Basic Methods Of Soliton Theory

This text is a introduction to the mathematical soliton theory with an emphasis on algebraic aspects, including an exposition of its background, the recent developments and concrete methods for the equations important to physics. The book is centered around general matrix soliton equation, which are of importance to the foundations and applications of soliton theory. It contains algebraized constructions of local conservation laws; Backlund-Darboux transformations; the integration technique by algebraic geometry and the inverse scattering method and the Hamiltonian theory with well grounded preliminaries. The general results are detailed for the Sine-Gordon, Nonlinear Schrodinger, Landau-Lifschitz equations and are interpreted from the point of view of algebraic geometry and the representaion theory by means of loop groups and Kac-Moody algebras, r-matrices and tau-functions.