The Complex WKB Method for Nonlinear Equations I

Linear Theory

Author: Victor P. Maslov

Publisher: Birkhäuser

ISBN:

Category: Science

Page: 304

View: 239

When this book was first published (in Russian), it proved to become the fountainhead of a major stream of important papers in mathematics, physics and even chemistry; indeed, it formed the basis of new methodology and opened new directions for research. The present English edition includes new examples of applications to physics, hitherto unpublished or available only in Russian. Its central mathematical idea is to use topological methods to analyze isotropic invariant manifolds in order to obtain asymptotic series of eigenvalues and eigenvectors for the multi-dimensional Schrödinger equation, and also to take into account the so-called tunnel effects. Finite-dimensional linear theory is reviewed in detail. Infinite-dimensional linear theory and its applications to quantum statistics and quantum field theory, as well as the nonlinear theory (involving instantons), will be considered in a second volume.

Mathematical Modelling of Heat and Mass Transfer Processes

Author: V.G. Danilov

Publisher: Springer Science & Business Media

ISBN:

Category: Mathematics

Page: 323

View: 343

In the present book the reader will find a review of methods for constructing a certain class of asymptotic solutions, which we call self-stabilizing solutions. This class includes solitons, kinks, traveling waves, etc. It can be said that either the solutions from this class or their derivatives are localized in the neighborhood of a certain curve or surface. For the present edition, the book published in Moscow by the Nauka publishing house in 1987, was almost completely revised, essentially up-dated, and shows our present understanding of the problems considered. The new results, obtained by the authors after the Russian edition was published, are referred to in footnotes. As before, the book can be divided into two parts: the methods for constructing asymptotic solutions ( Chapters I-V) and the application of these methods to some concrete problems (Chapters VI-VII). In Appendix a method for justification some asymptotic solutions is discussed briefly. The final formulas for the asymptotic solutions are given in the form of theorems. These theorems are unusual in form, since they present the results of calculations. The authors hope that the book will be useful to specialists both in differential equations and in the mathematical modeling of physical and chemical processes. The authors express their gratitude to Professor M. Hazewinkel for his attention to this work and his support.

Quantization Methods in the Theory of Differential Equations

Author: Vladimir E. Nazaikinskii

Publisher: CRC Press

ISBN:

Category: Mathematics

Page: 368

View: 933

This volume presents a systematic and mathematically rigorous exposition of methods for studying linear partial differential equations. It focuses on quantization of the corresponding objects (states, observables and canonical transformations) in the phase space. The quantization of all three types of classical objects is carried out in a unified way with the use of a special integral transform. This book covers recent as well as established results, treated within the framework of a universal approach. It also includes applications and provides a useful reference text for graduate and research-level readers.

Introduction to the General Theory of Singular Perturbations

Author: S. A. Lomov

Publisher: American Mathematical Soc.

ISBN:

Category: Mathematics

Page: 375

View: 739

This book is aimed at researchers and students in physics, mathematics, and engineering. It contains the first systematic presentation of a general approach to the integration of singularly perturbed differential equations describing nonuniform transitions, such as the occurrence of a boundary layer, discontinuities, boundary effects and so on. The method of regularization of singular perturbations presented here can be applied to the asymptotic integration of systems of ordinary and partial differential equations.

Asymptotic Methods for Wave and Quantum Problems

Author: M. V. Karasev

Publisher: American Mathematical Soc.

ISBN:

Category: Mathematics

Page: 284

View: 635

The collection consists of four papers in different areas of mathematical physics united by the intrinsic coherence of the asymptotic methods used. The papers describe both the known results and most recent achievements, as well as new concepts and ideas in mathematical analysis of quantum and wave problems. In the introductory paper ``Quantization and Intrinsic Dynamics'' a relationship between quantization of symplectic manifolds and nonlinear wave equations is described and discussed from the viewpoint of the weak asymptotics method (asymptotics in distributions) and the semiclassical approximation method. It also explains a hidden dynamic geometry that arises when using these methods. Three other papers discuss applications of asymptotic methods to the construction of wave-type solutions of nonlinear PDE's, to the theory of semiclassical approximation (in particular, the Whitham method) for nonlinear second-order ordinary differential equations, and to the study of the Schrodinger type equations whose potential wells are sufficiently shallow that the discrete spectrum contains precisely one point. All the papers contain detailed references and are oriented not only to specialists in asymptotic methods but also to a wider audience of researchers and graduate students working in partial differential equations and mathematical physics.

