Mathematical Structures of Nonlinear Science: An Introduction

1 Integrable Nonlinear Systems and their Perturbation.- 1.1 The Simplest Nonlinear Systems.- 1.2 Integration by Quadrature and Its Alternatives.- 1.3 Classical Mechanical Integrable Systems.- 1.4 New Ideas on Complete Integrability for Equilibrium Processes.- 1.5 Canonical Changes of Coordinates for the Mapping A.- 1.6 Bifurcation and the Integration of Nonlinear Ordinary and Partial Differential Equations.- 1.7 Qualitative Properties of Integrable Systems - Periodic and Quasiperiodic Motions of Dynamical Systems.- 1.8 Almost Periodic Motions of Dynamical Systems.- Appendix 1 Nonlinear Fredholm Operators.- Appendix 2 Bifurcation from Equilibria for Certain Infinite-Dimensional Dynamical Systems.- Appendix 3 Elementary Facts about the Linear Dirichlet Problem.- Appendix 4 On Besicovitch Almost Periodic Functions.- 2 General Principles for Nonlinear Systems.- 2.1 Differentiable Nonlinear Operators.- 2.2 Iteration of Nonlinear Operators.- 2.3 Nonlinear Fredholm Alternatives.- 2.4 The Idea of Nonlinear Desingularization.- 2.5 Variational Principles - New Ideas in the Calculus of Variations in the Large.- 2.6 Bifurcation.- 2.7 Bifurcation Into Folds.- 3 Some Connections between Global Differential Geometry and Nonlinear Analysis.- 3.1 Geodesics.- 3.2 Gauss Curvature and Its Extensions.- 3.3 Manifolds of Constant Gauss Positive Curvature.- 3.4 Mean Curvature.- 3.5 Simple Riemannian Metrics.- 3.6 Einstein Metrics.- 4 Vortices in Ideal Fluids.- 4.1 The Early History of Vortices in Fluids.- 4.2 Formulation of the Vortex Concept in Ideal Incompressible Fluids.- 4.3 Axisymmetric Vortex Motions with and without Swirl.- 4.4 Variational Principles for the Stream Function for Vortex Rings without Swirl.- 4.5 Leapfrogging of Vortices.- 4.6 Vortex Breakdown.- 4.7 Nonlinear Desingularization and Vortex Filaments.- 5 Mathematical Aspects of Superconductivity.- 5.1 The Simplest Nonlinear Yang-Mills Theory that Works.- 5.2 Physical Viewpoint.- 5.3 The Linear Approach to Superconductivity and Nonlinear Desingularization.- 5.4 Function Spaces for Symmetric Vortices.- 5.5 The Existence of Critical Points for I? Associated with Symmetric Vortices.- Problems.