This chapter is devoted to three additional topics.

The equations we have discussed have been those for which the inverse scattering transform is applicable. However, given any evolution equation, it is natural to ask whether it can be solved by the inverse scattering transform (1ST); in other words, how do we decide if a given equation is completely integrable? This question is still open but a promising conjecture concerns the so-called Painlevé property. We shall describe how the Painlevé equations arise, what they are and the conjecture itself.

If the evolution equation cannot be solved by the 1ST, but is close to one which can be (by virtue of a small parameter), we may adopt the following procedure. The 1ST method is formulated in the conventional way but the time evolution of the scattering data now involves the small parameter. This parameter can be used as the basis for generating an asymptotic solution of the inverse scattering problem, and hence of the original equation. We shall outline the development of this argument.

Finally, if neither of the above methods is applicable, or if a graphical representation of the solution is required, then we may use a numerical solution. Indeed, the original motivation for the 1ST came from a study of numerical solutions of the KdV equation. We shall, in the final section of this chapter, present some numerical methods suited to the solution of the initial-value problem for evolution equations.

The theory of solitons is attractive and exciting; it brings together many branches of mathematics, some of which touch on deep ideas. Several of its aspects are amazing and beautiful; we shall present some of them in this book. The theory is, nevertheless, related to even more areas of mathematics, and has even more applications to the physical sciences, than the number which are included here. It has an interesting history and a promising future. Indeed, the work of Kruskal and his associates which gave us the ‘inverse scattering transform’ – a grand title for soliton theory – is a major achievement of twentieth-century mathematics. Their work was stimulated by a physical problem together with some surprising computational results. This is a classic example of how numerical results lead to the development of new mathematics, just as observational and experimental results have done since the time of Archimedes.

This book has grown out of Solitons written by one of us (PGD). That book originated from lectures given to final-year mathematics honours students at the University of Bristol. Much of the material in this version has also been used as the basis for an introductory course on inverse scattering theory given to MSc students at the University of Newcastle upon Tyne. In both courses the aim was to present the essence of inverse scattering clearly, rather than to develop the theory rigorously and completely. That is also the overall aim of this book.

Abstract

Introduction

Many natural materials possess mechanisms by which they adjust the degree of their anisotropy in order to carry, more efficiently, the load to which they are being subjected. By the degree of anisotropy we mean the relative stiffness or compliance of the material in different directions. For example, some fibrous composites have a ratio of Young's modulus in their fiber direction to Young's modulus in their transverse direction of 200. These materials are said to be strongly anisotropic. On the other hand bone can be described as mildly anisotropic because the ratio of Young's moduli in different directions generally exceeds two. However, bone has a mechanism by which it changes the degree of its anisotropy to adjust to its environmental load and manmade fibrous composites do not generally possess these adaptive mechanisms. As an example of the effect of this adaptive mechanism in bone one can note the fact that the ratio of Young's moduli near the mid-shaft of the human femur is about two, but it decreases to near one (almost isotropic) near the joints.

INTRODUCTION

Nearby, but below, the order-disorder phase-transition temperature, the dynamical equations that govern elastic crystals with a microstructure exhibit all the necessary, if not sufficient, properties (essentially, nonlinearity and dispersion with a possible compensation between the two effects) to allow for the propagation of so-called solitary waves. Whether these are true solitons or not is a question that can be answered only in each case through analysis and/or numerical simulations. These waves, however, are supposed to represent domain walls in motion. The latter are layers of relatively small thickness which carry a strong nonuniformity in a relevant parameter between two adjacent phases or degenerate ground states (see, e.g., Maugin and Pouget, 1986).

This volume contains most of the invited lectures delivered as part of the Symposium on “Non-classical Continuum Mechanics, Abstract Techniques and Applications” held in the University of Durham from 2 – 12th July, 1986. The meeting was under the auspices of the London Mathematical Society with financial support generously provided by the Science and Engineering Research Council of Great Britain. To these organisations and also to the University of Durham (including staff of the Department of Mathematics and of Grey College) grateful thanks are extended on behalf of all participants and members of the Organising Committee. Sincere thanks are also due to Mrs P. Hampton (Heriot-Watt University) and Mrs S. Cuttle (University of Durham) who admirably coped with the secretarial and administrative burdens.

One objective of the Symposium was to provide the opportunity for interaction between two broad trends now discernible in continuum mechanics which respectively emphasise applications and rigorous mathematical analysis. The mutual enrichment of these two developments requires frequent exchange of information and thanks to the right note struck in the formal addresses this certainly occurred at the Durham meeting. Indeed, all contributors are to be commended for ensuring that their presentations were readily accessible to both groups thus helping to further the aims of the organisers. This in turn has resulted in the present volume containing articles which besides being accounts of recent progress should also be of wide interest to the specialist and non-specialist alike.