Summary: Three Sources and Three Component Parts
of the Concept of Dissipative Solitons
N. Akhmediev and A. Ankiewicz
Abstract We explain the notion of dissipative solitons within a historical perspec-
tive. We show that the ideas of the theory of dissipative solitons emerge from several
fields, including classical soliton theory, nonlinear dynamics, with its theory of bi-
furcations, and Prigogine's concept of self-organization. A new notion, emerging
from this three-part foundation, allows us to build the novel concept of the dissipa-
tive soliton. We also show that reductions to lower dimensional systems have to be
done carefully and should always include a comparison of the results with numerical
simulations of the original equations.
This book is a collection of works in various fields that have the common concept of
the "dissipative soliton" behind each specific topic. Before turning to the chapters,
written by experts in their fields, it is instructive to start with general "definitions"
and a little history of the terms. Our introductory chapter summarizes the main ideas
and explains what a "dissipative soliton" entails, in simple terms. The ideas have
been developed over a number of years and have appeared in various publications
[1, 2, 3], but a compact presentation of all these ideas in a single volume is important.
The term "soliton" was first introduced in 1965 by Zabusky and Kruskal