In the 25 years of its existence Soliton Theory has drastically expanded our understanding of “integrability” and contributed a lot to the reunification of Mathematics and Physics in the range from deep algebraic geometry and modern representation theory to quantum field theory and optical transmission lines.The book is a systematic introduction to the Soliton Theory with an emphasis on its background and algebraic aspects. It is the first one devoted to the general matrix soliton equations, which are of great importance for the foundations and the applications.Differential algebra (local conservation laws, Bäcklund–Darboux transforms), algebraic geometry (theta and Baker functions), and the inverse scattering method (Riemann–Hilbert problem) with well-grounded preliminaries are applied to various equations including principal chiral fields, Heisenberg magnets, Sin–Gordon, and Nonlinear Schrödinger equation.Contents:Introduction:Chiral Fields and Sin-Gordon EquationGeneralized Heisenberg Magnet and VNS EquationConservation Laws and Algebraic-Geometric Solutions:Local Conservation LawsGeneralized Lax EquationsAlgebraic-Geometric Solutions of Basic EquationsAlgebraic-Geometric Solutions of Sin-Gordon, NS, etcBäcklund Tranforms and Inverse Problem:Bäcklund TransformationsIntroduction to the Scattering TheoryApplications of the Inverse Problem MethodReadership: Mathematicians, mathematical physicists and graduate students familiar with basic notions from analysis and algebraic geometry.Key Features:Includes important contributions to the foundations of quantum mechanics and related topics by key Italian scholars such as Ascoli, Beltrametti, Bergia, Cini and LanzGroups together articles by scholars from dissimilar cultural backgrounds, allowing the reader to compare their different perspectivesContains interdisciplinary papers that supply interesting examples of a cross-fertilization of ideas