Weakly Nonlocal Solitary Waves a nd Beyond-All-Orders Asymptotics, Kluwer (1998)
Generalized Solitons and Hyperasymptotic Perturbation Theory


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)