Scaling Limits and Models in Physical Processes (Oberwolfach Seminars)

This is an introductory text, in two parts, on scaling limits and modelling in equations of mathematical physics. The first part is concerned with basic concepts of the kinetic theory of gases which is not only important in its own right but also as a prototype of a mathematical construct central to the theory of non-equilibrium phenomena in large systems. It also features a very readable historic survey of the field. The second part dwells on the role of integrable systems for modelling weakly nonlinear equations which contain the effects of both dispersion and nonlinearity. Starting with a historical introduction to the subject and a description of numerical techniques, it proceeds to a discussion of the derivation of the Korteweg de Vries and nonlinear Schrödinger equations, followed by a careful treatment of the inverse scattering theory for the Schrödinger operator. The book provides an up-to-date and detailed overview to this very active area of research and is intended as an accessible introduction for non-specialists and graduate students in mathematics, physics and engineering.