In recent years there have been important and far reaching developments in the study of nlinear waves and a class of nlinear wave equations which arise frequently in applications. The wide interest in this field comes from the understanding of special waves called 'solitons' and the associated development of a method of solution to a class of nlinear wave equations termed the inverse scattering transform (IST). Before these developments, very little was kwn about the solutions to such 'soliton equations'. The IST technique applies to both continuous and discrete nlinear Schrodinger equations of scalar and vector type. Also included is the IST for the Toda lattice and nlinear ladder network, which are well-kwn discrete systems. This book presents the detailed mathematical analysis of the scattering theory; soliton solutions are obtained and soliton interactions, both scalar and vector, are analyzed. Much of the material is t available in the previously-published literature.