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.