The Einstein Equations and the Large Scale Behavior of Gravitational Fields

50 Years Of The Cauchy Problem In General Relativity

Author: Piotr T. Chruściel

Publisher: Springer Science & Business Media

ISBN:

Category: Science

Page: 481

View: 670

Accompanying DVD-ROM contains the electronic proceedings of the summer school on mathematical general relativity and global properties of solutions of Einstein's equations held at Cargèse, Corsica, France, July 20-Aug. 10, 2002.

Solving Transcendental Equations

The Chebyshev Polynomial Proxy and Other Numerical Rootfinders, Perturbation Series, and Oracles

Author: John P. Boyd

Publisher: SIAM

ISBN:

Category: Mathematics

Page: 462

View: 261

Transcendental equations arise in every branch of science and engineering. While most of these equations are easy to solve, some are not, and that is where this book serves as the mathematical equivalent of a skydiver's reserve parachute--not always needed, but indispensible when it is. The author's goal is to teach the art of finding the root of a single algebraic equation or a pair of such equations.

Homogenization of Partial Differential Equations

Author: Vladimir A. Marchenko

Publisher: Springer Science & Business Media

ISBN:

Category: Mathematics

Page: 398

View: 109

A comprehensive study of homogenized problems, focusing on the construction of nonstandard models Details a method for modeling processes in microinhomogeneous media (radiophysics, filtration theory, rheology, elasticity theory, and other domains) Complete proofs of all main results, numerous examples Classroom text or comprehensive reference for graduate students, applied mathematicians, physicists, and engineers

Multiple Time Scale Dynamics

Author: Christian Kuehn

Publisher: Springer

ISBN:

Category: Mathematics

Page: 814

View: 132

This book provides an introduction to dynamical systems with multiple time scales. The approach it takes is to provide an overview of key areas, particularly topics that are less available in the introductory form. The broad range of topics included makes it accessible for students and researchers new to the field to gain a quick and thorough overview. The first of its kind, this book merges a wide variety of different mathematical techniques into a more unified framework. The book is highly illustrated with many examples and exercises and an extensive bibliography. The target audience of this book are senior undergraduates, graduate students as well as researchers interested in using the multiple time scale dynamics theory in nonlinear science, either from a theoretical or a mathematical modeling perspective.

Quantum-Statistical Models of Hot Dense Matter

Methods for Computation Opacity and Equation of State

Author: Arnold F. Nikiforov

Publisher: Springer Science & Business Media

ISBN:

Category: Science

Page: 428

View: 846

This book studies the widely used theoretical models for calculating properties of hot dense matter. Calculations are illustrated by plots and tables, and they are compared with experimental results. The purpose is to help understanding of atomic physics in hot plasma and to aid in developing efficient and robust computer codes for calculating opacity and equations of state for arbitrary material in a wide range of temperatures and densities.

A Geometric Approach to Thermomechanics of Dissipating Continua

Author: Lalao Rakotomanana

Publisher: Springer Science & Business Media

ISBN:

Category: Mathematics

Page: 265

View: 408

Across the centuries, the development and growth of mathematical concepts have been strongly stimulated by the needs of mechanics. Vector algebra was developed to describe the equilibrium of force systems and originated from Stevin's experiments (1548-1620). Vector analysis was then introduced to study velocity fields and force fields. Classical dynamics required the differential calculus developed by Newton (1687). Nevertheless, the concept of particle acceleration was the starting point for introducing a structured spacetime. Instantaneous velocity involved the set of particle positions in space. Vector algebra theory was not sufficient to compare the different velocities of a particle in the course of time. There was a need to (parallel) transport these velocities at a single point before any vector algebraic operation. The appropriate mathematical structure for this transport was the connection. I The Euclidean connection derived from the metric tensor of the referential body was the only connection used in mechanics for over two centuries. Then, major steps in the evolution of spacetime concepts were made by Einstein in 1905 (special relativity) and 1915 (general relativity) by using Riemannian connection. Slightly later, nonrelativistic spacetime which includes the main features of general relativity I It took about one and a half centuries for connection theory to be accepted as an independent theory in mathematics. Major steps for the connection concept are attributed to a series of findings: Riemann 1854, Christoffel 1869, Ricci 1888, Levi-Civita 1917, WeyJ 1918, Cartan 1923, Eshermann 1950.

