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?An exceptionally complete overview. There are numerous examples and the emphasis is on applications to almost all areas of science and engineering. There is truly something for everyone here. This reviewer feels that it is a very hard act to follow， and recommends it strongly. [This book] is a jewel.? ---Applied Mechanics Review (Review of First Edition) This expanded and revised second edition is a comprehensive and systematic treatment of linear and nonlinear partial differential equations and their varied applications. Building upon the successful material of the first book， this edition contains updated modern examples and applications from areas of fluid dynamics， gas dynamics， plasma physics， nonlinear dynamics， quantum mechanics， nonlinear optics， acoustics， and wave propagation. Methods and properties of solutions are presented， along with their physical significance， making the book more useful for a diverse readership. Nonlinear Partial Differential Equations for Scientists and Engineers， Second Edition is an exceptionally complete and accessible text/reference for graduate students， researchers， and professionals in mathematics， physics， and engineering. It may be used in graduate-level courses， as a self-study resource， or as a research reference.
Book Description
?An exceptionally complete overview of the latest developments in the field of PDEs. There are numerous examples and the emphasis is on applications to almost all areas of science and engineering. There is truly something for everyone here... The style is clear and to the point... This is a very useful book for all scientists dealing with nonlinear phenomena. This reviewer feels that it is a very hard act to follow， and recommends it strongly for individual students， practitioners， and libraries alike. The reviewer seldom gets so enthusiastic about the books he is asked to review， but [this book] is a jewel.? -- Applied Mechanics Review This book presents a comprehensive and systematic treatment of nonlinear partial differential equations and their varied applications. It contains methods and properties of solutions along with their physical significance. In an effort to make the book useful for a diverse readership， modern examples of applications and exercises are chosen from areas of fluid dynamics， gas dynamics， plasma physics， nonlinear dynamics， quantum mechanics， nonlinear optics， acoustics， and wave propagation. Topics and Features Thorough coverage of derivation and methods of solutions for all fundamental nonlinear model equations which include Korteweg-de Vries， Boussinesq， Fisher， nonlinear reaction-diffusion， Euler-Langrage， nonlinear Kline-Gordon， sine-Gordon， nonlinear Schrödinger， and Whitham equations Systematic presentation and explanation of conservation laws， weak solutions and shock waves Several nonlinear real-world models that include traffic flow， flood waves， chromatographic models， sediment transport in rivers， glacier flow， and roll waves Solitons and the Inverse Scattering Transform Special emphasis on nonlinear instability of dispersive waves with applications to water waves Over 450 worked examples and end-of-chapter exercises Nonlinear Partial Differential Equations for Scientists and Engineers is an exceptionally complete and accessible text/reference for graduates and professionals in mathematics， physics， science， and engineering. It is also suitable as a self-study/reference guide. Contents: 1. Linear Partial Differential Equations 1.1 Introduction 1.2 Basic Concepts and Definitions 1.3 Linear Superposition Principle 1.4 Some Important Classical Linear Model Equations 1.5 Classification of Second-Order Partial Differential Equations and the Method of Characteristics 1.6 The Method of Separation of Variables 1.7 Fourier Transforms and Initial Boundary-Value Problems 1.8 Applications of Multiple Fourier Transforms to Partial Differential Equations 1.9 Laplace Transforms and Initial Boundary-Value Problems 1.10 Hankel Transforms and Initial Boundary-Value Problems 1.11 Green’s Functions and Boundary-Value Problems 1.12 Exercises 2. Nonlinear Model Equations and Variational Principles 2.1 Introduction 2.2 Basic Concepts and Definitions 2.3 Some Nonlinear Model Equations 2.4 The Variational Principles and the Euler-Lagrange Equations 2.5 The Variational Principle for Nonlinear Klein-Gordon Equations 2.6 The Variational Principle for Nonlinear Water Waves 2.7 Exercises 3. First-Order Quasi-Linear Equations and The Method of Characteristics 3.1 Introduction 3.2 Classification of First-Order Equations 3.3 Construction of a First-Order Equation 3.4 Geometrical Interpretation