This is the first volume of a series of books that will describe current advances and past accompli shments of mathemat i ca 1 aspects of nonlinear sCience taken in the broadest contexts. This subject has been studied for hundreds of years, yet it is the topic in whi ch a number of outstandi ng di scoveri es have been made in the past two decades. Clearly, this trend will continue. In fact, we believe some of the great scientific problems in this area will be clarified and perhaps resolved. One of the reasons for this development is the emerging new mathematical ideas of nonlinear science. It is clear that by looking at the mathematical structures themselves that underlie experiment and observation that new vistas of conceptual thinking lie at the foundation of the unexplored area in this field. To speak of specific examples, one notes that the whole area of bifurcation was rarely talked about in the early parts of this century, even though it was discussed mathematically by Poi ncare at the end of the ni neteenth century. I n another di rect ion, turbulence has been a key observation in fluid dynamics, yet it was only recently, in the past decade, that simple computer studies brought to light simple dynamical models in which chaotic dynamics, hopefully closely related to turbulence, can be observed.

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 NonlinearDesingularization 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.