Weakly Nonlocal Solitary Waves a
nd Beyond-All-Orders Asymptotics, Kluwer (1998)
Generalized Solitons and Hyperasymptotic Perturbation Theory
Description
This book has several themes. The first is to describe weakly nonlocal solitary
waves, which radiate away from the core of the disturbance, but are nevertheless
very long-lived nonlinear disturbances. Half a dozen chapters describe specific
examples in water waves, particle physics, meteorology and oceanography, pulses
in fiber optics, and dynamical systems theory. For many species of nonlocal
solitary waves, the radiation is exponentially small in 1/e where e is a
perturbation parameter, thus lying "beyond-all-orders". A second theme of the
book is to describe hyperasymptotic perturbation theory and other extensions of
standard perturbation methods which have been developed, mostly in the last ten
years, to compute such exponentially small corrections to asymptotic series. A
third theme is developed in three chapters: Chebyshev and Fourier numerical
methods for computing solitary waves. Special emphasis is given to
steadily-translating coherent structures, a difficult numerical problem even
today. A fourth theme is to briefly describe a large number of non-soliton
problems in quantum physics, hydrodynamics, instability theory and others where
"beyond-all-orders" corrections arise, and where the perturbative and numerical
methods described earlier are essential.
The book is aimed at graduate students and postgraduate researchers in applied
mathematics, physics, meteorology, oceanography, or one of the many engineering
fields where solitary waves are important. However, because the range of
applications is so broad, it is has been written so that an undergraduate
physical science or engineering background should suffice to follow most
sections. (However, a little previous exposure to the theory of solitary waves
and perturbation theory is helpful.)
The application chapters are (largely) self-contained so that after reading the
introductory chapter, one can jump directly to the chapter on Rossby waves, or
breathers, or water waves, and follow at least the main ideas.
closely related review
article "The Devil's Invention: Asymptotics, Superasymptotic and Hyperas
ymptotic Series", Acta Applicandae vol. 56, pgs. 1-98 (1999).
Table of Contents (.pdf file, 0.045 MB)
Bibliography (.pdf file, 0.185 MB)