Continuum Thermomechanics

Author: Alfredo Bermúdez de Castro

Publisher: Springer Science & Business Media

ISBN:

Category: Science

Page: 209

View: 254

The general goal of this book is to deduce rigorously, from the first principles, the partial differential equations governing the thermodynamic processes undergone by continuum media under forces and heat. Solids and fluids are considered in a unified framework. Reacting mixtures of fluids are also included for which general notions of thermodynamics are recalled, such as the Gibbs equilibrium theory. Linear approximate models are mathematically obtained by calculating the derivatives of the constitutive response functions. They include the classical models for linear vibrations of thermoelastic solids and also for wave propagation in fluids (dissipative and non-dissipative acoustics and internal gravity waves).

Historical Developments in Singular Perturbations

Author: Robert E. O'Malley

Publisher: Springer

ISBN:

Category: Mathematics

Page: 256

View: 470

This engaging text describes the development of singular perturbations, including its history, accumulating literature, and its current status. While the approach of the text is sophisticated, the literature is accessible to a broad audience. A particularly valuable bonus are the historical remarks. These remarks are found throughout the manuscript. They demonstrate the growth of mathematical thinking on this topic by engineers and mathematicians. The book focuses on detailing how the various methods are to be applied. These are illustrated by a number and variety of examples. Readers are expected to have a working knowledge of elementary ordinary differential equations, including some familiarity with power series techniques, and of some advanced calculus. Dr. O'Malley has written a number of books on singular perturbations. This book has developed from many of his works in the field of perturbation theory.

Partial Differential Equations II

Elements of the Modern Theory. Equations with Constant Coefficients

Author: Yu.V. Egorov

Publisher: Springer Science & Business Media

ISBN:

Category: Mathematics

Page: 266

View: 622

This book, the first printing of which was published as Volume 31 of the Encyclopaedia of Mathematical Sciences, contains a survey of the modern theory of general linear partial differential equations and a detailed review of equations with constant coefficients. Readers will be interested in an introduction to microlocal analysis and its applications including singular integral operators, pseudodifferential operators, Fourier integral operators and wavefronts, a survey of the most important results about the mixed problem for hyperbolic equations, a review of asymptotic methods including short wave asymptotics, the Maslov canonical operator and spectral asymptotics, a detailed description of the applications of distribution theory to partial differential equations with constant coefficients including numerous interesting special topics.

Controlled Nucleosynthesis

Breakthroughs in Experiment and Theory

Author: Stanislav Adamenko

Publisher: Springer Science & Business Media

ISBN:

Category: Science

Page: 782

View: 613

This book ushers in a new era of experimental and theoretical investigations into collective processes, structure formation, and self-organization of nuclear matter. It reports the results of experiments wherein for the first time the nuclei constituting our world (those displayed in Mendeleev's table as well as the super-heavy ones) have been artificially created. Pioneering breakthroughs are described, achieved at the "Proton-21" Laboratory, Kiev, Ukraine in a variety of new physical and technological directions.

MathPhys Odyssey 2001

Integrable Models and Beyond In Honor of Barry M. McCoy

Author: Masaki Kashiwara

Publisher: Springer Science & Business Media

ISBN:

Category: Mathematics

Page: 476

View: 729

'MathPhys Odyssey 2001' will serve as an excellent reference text for mathematical physicists and graduate students in a number of areas.; Kashiwara/Miwa have a good track record with both SV and Birkhauser.

Coherent Transform, Quantization and Poisson Geometry

Author: Mikhail Vladimirovich Karasev

Publisher: American Mathematical Soc.

ISBN:

Category: Mathematics

Page: 360

View: 206

This volume contains three extensive articles written by Karasev and his pupils. The topics covered include the following: coherent states and irreducible representations for algebras with non-Lie permutation relations, Hamilton dynamics and quantization over stable isotropic submanifolds, and infinitesimal tensor complexes over degenerate symplectic leaves in Poisson manifolds. The articles contain many examples (including from physics) and complete proofs.

The Kepler Problem

Group Theotretical Aspects, Regularization and Quantization, with Application to the Study of Perturbations

Author: Bruno Cordani

Publisher: Springer Science & Business Media

ISBN:

Category: Mathematics

Page: 439

View: 808

"The accompanying CD-ROM contains mainly the Microsoft Windows program KEPLER which calculates the effects of any perturbation of the Kepler problem and plots the resulting trajectories." -- p. [4] of cover.

Partial Differential Equations VII

Spectral Theory of Differential Operators

Author: M.A. Shubin

Publisher: Springer Science & Business Media

ISBN:

Category: Mathematics

Page: 274

View: 399

This EMS volume contains a survey of the principles and advanced techniques of the spectral theory of linear differential and pseudodifferential operators in finite-dimensional spaces. Also including a special section of Sunada's recent solution of Kac's celebrated problem of whether or not "one can hear the shape of a drum".