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

" . . . 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.

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Content:
Front Matter....Pages i-xix
Front Matter....Pages 1-1
Introduction....Pages 3-28
Front Matter....Pages 29-29
The Method of Multiple Scales and the ?-Power Series....Pages 31-47
Hyperasymptotic Perturbation Theory....Pages 48-79
Matched Asymptotic Expansions in the Complex Plane....Pages 80-105
Stokes’ Expansion, Resonance & Polycnoidal Waves....Pages 106-131
Theorems and Proofs: Existence, Non-Existence and Symmetry....Pages 132-138
Front Matter....Pages 139-139
Pseudospectral and Galerkin Methods....Pages 141-171
Nonlinear Algebraic Equations....Pages 172-223
Special Algorithms for Exponentially Small Phenomena....Pages 224-240
Front Matter....Pages 241-241
Water Waves: Fifth-Order Korteweg-Devries Equation....Pages 243-278
Rossby & Internal Gravity Waves: Nonlocal Higher Modes....Pages 279-305
The ?4 Breather....Pages 306-324
Envelope Solitary Waves: Third Order Nonlinear Schroedinger Equation and the Klein-Gordon Equation....Pages 325-365
Temporal Analogues: Separatrix Splitting & the Slow Manifold....Pages 366-386
Micropterons....Pages 387-430
Front Matter....Pages 431-431
Radiative Decay of Weakly Nonlocal Solitary Waves....Pages 433-454
Non-Soliton Exponentially Small Phenomena....Pages 455-478
The Future....Pages 479-481
Back Matter....Pages 482-596

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Content:
Front Matter....Pages i-xix
Front Matter....Pages 1-1
Introduction....Pages 3-28
Front Matter....Pages 29-29
The Method of Multiple Scales and the ?-Power Series....Pages 31-47
Hyperasymptotic Perturbation Theory....Pages 48-79
Matched Asymptotic Expansions in the Complex Plane....Pages 80-105
Stokes’ Expansion, Resonance & Polycnoidal Waves....Pages 106-131
Theorems and Proofs: Existence, Non-Existence and Symmetry....Pages 132-138
Front Matter....Pages 139-139
Pseudospectral and Galerkin Methods....Pages 141-171
Nonlinear Algebraic Equations....Pages 172-223
Special Algorithms for Exponentially Small Phenomena....Pages 224-240
Front Matter....Pages 241-241
Water Waves: Fifth-Order Korteweg-Devries Equation....Pages 243-278
Rossby & Internal Gravity Waves: Nonlocal Higher Modes....Pages 279-305
The ?4 Breather....Pages 306-324
Envelope Solitary Waves: Third Order Nonlinear Schroedinger Equation and the Klein-Gordon Equation....Pages 325-365
Temporal Analogues: Separatrix Splitting & the Slow Manifold....Pages 366-386
Micropterons....Pages 387-430
Front Matter....Pages 431-431
Radiative Decay of Weakly Nonlocal Solitary Waves....Pages 433-454
Non-Soliton Exponentially Small Phenomena....Pages 455-478
The Future....Pages 479-481
Back Matter....Pages 482-596
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