Preface

Summary

The last two decades have produced major advances in the mathematical theory of nonlinear wave phenomena and their applications. In an effort to acquaint researchers in applied mathematics, physics, and engineering and to stimulate further research, an NSF-CBMS regional research conference on Nonlinear Waves and Integrable Systems was convened at East Carolina University in June, 1982. Many distinguished applied mathematicians and scientists from all over the world participated in the conference, and provided a digest of recent developments, open questions, and unsolved problems in this rapidly growing and important field.

As a follow-up project, this book has developed from manuscripts submitted by renowned applied mathematicians and scientists who have made important contributions to the subject of nonlinear waves. This publication brings together current developments in the theory and applications of nonlinear waves and solitons that are likely to determine fruitful directions for future advanced study and research.

The book has been divided into three parts. Part I, entitled Nonlinear Waves in Fluids, consists of seven chapters. Nonlinear Waves in Plasmas are the contents of Part II, which has five chapters. Part III contains six chapters on current results and extensions of the inverse scattering transform and of evolution equations. Included also is recent progress on statistical mechanics of the sine-Gordon field.

The opening chapter, by M.S. Longuet-Higgins, is devoted to recent progress in the analytical representation of overturning waves. Among the forms suggested for the fluid flow are, for the tip of the jet, a rotating Dirichlet hyperbola, and, for the tube, a “-ellipse” or a parametric cubic. All these have been expressed in a semi-Lagrangian form. The semi-Lagrangian form for the rotating hyperbola is derived by a new and simpler method, and certain integral invariants are obtained which have the dimensions of mass, angular momentum and energy. The relation of these to the previously known constants of integration is discussed, and directions for further generalizations are indicated. Also, a new class of polynomial solutions of the semi- Lagrangian boundary conditions is derived. These, or their generalizations, may be of use when combining different solutions so as to form a complete description of the overturning wave.