**Translations of Mathematical Monographs**

2001;
285 pp;
Hardcover

MSC: Primary 35; 76;

**Print ISBN: 978-0-8218-2109-1
Product Code: MMONO/202**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

**Electronic ISBN: 978-1-4704-4627-7
Product Code: MMONO/202.E**

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

# Geometric Asymptotics for Nonlinear PDE. I

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*V. P. Maslov; G. A. Omel’yanov*

The study of asymptotic solutions to nonlinear systems of partial
differential equations is a very powerful tool in the analysis of such
systems and their applications in physics, mechanics, and engineering.
In the present book, the authors propose a new powerful method of
asymptotic analysis of solutions, which can be successfully applied in
the case of the so-called "smoothed shock waves", i.e., nonlinear waves
which vary fast in a neighborhood of the front and slowly outside of
this neighborhood. The proposed method, based on the study of geometric
objects associated to the front, can be viewed as a generalization of
the geometric optics (or WKB) method for linear equations. This volume
offers to a broad audience a simple and accessible presentation of this
new method.

The authors present many examples originating from problems of
hydrodynamics, nonlinear optics, plasma physics, mechanics of continuum,
and theory of phase transitions (problems of free boundary). In the
examples, characterized by smoothing of singularities due to dispersion
or diffusion, asymptotic solutions in the form of distorted solitons,
kinks, breathers, or smoothed shock waves are constructed. By a unified
rule, a geometric picture is associated with each physical problem that
allows for obtaining tractable asymptotic formulas and provides a
geometric interpretation of the physical process. Included are many
figures illustrating the various physical effects.

#### Readership

Graduate students, research and applied mathematicians, physicists, specialists in theoretical mechanics interested in partial differential equations and fluid mechanics.

#### Reviews & Endorsements

The book contains many interesting new asymptotic formulas for an extensive number of physically meaningful equations. Moreover, the application of the methods developed here goes beyond the models discussed in the book and could clearly stimulate further research in the area. The book is highly recommended to specialists in the theory of nonlinear partial differential equations and their applications in various domains of science.

-- Bulletin of the LMS

The introduction gives an impressive description of the state of the art in the asymptotic theory of nonlinear waves. This monograph is of great interest for specialists in partial differential equations and mathematical physics.

-- Bulletin of the Belgian Mathematical Society

#### Table of Contents

# Table of Contents

## Geometric Asymptotics for Nonlinear PDE. I

- Cover Cover11
- Title page v6
- Contents vii8
- Introduction 110
- Waves in one-dimensional nonlinear media 1120
- Nonlinear waves in multidimensional media 5766
- Asymptotic solutions of some pseudodifferential equations and dynamical systems with small dispersion 133142
- Problems with a free boundary 159168
- Multi-phase asymptotic solutions 193202
- Asymptotics of stationary solutions to the Navier-Stokes equations describing stretched vortices 241250
- List of equations 271280
- Bibliography 275284
- Back Cover Back Cover1297