The theory of nonlinear wave equations in the
absence of shocks began in the 1960s. Despite a great deal of recent
activity in this area, some major issues remain unsolved, such as
sharp conditions for the global existence of solutions with arbitrary
initial data, and the global phase portrait in the presence of
periodic solutions and traveling waves.
This book, based on lectures presented by the author at George Mason
University in January 1989, seeks to present the sharpest results to
date in this area. The author surveys the fundamental qualitative
properties of the solutions of nonlinear wave equations in the absence
of boundaries and shocks. These properties include the existence and
regularity of global solutions, strong and weak singularities,
asymptotic properties, scattering theory and stability of solitary
waves. Wave equations of hyperbolic, Schrödinger, and KdV type are
discussed, as well as the Yang–Mills and the Vlasov–Maxwell
equations.
The book offers readers a broad overview of the field and
an understanding of the most recent developments, as well as the
status of some important unsolved problems. Intended for mathematicians
and physicists interested in nonlinear waves, this book would be
suitable as the basis for an advanced graduate-level course.