, on the other hand.
This rich theory has been developed, during the past 20 years, simultaneously by several analysts and geometers, in particular by Krichever, Novikov, Dubrovin, Lax, McKean, Adler, van Moerbeke, Mumford, Akhieser, Gelfan’d, Dikii, Marchenko, the authors of the present monograph, and many others. Apart from its usefulness for integrating various types of nonlinear equations relevant in mathematical physics, the link between integrable systems and algebraic curves has led to spectacular solutions of long-standing problems in the classification theory of compact Riemann surfaces and complex abelian varieties. In fact, the analytic solution of the classical Riemann-Schottky problem of characterizing period matrices of Riemann surfaces among all Riemann matrices by equations in theta constants (Arbarello-De Concini, Welters (1983)) as well as the affirmative answer to the more general Novikov conjecture that the important Kadomtsev-Petviashvili equation (KP) also characterizes theta functions of Riemann surfaces (Shiota (1985)) are substantially based upon this so-called Krichever correspondence.
The present monograph, written by some of the pioneers and main contributors in this fairly new area, is perhaps the first coherent, comprehensive and self-contained exposition of the now far-developed algebro-geometric approach to nonlinear integrable systems. The contents consist of eight chapters, each of which being divided into several sections and subsections. Chapter 1, the introduction, sketches the thrilling history of the search for solutions of the KdV-equation, and gives a detailed outline of the contents of the following chapters, together with historical remarks and references to the original literature.
Chapter 2 provides, mainly for non-specialists, the background material from the theory of compact Riemann surfaces, their Jacobian varieties, and theta functions. This purely algebro-geometric (or complex-analytic) survey contains -- of course -- no proofs, but it is nicely arranged and completely sufficient for the following applications to nonlinear differential equations.
Chapter 3 deals with the finite-gap solutions of the Kadomtsev- Petviashvili and the Korteweg-de Vries equations. As a first application of theta functions of Riemann surfaces to integrable systems, it is shown how the so-called Akhieser-Baker functions on curves lead to solutions of these equations. Along this way, also Hill’s equation, Lamé’s equation, and the analytic properties of Bloch functions are discussed.
Chapter 4 is devoted to the higher-dimensional case. By means of vector- valued Akhieser-Baker functions on Riemann surfaces, the authors describe the finite-gap solutions of the nonlinear Schrödinger equation, the sine-Gordon equation, and other related systems.
Chapter 5 explains how the Schottky uniformization of Riemann surfaces and the Poincaré theta series can equally be used to construct solutions of KdV-like equations.
Chapter 6 discusses the various classical tops via the algebro-geometric approach, essentially by applying the classical theta function relations, through this approach, to concrete Lax equations and their associated Akhieser-Baker functions. The main explicit results concern the Kovalevskaya top, the Goryachev-Chaplygin top, the $XYZ$ Landau-Lifschitz equation, and other well-known integrable systems of classical mechanics.
Chapter 7 concerns the problem of interpreting the superposition of finite-gap solutions of integrable nonlinear equations by algebro- geometric constructions and principles. This is done by using multi- sheeted coverings of Riemann surfaces, the Weierstrass reduction theory, and relations between theta constants with respect to coverings.
The concluding Chapter 8 describes one of the most remarkable applications of the algebro-geometric method in nonlinear dynamics, namely the construction of an exact solution of the Peierls-Fröhlich problem in solid-state physics.
With regard to these contents, it is fair to say that the present monograph includes, apart from a careful introduction to, and from an enlightening illustration of the algebro-geometric method in nonlinear dynamics, a vast account of recent research achievements obtained by the authors themselves. The text provides an up-to-date picture of the present state of the topic, and thus should be highly welcome to mathematicians and physicists dealing with dynamical systems, quantum physics, and/or complex algebraic geometry.