Hyperbolic Equations and Frequency Interactions

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Brand new. We distribute directly for the publisher. 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 Schr??dinger 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. Read more Show Less

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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.

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Editorial Reviews

Booknews
This volume contains lecture notes from the Graduate Summer School program on nonlinear wave phenomena, which took place in Park City, Utah, July 1995. The five write-ups of courses discuss geometric optics as an asymptotic limit of high-frequency phenomena, the main structure of the wavelet theory, nonlinear Schrodinger equations, waves and radiative transport equations, and the stability and instability properties of an ideal fluid. Of interest to graduate students and research mathematicians working in partial differential equations. No index. Annotation c. by Book News, Inc., Portland, Or.
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Product Details

Table of Contents

Preface
Introduction 1
Nonlinear Schrodinger Equations 3
Harmonic Analysis, Wavelets and Applications 159
Lectures on Stability and Instability of an Ideal Fluid 227
Waves and Transport 305
Lectures on Geometric Optics 383
References 465
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