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  3. Soliton Equations and Their Algebro-Geometric Solutions
  • Cited by 52
Publisher:
Cambridge University Press
Online publication date:
January 2010
Print publication year:
2008
Online ISBN:
9780511543203
DOI:
https://doi.org/10.1017/CBO9780511543203
Subjects:
Mathematics (general), Mathematics, Differential and Integral Equations, Dynamical Systems and Control Theory, Mathematical Physics
Series:
Cambridge Studies in Advanced Mathematics (114)
Information

Book description

As a partner to Volume 1: Dimensional Continuous Models, this monograph provides a self-contained introduction to algebro-geometric solutions of completely integrable, nonlinear, partial differential-difference equations, also known as soliton equations. The systems studied in this volume include the Toda lattice hierarchy, the Kac-van Moerbeke hierarchy, and the Ablowitz-Ladik hierarchy. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The theory presented includes trace formulas, algebro-geometric initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses basic techniques from the theory of difference equations and spectral analysis, some elements of algebraic geometry and especially, the theory of compact Riemann surfaces. The presentation is constructive and rigorous, with ample background material provided in various appendices. Detailed notes for each chapter, together with an exhaustive bibliography, enhance understanding of the main results.

Reviews

Reviews of Volume 1:'… this is a book that I would recommend to any student of mine, for clarity and completeness of exposition … Any expert as well would enjoy the book and learn something stimulating from the sidenotes that point to alternative developments. We look forward to volumes two and three!'

Source: Mathematical Reviews

'The book is very well organized and carefully written. It could be particularly useful for analysts wanting to learn new methods coming from algebraic geometry.'

Source: EMS Newsletter

Review of Volume 2:'As with the first part, the book is very well written and carefully organised and it is a pleasure to read it.'

Source: EMS Newsletter

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