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Geometric Asymptotics for Nonlinear PDE. I
V. P. Maslov, Moscow State University, Russia, and G. A. Omel'yanov, Moscow Institute of Electronic Engineering, Russia
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Translations of Mathematical Monographs
2001; 285 pp; hardcover
Volume: 202
ISBN-10: 0-8218-2109-1
ISBN-13: 978-0-8218-2109-1
List Price: US$109
Member Price: US$87.20
Order Code: MMONO/202
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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

"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

  • Introduction
  • Waves in one-dimensional nonlinear media
  • Nonlinear waves in multidimensional media
  • Asymptotic solutions of some pseudodifferential equations and dynamical systems with small dispersion
  • Problems with a free boundary
  • Multi-phase asymptotic solutions
  • Asymptotics of stationary solutions to the Navier-Stokes equations describing stretched vortices
  • List of equations
  • Bibliography
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