The research topic for this IAS/PCMI Summer Session was nonlinear wave phenomena. Mathematicians from the more theoretical areas of PDEs were brought together with those involved in applications. The goal was to share ideas, knowledge, and perspectives. How waves, or 'frequencies', interact in nonlinear phenomena has been a central issue in many of the recent developments in pure and applied analysis. It is believed that wavelet theory - with its simultaneous localization in both physical and frequency space and its lacunarity - is and will be a fundamental new tool in the treatment of the phenomena.Included in this volume are write-ups of the 'general methods and tools' courses held by Jeff Rauch and Ingrid Daubechies. Rauch's article discusses geometric optics as an asymptotic limit of high-frequency phenomena. He shows how nonlinear effects are reflected in the asymptotic theory. In the article ""Harmonic Analysis, Wavelets and Applications"" by Daubechies and Gilbert the main structure of the wavelet theory is presented.Also included are articles on the more 'specialized' courses that were presented, such as ""Nonlinear Schrodinger Equations"" by Jean Bourgain and ""Waves and Transport"" by George Papanicolaou and Leonid Ryzhik.
Susan Friedlander provides a written version of her lecture series ""Stability and Instability of an Ideal Fluid"", given at the Mentoring Program for Women in Mathematics, a preliminary program to the Summer Session. This Summer Session brought together students, fellows, and established mathematicians from all over the globe to share ideas in a vibrant and exciting atmosphere. This book presents the compelling results.
Nonlinear Schrodinger equations: Introduction by J. Bourgain Generalities and initial value problems by J. Bourgain The initial value problem (continued) by J. Bourgain A digressioin: The initial value problem for the KdV equation by J. Bourgain 1D invariant Gibbs measures by J. Bourgain Invariant measures (2D) by J. Bourgain Quasi-periodic solutions of Hamiltonian PDE by J. Bourgain Time periodic solutions by J. Bourgain Time quasi-periodic solutions by J. Bourgain Normal forms by J. Bourgain Applications of symplectic capacities to Hamiltonian PDE by J. Bourgain Remarks on longtime behaviour of the flow of Hamiltonian PDE by J. Bourgain Harmonic analysis, wavelets and applications: Introduction by I. C. Daubechies and A. C. Gilbert Constructing orthonormal wavelet bases: Multiresolution analysis by I. C. Daubechies and A. C. Gilbert Wavelet bases: Construction and algorithms by I. C. Daubechies and A. C. Gilbert More wavelet bases by I. C. Daubechies and A. C. Gilbert Wavelets in other functional spaces by I. C. Daubechies and A. C. Gilbert Pointwise convergence for wavelet expansions by I. C. Daubechies and A. C. Gilbert Two-dimensional wavelets and operators by I. C. Daubechies and A. C. Gilbert Wavelets and differential equations by I. C. Daubechies and A. C. Gilbert References by I. C. Daubechies and A. C. Gilbert Lectures on stability and instability of an ideal fluid: Introduction by S. Friedlander Equations of motion by S. Friedlander Initial-boundary value problem by S. Friedlander The type of the Euler equations by S. Friedlander Vorticity by S. Friedlander Steady flows by S. Friedlander Stability/instability of an equilibrium state by S. Friedlander Two-dimensional spectral problem by S. Friedlander ""Arnold"" criterion for nonlinear stability by S. Friedlander Plane parallel shear flow by S. Friedlander Instability in a vorticity norm by S. Friedlander Sufficient condition for instability by S. Friedlander Exponential stretching by S. Friedlander Integrable flows by S. Friedlander Baroclinic instability by S. Friedlander Nonlinear instability by S. Friedlander References by S. Friedlander Waves and transport: Introduction by G. Papanicolaou and L. Ryzhik The Schrodinger equation by G. Papanicolaou and L. Ryzhik Symmetric hyperbolic systems by G. Papanicolaou and L. Ryzhik Waves in random media by G. Papanicolaou and L. Ryzhik The diffusion approximation by G. Papanicolaou and L. Ryzhik The geophysical applications by G. Papanicolaou and L. Ryzhik References by G. Papanicolaou and L. Ryzhik Lectures on geometric optics: Introduction by J. Rauch and M. Keel Basic linear existence theorems by J. Rauch and M. Keel Examples of propagation of singularities and of energy by J. Rauch and M. Keel Elliptic geometric optics by J. Rauch and M. Keel Linear hyperbolic geometric optics by J. Rauch and M. Keel Basic nonlinear existence theorems by J. Rauch and M. Keel One phase nonlinear geometric optics by J. Rauch and M. Keel Justification of one phase nonlinear geometric optics by J. Rauch and M. Keel References by J. Rauch and M. Keel.
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