Integrable Hamiltonian hierarchies: spectral and geometric methods

V.S. Gerdjikov, G. Vilasi, A.B. Yanovski (auth.), V.S. Gerdjikov, G. Vilasi, A.B. Yanovski (eds.)3540770534, 9783540770534, 9783540770541, 3540770542

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.

Table of contents :
Front Matter….Pages I-XII
Front Matter….Pages 1-1
Introduction….Pages 3-36
The Lax Representation and the AKNS Approach….Pages 37-70
The Direct Scattering Problem for theZakharov–Shabat System….Pages 71-95
The Inverse Scattering Problem for the Zakharov–Shabat System….Pages 97-132
The Generalized Fourier Transforms….Pages 133-173
Fundamental Properties of the solvable NLEEs….Pages 175-209
Hierarchies of Hamiltonian structures….Pages 211-245
The NLEEs and the Gauge Transformations….Pages 247-314
The Classical r -Matrix Method….Pages 315-354
Front Matter….Pages 355-355
Introduction….Pages 357-371
Smooth Manifolds….Pages 373-406
Hamiltonian Dynamics….Pages 407-458
Vector-Valued Differential Forms….Pages 459-471
Integrability and Nijenhuis Tensors….Pages 473-513
Poisson–Nijenhuis structures Related to the Generalized Zakharov–Shabat System….Pages 515-545
Linear Bundles of Lie Algebras and Compatible Poisson Structures….Pages 547-611
Back Matter….Pages 613-643