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Bäcklund and Darboux Transformations. The Geometry of Solitons
Edited by: Alan Coley, Dalhousie University, Halifax, NS, Canada, Decio Levi, University of Rome III, Italy, Robert Milson, Dalhousie University, Halifax, NS, Canada, Colin Rogers, University of New South Wales, Sydney, NSW, Australia, and Pavel Winternitz, Université de Montréal, QC, Canada
A co-publication of the AMS and Centre de Recherches Mathématiques.

CRM Proceedings & Lecture Notes
2001; 436 pp; softcover
Volume: 29
ISBN-10: 0-8218-2803-7
ISBN-13: 978-0-8218-2803-8
List Price: US$138
Member Price: US$110.40
Order Code: CRMP/29
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This book is devoted to a classical topic that has undergone rapid and fruitful development over the past 25 years, namely Bäcklund and Darboux transformations and their applications in the theory of integrable systems, also known as soliton theory.

The book consists of two parts. The first is a series of introductory pedagogical lectures presented by leading experts in the field. They are devoted respectively to Bäcklund transformations of Painlevé equations, to the dressing method and Bäcklund and Darboux transformations, and to the classical geometry of Bäcklund transformations and their applications to soliton theory. The second part contains original contributions that represent new developments in the theory and applications of these transformations.

Both the introductory lectures and the original talks were presented at an International Workshop that took place in Halifax, Nova Scotia (Canada). This volume covers virtually all recent developments in the theory and applications of Bäcklund and Darboux transformations.

Titles in this series are co-published with the Centre de Recherches Mathématiques.


Graduate students and research mathematicians interested in dynamical systems, ergodic theory, and partial differential equations.

Table of Contents

Introductory Lectures
  • V. I. Gromak -- Bäcklund transformations of the higher order Painlevé equations
  • D. Levi and O. Ragnisco -- Dressing method and Bäcklund and Darboux transformations
  • C. Rogers and W. K. Schief -- The classical geometry of Bäcklund transformations. Introduction to applications in soliton theory
  • W. K. Schief -- An introduction to integrable difference and differential geometries: Affine spheres, their natural generalization and discretization
Original Contributions
  • Yu. Aminov and A. Sym -- On Bianchi and Bäcklund transformations of two dimensional surfaces in four dimensional Euclidean space
  • I. M. Anderson, M. E. Fels, and C. G. Torre -- Group invariant solutions without transversality and the principle of symmetric criticality
  • H. Aratyn, E. Nissimov, and S. Pacheva -- Multi-component matrix KP hierarchies as symmetry-enhanced scalar KP hierarchies and their Darboux-Bäcklund solutions
  • J. L. Cieśliński -- The Darboux-Bäcklund transformation and Clifford algebras
  • P. A. Clarkson, E. L. Mansfield, and H. N. Webster -- On discrete Painlevé equations as Bäcklund transformations
  • J. N. Clelland -- A Bäcklund transformation for timelike surfaces of constant mean curvature in \(\mathbb R^{1,2}\)
  • A. V. Corro, W. Ferreira, and K. Tenenblat -- On Ribaucour transformations
  • A. Doliwa -- The Ribaucour congruences of spheres within Lie sphere geometry
  • N. M. Ercolani -- Bäcklund transformations for the reduced Maxwell-Bloch equations
  • E. V. Ferapontov -- Transformations of quasilinear systems originating from the projective theory of congruences
  • E. V. Ferapontov and A. M. Grundland -- Bäcklund links between different analytic descriptions of constant mean curvature surfaces
  • F. Finkel -- On the integrability of Weingarten surfaces
  • F. Finkel and A. S. Fokas -- A new immersion formula for surfaces on Lie algebras and integrable equations
  • J. D. Finley III -- Difficulties with the SDiff(2) Toda equation
  • M. Havlíček, S. Pošta, and P. Winternitz -- Superposition formulas based on nonprimitive group action
  • R. H. Heredero, D. Levi, M. A. Rodríguez, and P. Winternitz -- Symmetries of differential difference equations and Lie algebra contractions
  • J. Hietarinta -- Bäcklund transformations from the bilinear viewpoint
  • L. Hlavatý -- Towards the Lax formulation of SU(2) principal models with nonconstant metric
  • C. A. Hoenselaers and S. Micciché -- Transcendental solutions of the sine-Gordon equation
  • T. Ioannidou, B. Piette, and W. J. Zakrzewski -- Three dimensional skyrmions and harmonic maps
  • B. G. Konopelchenko and G. Landolfi -- Induced surfaces and their integrable deformations
  • M. Kovalyov -- Properties of a class of slowly decaying oscillatory solutions of KdV
  • S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta, and K. M. Tamizhmani -- Blending two discrete integrability criteria: Singularity confinement and algebraic entropy
  • W.-X. Ma and X. Geng -- Bäcklund transformations of soliton systems from symmetry constraints
  • P. Mathieu -- Open problems for the super KdV equations
  • R. Milson -- Combinatorial aspects of the Darboux transformation
  • M. Musette, R. Conte, and C. Verhoeven -- Bäcklund transformation and nonlinear superposition formula of the Kaup-Kupershmidt and Tzitzéica equations
  • P. J. Olver, J. A. Sanders, and J. P. Wang -- Classification of symmetry-integrable evolution equations
  • E. G. Reyes -- Integrability of evolution equations and pseudo-spherical surfaces
  • C. Rogers and W. K. Schief -- Infinitesimal Bäcklund transformations of \(K\)-nets. The 2 + 1-dimensional Sinh-Gordon system
  • W. K. Schief -- Isothermic surfaces and the Calapso equation: The full Monty
  • R. Schmid -- Bäcklund transformations induced by symmetries. Application: Discrete mKdV
  • H. Steudel -- Darboux transformation for a spectral problem quadratic in the spectral parameter
  • Z. Thomova and P. Winternitz -- Separation of variables and Darboux transformations
  • P. Winternitz -- Bäcklund transformations as nonlinear ordinary differential, or difference equations with superposition formulas
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