Classical and Quantum Dynamics of Integrable Nonlinear Field Theories

Inverse scattering method leads to complete integration and explicit construction of soliton type solutions in many important nonlinear equations and has a wide range of applications in almost all areas of physics. Many examples of relativistic two-dimensional classical field equations can also be completely solved by this inverse scattering technique. This chapter summarizes and briefly reviews the important nonlinear field theories which are solvable by the inverse scattering method. It gives a list of relevant field equations and exhibits the wide variety of field theories which can be completely solved by this powerful method. The Sine-Gordon equation played a central role in the development of semiclassical methods. It is in fact the first example of a nontrivial relativistic quantum field theory which allows for an exact quantum solution. The relevance of classical solutions to the quantum theory is established through semiclassical quantization methods.