Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics: Generalized Solitons and Hyperasymptotic Perturbation Theory / Edition 1

by John P. Boyd

Overview

" . . . if a physical system is capable of supporting solitary wave motions then such motions will invariably arise from quite general excitations. " - T. Maxworthy (1980), pg. 52. The discover of nonlocal solitary waves is unknown and anonymous, but he or she lived in the dry north of Australia many millenia before the birth of writing. There, on the shores of the Gulf of Carpentaria, vast cylinders of cloud roll from northeast to southwest most mornings. Perhaps 300 meters in diameter, perhaps 500 meters above the ocean, these cylinders of cloud stretch from horizon to horizon. As the cloud evaporates on the trailing edge of the wave and condenses on the leading edge, the cylinder appears to roll backwards even as it propagates inland at perhaps 10-20 meters per second. Often, a whole train of cloud-cylinders propagates from Cape Yorke Penisula across the Gulf towards the southwest across modern Burketown, perhaps as much as 500 km inland into the Northern Territory. Modern-day Australians call it the "Morning Glory". What the discover called it, so many centuries before the invention of hieroglyphics, the foundation of Ur and the coronation of the First Dynasty of China, we do not know. But unless he was very different from us, he felt awe. Physicists (Smith, 1988, and Rottman and Einaudi, 1!:!93) have identified the Morning Glory as a solitary wave.

Product Details

ISBN-13:	9780792350729
Publisher:	Springer US
Publication date:	05/31/1998
Series:	Mathematics and Its Applications , #442
Edition description:	1998
Pages:	596
Product dimensions:	6.10(w) x 9.25(h) x 0.36(d)

Table of Contents

I Overview.- 1 Introduction.- II Analytical Methods.- 2 The Method of Multiple Scales and the E-Power Series.- 3 Hyperasymptotic Perturbation Theory.- 4 Matched Asymptotic Expansions in The Complex Plane.- 5 Stokes’ Expansion, Resonance & Polycnoidal Waves.- 6 Theorems and Proofs: Existence Non-Existence & Symmetry.- III Numerical Methods.- 7 Pseudospectral and Galerkin Methods.- 8 Nonlinear Algebraic Equations.- 9 Special Algorithms for Exponentially Small Phenomena.- IV Applications.- 10 Water Waves: Fifth-Order Korteweg-Devries Equation.- 11 Rossby & Internal Gravity Waves: Nonlocal Higher Modes.- 12 The—4 Breather.- 13 Envelope Solitary Waves: Third Order Nonlinear Schroedinger Equation and the Klein-Gordon Equation.- 14 Temporal Analogues: Separatrix Splitting &The Slow Manifold.- 15 Micropterons.- V Radiative Decay &Other Exponentially Small Phenomena.- 16 Radiative Decay Of Weakly Nonlocal Solitary Waves.- 17 Non-Soliton Exponentially Small Phenomena.- 18 The Future.- References.

Editorial Reviews

Describes a class of waves that radiate away from the core of the disturbance yet are long-lived nonlinear disturbances. Also discussed are the hyperasympototic perturbation theory and the use of Chebysev and Fourier numerical methods to computer solitary waves. A number of non-soliton problems in quantum physics, hydrodynamics, and instability theory are featured. The final chapters examine matched asymptotic expansions in the complex plane, the small denominator problem in Stockes expansions, multiple scale expansions in powers of the hyperbolic secant and tangent functions, and hyperasymptotic perturbation theory. Annotation c. by Book News, Inc., Portland, Or.

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