1 Soliton Theory and Modern Physics.- 2 Inverse Scattering Methods.- 3 Bäcklund Transformations and Darboux Transformations.- 4 Classical Integrable Systems.- 5 Symmetry.- 6 Kac-Moody Algebras and Integrable Systems.- 7 Soliton and Differential Geometry.- 8 Numerical Study of Nonlinear Waves.- 9 Solitons in the Theory of Gravitational Waves.- References.

Soliton theory is an important branch of applied mathematics and mathematical physics. An active and productive field of research, it has important applications in fluid mechanics, nonlinear optics, classical and quantum fields theories etc..This book presents a broad view of soliton theory. It gives an expository survey of the most basic ideas and methods, such as physical background, inverse scattering, Bäcklund transformations, finite-dimensional completely integrable systems, symmetry, Kac-Moody algebra, solitons and differential geometry, numerical analysis for nonlinear waves, and gravitational solitons. Besides the essential points of the theory, several applications are sketched and some recent developments, partly by the author and his collaborators, are presented.This book has been written for specialists, as well as for teachers and students in mathematics and physics.