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Geometric Asymptotics for Nonlinear PDE. I
 
V. P. Maslov Moscow State University, Moscow, Russia
G. A. Omel’yanov Moscow Institute of Electronic Engineering, Moscow, Russia
Geometric Asymptotics for Nonlinear PDE. I
Hardcover ISBN:  978-0-8218-2109-1
Product Code:  MMONO/202
List Price: $165.00
MAA Member Price: $148.50
AMS Member Price: $132.00
eBook ISBN:  978-1-4704-4627-7
Product Code:  MMONO/202.E
List Price: $155.00
MAA Member Price: $139.50
AMS Member Price: $124.00
Hardcover ISBN:  978-0-8218-2109-1
eBook: ISBN:  978-1-4704-4627-7
Product Code:  MMONO/202.B
List Price: $320.00 $242.50
MAA Member Price: $288.00 $218.25
AMS Member Price: $256.00 $194.00
Geometric Asymptotics for Nonlinear PDE. I
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Geometric Asymptotics for Nonlinear PDE. I
V. P. Maslov Moscow State University, Moscow, Russia
G. A. Omel’yanov Moscow Institute of Electronic Engineering, Moscow, Russia
Hardcover ISBN:  978-0-8218-2109-1
Product Code:  MMONO/202
List Price: $165.00
MAA Member Price: $148.50
AMS Member Price: $132.00
eBook ISBN:  978-1-4704-4627-7
Product Code:  MMONO/202.E
List Price: $155.00
MAA Member Price: $139.50
AMS Member Price: $124.00
Hardcover ISBN:  978-0-8218-2109-1
eBook ISBN:  978-1-4704-4627-7
Product Code:  MMONO/202.B
List Price: $320.00 $242.50
MAA Member Price: $288.00 $218.25
AMS Member Price: $256.00 $194.00
  • Book Details
     
     
    Translations of Mathematical Monographs
    Volume: 2022001; 285 pp
    MSC: Primary 35; 76

    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.

  • Table of Contents
     
     
    • Chapters
    • 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
  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2022001; 285 pp
MSC: Primary 35; 76

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.

  • Chapters
  • 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
  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.