Bäcklund and Darboux transformations. Geometry and modern applications in soliton theory. (English) Zbl 1019.53002

Rogers, C.; Schief, W. K.

Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press. xvii, 413 p. (2002).

The aim of this monograph is to use Bäcklund-Darboux transformations to provide an introductory text in modern soliton theory, connected with the classical differential geometry of surfaces. The first chapter is devoted to the pseudo-spherical surfaces, the classical Bäcklund transformations and the Bianchi system. The second chapter is concerned with how certain motions of privileged curves and surfaces can lead to solitonic equations. In chapter 3, the classical surfaces of Tzitzeica (the Romanian geometer who initiated affine geometry) have an underlying soliton connection.
Chapter 4 is devoted to the nonlinear Schrödinger Equation and chapter 5 is concerned with another class of surfaces which have a soliton connection, namely isothermic surfaces. Chapter 6 introduces the key Sym-Tafel formula for the construction of soliton surfaces associated with an su(2) linear representation. Chapter 7 establishes the important connection between Bäcklund transformations and matrix Darboux transformations. Chapter 8 deals with the geometric properties of important soliton system which admit non-isospectral linear representation. Chapter 9 describes developments in soliton theory which are linked to the geometry of projective-minimal and isothermal-asymptotic surfaces.
The book contains also some appendices and it is very useful for those interested in this subject.

Reviewer: Cornelia-Livia Bejan (Iasi)

MSC:

53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
37K35	Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
58J35	Heat and other parabolic equation methods for PDEs on manifolds

Keywords:

Bäcklund transformations; Darboux transformations; soliton theory