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KdV & KAM. (English) Zbl 1032.37001
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 45. Berlin: Springer. xii, 279 p. (2003).
This monograph is concerned with a natural question that arises once a partial differential equation – such as the Korteweg-de Vries (KdV) equation considered here – is understood as an infinite dimensional completely integrable Hamiltonian system. That question is whether the famous Kolmogorov-Arnold-Moser (KAM) Theorem for finite dimensional Hamiltonian systems can be extended to the infinite dimensional case.
Indeed, there have recently been published other texts dealing with KAM theory for partial differential equations, by J. Bourgain [Global solutions of nonlinear Schrödinger equations, Colloquium Publications, American Mathematical Society, Providence, RI (1999; Zbl 0933.35178)], W. Craig [Problèmes de petits diviseurs dans les équations aux dérivées partielles, Panoramas et Synthèses, 9, Societé Mathématique de France, Paris (2000; Zbl 0977.35014)] and S. B. Kuksin [Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, 19, Oxford University Press, Oxford (2000; Zbl 0960.35001)]. Here an abstract KAM Theorem is presented which is then used to derive a KdV perturbation theory.
The book under review is extremely well written. It is intended to be self-contained and includes a number of appendices (in fact some eighty pages of appendices) which cover a variety of supporting topics, for example: Birkhoff normal forms, the spectra of Schrödinger operators on a finite interval, and symplectic geometry in infinite dimensions. In this way the book is made much more accesible than it might otherwise have been. The authors also intend that each chapter might be read independently; whilst they have certainly been successful in this, there remains the question of whether the resulting repetition is worth it or not. On balance it probably is, and in either case this is a minor quibble in an otherwise excellent text.
Chapter I provides a summary of the results presented in the main chapters of the book. Chapter II provides a detailed review of classical (finite dimensional) KAM theory, including ideas – such as Birkhoff normal forms – that in later chapters are extended from the finite- to the infinite dimensional case. Chapter III introduces global Birkhoff coordinates for KdV (summarised in Theorem 12.6). In Chapter IV, the main theorems on persistence under small Hamiltonian perturbations of quasiperiodic solutions of the KdV equation and of the fifth-order KdV equation (Theorems 13.1 and 13.2) are given; the authors are in fact conscious throughout of the possibility of extending their results to all equations of KdV hierarchy. The proof of these theorems is an application of an infinite-dimensional version of KAM theory that is concerned with the persistence of finite-dimensional invariant tori: the authors’ KAM Theorem is presented in Chapter IV and then restated (as Theorem 18.1, and with a slight change to one of the hypotheses) and proved in Chapter V. Chapter VI is concerned with a Lemma due to Kuksin, used by the authors in the proof of their KAM Theorem.
This book should certainly be of interest to anyone working in the areas of finite dimensional Hamiltonian dynamics or completely integrable partial differential equations, even if not working directly on the topic considered here.

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
35Q53 KdV equations (Korteweg-de Vries equations)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
37K55 Perturbations, KAM for infinite-dimensional Hamiltonian and Lagrangian systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion