**Graduate Studies in Mathematics**

Volume: 182;
2017;
575 pp;
Hardcover

MSC: Primary 35; 37;

**Print ISBN: 978-1-4704-3613-1
Product Code: GSM/182**

List Price: $99.00

AMS Member Price: $79.20

MAA Member Price: $89.10

**Electronic ISBN: 978-1-4704-4228-6
Product Code: GSM/182.E**

List Price: $99.00

AMS Member Price: $79.20

MAA Member Price: $89.10

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#### Supplemental Materials

# Nonlinear PDEs: A Dynamical Systems Approach

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*Guido Schneider; Hannes Uecker*

This is an introductory textbook about
nonlinear dynamics of PDEs, with a focus on problems over unbounded
domains and modulation equations. The presentation is
example-oriented, and new mathematical tools are developed step by
step, giving insight into some important classes of nonlinear PDEs and
nonlinear dynamics phenomena which may occur in PDEs.

The book consists of four parts. Parts I and II are introductions
to finite- and infinite-dimensional dynamics defined by ODEs and by
PDEs over bounded domains, respectively, including the basics of
bifurcation and attractor theory. Part III introduces PDEs on the real
line, including the Korteweg-de Vries equation, the Nonlinear
Schrödinger equation and the Ginzburg-Landau equation. These examples
often occur as simplest possible models, namely as amplitude or
modulation equations, for some real world phenomena such as nonlinear
waves and pattern formation. Part IV explores in more detail the
connections between such complicated physical systems and the reduced
models. For many models, a mathematically rigorous justification by
approximation results is given.

The parts of the book are kept as self-contained as possible. The
book is suitable for self-study, and there are various possibilities
to build one- or two-semester courses from the book.

#### Readership

Graduate students and researchers interested in nonlinear dynamics of PDEs.

#### Reviews & Endorsements

This book as a whole is more than the sum of its chapters and deserves being slowly and thoughtfully read from the beginning to the end.

-- Michael Zaks, Mathematical Reviews

This is an excellent text which can be used for several graduate courses in mathematics departments.

-- Dmitry Pelinovsky, Zentralblatt MATH

The combination of rigor with simultaneous attention to associated real physical systems makes it particularly appealing.

-- William Satzer, MAA Reviews

#### Table of Contents

# Table of Contents

## Nonlinear PDEs: A Dynamical Systems Approach

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface xi12
- Chapter 1. Introduction 116
- Part I Nonlinear dynamics in \R^{𝑑} 1429
- Part II Nonlinear dynamics in countably many dimensions 131146
- Part III PDEs on the infinite line 204219
- Part IV Modulation theory and applications 314329
- Chapter 10. Dynamics of pattern and the GL equation 315330
- 10.1. Introduction 316331
- 10.2. The Swift-Hohenberg equation 319334
- 10.3. The universality of the GL equation 332347
- 10.4. An abstract approximation result 337352
- 10.5. Reaction-Diffusion systems 347362
- 10.6. Convection problems 354369
- 10.7. The Couette-Taylor problem 370385
- 10.8. Attractors for pattern forming systems 378393
- 10.9. Further remarks 395410

- Chapter 11. Wave packets and the NLS equation 401416
- Chapter 12. Long waves and their modulation equations 451466
- Chapter 13. Center manifold reduction and spatial dynamics 473488
- Chapter 14. Diffusive stability 497512
- Bibliography 541556
- List of symbols 567582
- Index 569584

- Back Cover Back Cover1593