Integrable Hamiltonian Hierarchies: Spectral and Geometric Methods

ISBN-13: 9783540770534 / Angielski / Twarda / 2008 / 643 str.

Vladimir Gerdjikov; Gaetano Vilasi; Alexandar Borrisov Yanovski

Opis

In the past decades now a famous class of evolution equations has been discovered and intensively studied, a class including the nowadays celebrated Korteweg-de Vries equation, sine-Gordon equation, nonlinear Schr] odinger equation, etc. The equations from this class are known also as the soliton equations or equations solvable by the so- called Inverse Scattering Tra- form Method. They possess a number of interesting properties, probably the most interesting from the geometric point of view of being that most of them are Liouville integrable Hamiltonian systems. Because of the importance of the soliton equations, a dozen monographs have been devoted to them. H- ever, the great variety of approaches to the soliton equations has led to the paradoxical situation that specialists in the same ?eld sometimes understand eachotherwithdi?culties. Wediscovereditourselvesseveralyearsagoduring a number of discussions the three of us had. Even though by friendship binds us, we could not collaborate as well as we wanted to, since our individual approach to the ?eld of integrable systems (?nite and in?nite dimensional) is quite di?erent. We have become aware that things natural in one approach are di?cult to understand for people using other approaches, though the - jects are the same, in our case the Recursion (generating) Operators and theirapplicationsto?niteandin?nitedimensional(notnecessarilyintegrable) Hamiltonian systems."

Recenzje

From the reviews:

"The book under review provides a wide overview on the so-called soliton equations ... . the book contains many topics that experts will find interesting, it is well organized and is an excellent research monograph on soliton equations. ... it is accessible to a wide range of readers, from the experts in the field, to researchers who are interested in nonlinear evolution equations and PhD students who would like to learn more about this field." (A. Yoshioka, Il Nuovo Saggiatore, Vol. 29 (5-6), 2013)

"Will be of great use to anybody who is interested in the AKNS method ... . The bibliographic remarks given at the end of each section are quite useful if one wants to be oriented in the existing trends of the theory. ... succeeded in bringing together different approaches while keeping the main focus on Recursion Operators. ... It should be mentioned that the book is written in a reader-friendly way ... . recommended to PhD students who want to learn more about the subject." (Giuseppe Marmo, Bulletin of the London Mathematical Society, Vol. 44, 2012)

"This is a book on integrable partial differential equations. ... scope of this book is large. It is a reference book. It can serve as an advance graduate textbook. ... A signal feature of this book is the number of references, allowing one to review and study other examples and related aspects. ... Overall, this work is impressive in what it covers and is of value to students and researchers in this area." (D. J. Kaup, Bulletin of the London Mathematical Society, December, 2009)

"This monograph aims to discuss about the inverse scattering transform and the bi-Hamiltonian theory of soliton equations in both analytical and geometric approaches. ... The book is well organized and clearly written. ... The interested readers ... can easily find various relevant theories and their references. It is an excellent research monograph summarizing and evaluating our growing body of research on soliton theory and integrable Hamiltonian systems." (Ma Wen-Xiu, Zentralblatt MATH, Vol. 1167, 2009)

"The book under review is divided into two parts. ... devoted to the recollection of many by-now classical results concerning the application of spectral methods to integrable Hamiltonian hierarchies. ... The two parts are written with the intention of being understandable to nonspecialists in the field and are self-contained. Each chapter ends with a rich bibliographical review which emphasizes the contribution of the authors to the field." (Simonetta Abenda, Mathematical Reviews, Issue 2009 m)

Spis treści

Integrable Hamiltonian Hierarchies: Spectral Methods.- The Lax Representation and the AKNS Approach.- The Direct Scattering Problem for theZakharov–Shabat System.- The Inverse Scattering Problem for the Zakharov–Shabat System.- The Generalized Fourier Transforms.- Fundamental Properties of the solvable NLEEs.- Hierarchies of Hamiltonian structures.- The NLEEs and the Gauge Transformations.- The Classical r-Matrix Method.- Integrable Hamiltonian Hierarchies: Geometric Theory of the Recursion Operators.- Smooth Manifolds.- Hamiltonian Dynamics.- Vector-Valued Differential Forms.- Integrability and Nijenhuis Tensors.- Poisson–Nijenhuis structures Related to the Generalized Zakharov–Shabat System.- Linear Bundles of Lie Algebras and Compatible Poisson Structures.

Blurb

This book presents a detailed derivation of the spectral properties of the Recursion Operators allowing one to derive all the fundamental properties of the soliton equations and to study their Hamiltonian hierarchies. Thus it is demonstrated that the inverse scattering method for solving soliton equations is a nonlinear generalization of the Fourier transform.

The book brings together the spectral and the geometric approaches and as such
will be useful to a wide readership: from researchers in the field of nonlinear
completely integrable evolution equations to graduate and post-graduate students.