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Analytic-bilinear approach to integrable hierarchies. (English) Zbl 0942.37048

Mathematics and its Applications (Dordrecht). 493. Dordrecht: Kluwer Academic Publishers. xii, 264 p. (1999).
The approach and flair of this book is best illustrated by reporting verbatim, almost in full, the author’s Preface.
“The subject of this book is the hierarchies of integrable equations connected with the one-component and multicomponent loop groups. …”
“The Sato Grassmannian approach, and other approaches standard in this context, reveal deep mathematical structures in the base of the integrable hierarchies. These approaches concentrate mostly on the algebraic picture, and they use a language suitable for applications to quantum field theory.
Another well-known approach, the \(\overline{\partial}\)-dressing method, developed by S. V. Manakov and V. E. Zakharov, is oriented mostly to particular systems and exact classes of their solutions. There is more emphasis on analytic properties, and the technique is connected with standard complex analysis. The language of the \(\overline{\partial}\)-dressing method is suitable for applications to integrable nonlinear PDEs, integrable nonlinear discrete equations and, as recently discovered, for the applications of integrable systems to continuous and discrete geometry.
The primary motivation of the author was to formalize the approach to integrable hierarchies that was developed in the context of the \(\overline{\partial}\)-dressing method, preserving the analytic structures characteristic for this method, but omitting the peculiarities of the constructive scheme. And it was desirable to find a starting point that could be included in the Grassmannian approach, thus filling the gap between these two methods and making the exchange of ideas possible. The result of these efforts is the analytic-bilinear approach to integrable hierarchies, which was developed by the author in collaboration with B. G. Konopelchenko. This book is an attempt to present the approach in a consistent and coherent manner.
The starting point of the analytic-bilinear approach is the Hirota bilinear identity for the Cauchy-Baker-Akhiezer (CBA) function (kernel). The notion of the CBA function was introduced by P. G. Grinevich and A. Yu. Orlov in the framework of the algebro-geometric integration technique; similar function for the \(\overline{\partial}\)-dressing method was constructed by S. V. Manakov and the author. …”
“Beginning with the Hirota bilinear identity for the CBA function and using mostly methods of standard complex analysis, we develop a picture of the generalized hierarchy. The primary objects are integrable discrete equations, which correspond to the group of rational loops. The expansion of these equations into powers of lattice parameters leads to the hierarchies of integrable PDEs, which include the basic hierarchy (the one-component KP hierarchy or the multicomponent KP hierarchy), the hierarchy of modified equations, and the hierarchy of singular manifold equations. Different types of objects connected with integrable systems (the Darboux transformations, the Bäcklund transformations, the Miura maps, the \(\tau\)-function and addition formulae for it) are naturally incorporated into the generalized hierarchy.
The author believes that the analytic-bilinear approach gives a consistent and technically simple description of integrable hierarchies, and shows an easy and direct way to the understanding of rather complicated structures. This book, in the author’s opinion, could be useful to students and specialists, interested in the theory of integrable systems. It provides interesting supplementary material for a course in complex analysis. The author also hopes that even experts in the field of integrable systems will find something new for them in this book.”
This book is however not easy reading even for specialists (let alone students), unless they are familiar with previous developments, such as the theory and terminology of loop groups. It does contain quite a lot of interesting results, especially on many types of Kadomtsev-Petviashvili (KP) equations, and on integrable hierarchies of such equations. As an indication of the level of previous understanding of the subject that is assumed of the reader, no reference to papers of historical importance are provided, for instance no paper by Kadomtsev and Petviashvili, or by Hirota, is quoted (yet on p. 2 the author writes: “It is no exaggeration to say that almost every formula in this book is derived from the Hirota bilinear identity for the Cauchy kernel (CBA function).”).
The reviewer only spotted few misprints, including unfortunately (a trivial) one just in the version (1.2) of the Hirota bilinear identity (\(d\lambda\) should be replaced by \(dv\)) and (another trivial) one in the subsequent (unnumbered) formula \((g(\mu)\) should be replaced by \(g(\mu)^{-1}\)).
The compact presentation of so many interesting results under a single cover makes this book a desirable addition to one’s library; unfortunately its price entails that few scholars will be able to avail themselves personally of this possibility.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q58 Other completely integrable PDE (MSC2000)
35Q53 KdV equations (Korteweg-de Vries equations)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
37K60 Lattice dynamics; integrable lattice equations
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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