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Bäcklund and Darboux Transformations. The Geometry of Solitons

About this Title

Alan Coley, Dalhousie University, Halifax, NS, Canada, Decio Levi, University of Rome III, Rome, Italy, Robert Milson, Dalhousie University, Halifax, NS, Canada, Colin Rogers, University of New South Wales, Sydney, NSW, Australia and Pavel Winternitz, Université de Montréal, Montréal, QC, Canada, Editors

Publication: CRM Proceedings and Lecture Notes
Publication Year: 2001; Volume 29
ISBNs: 978-0-8218-2803-8 (print); 978-1-4704-3943-9 (online)
DOI: https://doi.org/10.1090/crmp/029
MathSciNet review: MR1870397
MSC: Primary 37-06; Secondary 35A22, 35A30, 35Q53, 37K10, 37K25, 37K35, 53-06

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Table of Contents

Front/Back Matter

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Introductory Lectures

• Bäcklund transformations of the higher order Painlevé equations
• Dressing method and Bäcklund and Darboux transformations
• The classical geometry of Bäcklund transformations. Introduction to applications in soliton theory
• An introduction to integrable difference and differential geometries: Affine spheres, their natural generalization and discretization

Original Contributions

• On Bianchi and Bäcklund transformations of two dimensional surfaces in four dimensional Euclidean space
• Group invariant solutions without transversality and the principle of symmetric criticality
• Multi-component matrix KP hierarchies as symmetry-enhanced scalar KP hierarchies and their Darboux-Bäcklund solutions
• The Darboux-Bäcklund transformation and Clifford algebras
• On discrete Painlevé equations as Bäcklund transformations
• A Bäcklund transformation for timelike surfaces of constant mean curvature in $\mathbb R^{1,2}$
• On Ribaucour transformations
• The Ribaucour congruences of spheres within Lie sphere geometry
• Bäcklund transformations for the reduced Maxwell-Bloch equations
• Transformations of quasilinear systems originating from the projective theory of congruences
• Bäcklund links between different analytic descriptions of constant mean curvature surfaces
• On the integrability of Weingarten surfaces
• A new immersion formula for surfaces on Lie algebras and integrable equations
• Difficulties with the SDiff(2) Toda equation
• Superposition formulas based on nonprimitive group action
• Symmetries of differential difference equations and Lie algebra contractions
• Bäcklund transformations from the bilinear viewpoint
• Towards the Lax formulation of SU(2) principal models with nonconstant metric
• Transcendental solutions of the sine-Gordon equation
• Three dimensional skyrmions and harmonic maps
• Induced surfaces and their integrable deformations
• Properties of a class of slowly decaying oscillatory solutions of KdV
• Blending two discrete integrability criteria: Singularity confinement and algebraic entropy
• Bäcklund transformations of soliton systems from symmetry constraints
• Open problems for the super KdV equations
• Combinatorial aspects of the Darboux transformation
• Bäcklund transformation and nonlinear superposition formula of the Kaup-Kupershmidt and Tzitzéica equations
• Classification of symmetry-integrable evolution equations
• Integrability of evolution equations and pseudo-spherical surfaces
• Infinitesimal Bäcklund transformations of $K$-nets. The 2 + 1-dimensional Sinh-Gordon system
• Isothermic surfaces and the Calapso equation: The full Monty
• Bäcklund transformations induced by symmetries. Application: Discrete mKdV
• Darboux transformation for a spectral problem quadratic in the spectral parameter
• Separation of variables and Darboux transformations
• Bäcklund transformations as nonlinear ordinary differential, or difference equations with superposition formulas

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