Backlund and Darboux Transformations: The Geometry of Solitons

This book is devoted to a classical topic that has undergone rapid and fruitful development over the past 25 years, namely Backlund 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 Backlund transformations of Painleve equations, to the dressing method and Backlund and Darboux transformations, and to the classical geometry of Backlund 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 Backlund and Darboux transformations.

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