of a First-Order Equation 3.5 The Method of Characteristics and General Solutions 3.6 Exercises 4. First-Order Nonlinear Equations and Their Applications 4.1 Introduction 4.2 The Generalized Method of Characteristics 4.3 Complete Integrals of Certain Special Nonlinear Equations 4.4 Examples of Applications to Analytical Dynamics 4.5 Applications to Nonlinear Optics 4.6 Exercises 5. Conservation Laws and Shock Waves 5.1 Introduction 5.2 Conservation Laws 5.3 Discontinuous Solutions and Shock Waves 5.4 Weak or Generalized Solutions 5.5 Exercises 6. Kinematic Waves and Specific Real-World Nonlinear Problems 6.1 Introduction 6.2 Kinematic Waves 6.3 Traffic Flow Problems 6.4 Flood Waves in Long Rivers 6.5 Chromatographic Models and Sediment Transport in Rivers 6.6 Glacier Flow 6.7 Roll Waves and Their Stability Analysis 6.8 Simple Waves and Riemann’s Invariants 6.9 The Nonlinear Hyperbolic System and Riemanns Invariants 6.10 Generalized Simple Waves and Generalized Riemanns Invariants 6.11 Exercises 7. Nonlinear Dispersive Waves and Whitman’s Equations 7.1 Introduction 7.2 Linear Dispersive Waves 7.3 Initial Value Problems and Asymptotic Solutions 7.4 Nonlinear Dispersive Waves and Whitham’s Equations 7.5 Whitham’s Theory of Nonlinear Dispersive Waves 7.6 Whitham’s Averaged Variational Principle 7.7 The Whitham Instability Analysis and Its Applications to Water Waves 7.8 Exercises 8. Nonlinear Diffusion-Reaction Phenomena， Burgers’ and Fisher’s Equations 8.1 Introduction 8.2 Burgers’ Equation and the Plane-Wave Solution 8.3 Traveling Wave Solutions and Shock-Wave Structure 8.4 The Cole-Hopf Transformation and the Exact Solution of the Burgers Equation 8.5 Asymptotic Behavior of the Exact Solution of the Burgers Equation 8.6 N Wave Solution 8.7 Burgers’ Initial and Boundary-Value Problems 8.8 Fisher’s Equation and Diffusion-Reaction Processes 8.9 Traveling-Wave Solutions and Stability Analysis 8.10 Perturbation Solutions of the Fisher Boundary-Value Problem 8.11 Similarity Methods and Similarity Solutions of Diffusion Equations 8.12 Nonlinear Reaction-Diffusion Equations 8.13 Brief Summary of Recent Work with References 8.14 Exercises 9. Solitons and the Inverse Scattering Transform 9.1 Introduction 9.2 History of the Soliton and Soliton Interactions 9.3 The Boussinesq and Korteweg-de Vries (KdV) Equations 9.4 Solutions of the KdV Equations， Solitons and Cnoidal Waves 9.5 The Lie Group Method and Similarity and Rational Solutions of the KdV Equation 9.6 Conservation Laws and Nonlinear Transformations 9.7 The Inverse Scattering Transform (IST) Method 9.8 Backlund Transformations and Nonlinear Superposition Principle 9.9 The Lax Formulation， Its KdV Hierarchy， and the Zakharov and Shabat (ZS) Scheme 9.10 The AKNS Method 9.11 Exercises 10. The Nonlinear Schrodinger Equation and Solitary Waves 10.1 Introduction 10.2 The One-Dimensional Linear Schrodinger Equation 10.3 The Derivation of the Nonlinear Schrodinger Equation and Solitary Waves 10.4 Properties of the Solutions of the Nonlinear Schrodinger Equation 10.5 Conservation Laws for the NLS Equation 10.6 The Inverse Scattering Method for the Nonlinear Schrodinger Equation 10.7 Examples of Physical Applications in Fluid Dynamics and Plasma Physics 10.8 Applications to Nonlinear Optics 10.9 Exercises 11. Nonlinear Klein-Gordon and Sine-Gordon Equations 11.1 Introduction 11.2 The One-Dimension Linear Klein-Gordon Equation 11.3 The Two-Dimensional Linear Klein-Gordon Equation 11.4 The Three-Dimensional Linear Klein-Gordon Equation 11.5 The Nonlinear Klein-Gordon Equation and Averaging Techniques 11.6 The Klein-Gordon Equation and the Whitham Averaged Variational Principle 11.7 The Sine-Gordon Equation， Soliton and Anti-Soliton Solutions 11.8 The Solution of the Sine-Gordon Equation by Separation of Variables 11.9 The Backlund Transformations for the Sine-Gordon Equation 11.10 The Solution of the Sine-Gordon Equation by the Inverse Scattering Method 11.11 The Similarity Method for the Sine-Gordon Equation 11.12 Nonlinear Optics and the Sine-Gordon Equation 11.13 Exercises 12. Asymptotic Methods and Nonlinear Evolution Equations 12.1 Introduction 12.2 The Reductive Perturbation Method and Quasilinear Hyperbolic Systems 12.3 Quasilinear Dissipative Systems 12.4 Weakly Nonlinear Dispersive Systems and the Korteweg-de Vries Equation 12.5 Strongly Nonlinear Dispersive Systems and the Nonlinear Schrodinger Equation 12.6 The Perturbation Method of Ostrovsky and Pelinosky 12.7 The Method of Multiple Scales 12.8 Method of Multiple Scales for the Case of the Long-Wave Approximation Answers and Hints to Selected Exercises Bibliography Index
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