**Graduate Studies in Mathematics**

Volume: 133;
2012;
363 pp;
Hardcover

MSC: Primary 35; 78; 93;

**Print ISBN: 978-0-8218-7291-8
Product Code: GSM/133**

List Price: $68.00

AMS Member Price: $54.40

MAA Member Price: $61.20

**Electronic ISBN: 978-0-8218-8508-6
Product Code: GSM/133.E**

List Price: $64.00

AMS Member Price: $51.20

MAA Member Price: $57.60

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# Hyperbolic Partial Differential Equations and Geometric Optics

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*Jeffrey Rauch*

This book introduces graduate students and
researchers in mathematics and the sciences to the multifaceted
subject of the equations of hyperbolic type, which are used, in
particular, to describe propagation of waves at finite speed.

Among the topics carefully presented in the book are nonlinear
geometric optics, the asymptotic analysis of short wavelength
solutions, and nonlinear interaction of such waves. Studied in detail
are the damping of waves, resonance, dispersive decay, and solutions
to the compressible Euler equations with dense oscillations created by
resonant interactions. Many fundamental results are presented for the
first time in a textbook format. In addition to dense oscillations,
these include the treatment of precise speed of propagation and the
existence and stability questions for the three wave interaction
equations.

One of the strengths of this book is its careful motivation of
ideas and proofs, showing how they evolve from related, simpler
cases. This makes the book quite useful to both researchers and
graduate students interested in hyperbolic partial differential
equations. Numerous exercises encourage active participation of the
reader.

The author is a professor of mathematics at the University of
Michigan. A recognized expert in partial differential equations, he
has made important contributions to the transformation of three areas
of hyperbolic partial differential equations: nonlinear microlocal
analysis, the control of waves, and nonlinear geometric optics.

#### Readership

Graduate students and research mathematicians interested in partial differential equations.

#### Table of Contents

# Table of Contents

## Hyperbolic Partial Differential Equations and Geometric Optics

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface xi12 free
- Chapter 1. Simple examples of propagation 122 free
- Chapter 2. The linear Cauchy problem 4364
- Chapter 3. Dispersive behavior 91112
- Chapter 4. Linear elliptic geometric optics 123144
- Chapter 5. Linear hyperbolic geometric optics 141162
- Chapter 6. The nonlinear Cauchy problem 215236
- Chapter 7. One phase nonlinear geometric optics 259280
- Chapter 8. Stability for one phase nonlinear geometric optics 277298
- Chapter 9. Resonant interaction and quasilinear systems 291312
- Chapter 10. Examples of resonance in one dimensional space 317338
- Chapter 11. Dense oscillations for the compressible Euler equations 333354
- Bibliography 351372
- Index 359380 free
- Back Cover Back Cover1386