Soliton theory: a survey of results.

*(English)*Zbl 0697.35126
Nonlinear Science: Theory and Applications. Manchester etc.: Manchester University Press. vii, 449 p. £65.00 (1989).

[The articles of this volume will not be indexed individually.]

The theory of solitons deals with “travelling wave” solutions of some specific nonlinear evolution equations. Physically such solutions may behave like moving particles.

One of the most remarkable representative of such an equation is the Korteweg-de Vries (KdV) equation, and from numerical experiments by Zabusky and Kruskal it turned out that there should be travelling wave solutions of such an equation. The explicit integration of the KdV equation by Kruskal and his coworkers was the beginning of the mathematical theory of solitons.

It is an aim of this book to give a survey on some (more “elementary”) methods used for the integration of differential equations which are special examples for completely integrable Hamiltonian systems. There are for example the inverse scattering method, or a method using Bäcklund transformations or others, coming from differential geometry.

In a second part there are discussed selected applications like the soliton laser, propagation of solitons in the Andaman sea, general relativity, or soliton models of protein dynamics.

This shows that though the treated equations are very exceptional in the class of Hamiltonian systems, they even have important applications. The book then continues with topics on Hamiltonian systems and with the question of complete integrability. It has 16 chapters written by several authors, each of them reporting on his working field. At the end of each chapter there are numerous references for further reading.

The book provides a consistent introduction into the subject and should be recommended to everybody who is working in this field, but also interesting details are given to physicists and mathematicians who want to learn something on the subject of solitons.

The theory of solitons deals with “travelling wave” solutions of some specific nonlinear evolution equations. Physically such solutions may behave like moving particles.

One of the most remarkable representative of such an equation is the Korteweg-de Vries (KdV) equation, and from numerical experiments by Zabusky and Kruskal it turned out that there should be travelling wave solutions of such an equation. The explicit integration of the KdV equation by Kruskal and his coworkers was the beginning of the mathematical theory of solitons.

It is an aim of this book to give a survey on some (more “elementary”) methods used for the integration of differential equations which are special examples for completely integrable Hamiltonian systems. There are for example the inverse scattering method, or a method using Bäcklund transformations or others, coming from differential geometry.

In a second part there are discussed selected applications like the soliton laser, propagation of solitons in the Andaman sea, general relativity, or soliton models of protein dynamics.

This shows that though the treated equations are very exceptional in the class of Hamiltonian systems, they even have important applications. The book then continues with topics on Hamiltonian systems and with the question of complete integrability. It has 16 chapters written by several authors, each of them reporting on his working field. At the end of each chapter there are numerous references for further reading.

The book provides a consistent introduction into the subject and should be recommended to everybody who is working in this field, but also interesting details are given to physicists and mathematicians who want to learn something on the subject of solitons.

Reviewer: G.Jank

##### MSC:

35Q99 | Partial differential equations of mathematical physics and other areas of application |

35-06 | Proceedings, conferences, collections, etc. pertaining to partial differential equations |

00Bxx | Conference proceedings and collections of articles |

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |