Nonlinear Partial Differential Equations for Scientists and Engineers......Page 1
Contents......Page 6
Overview......Page 11
Audience and Organization......Page 12
Salient Features......Page 14
Acknowledgements......Page 15
1.1 Introduction......Page 16
1.2 Basic Concepts and Definitions......Page 17
1.3 The Linear Superposition Principle......Page 19
1.4 Some Important Classical Linear Model Equations......Page 22
1.5 The Classification of Second-Order Linear Equations and The Method of Characteristics......Page 25
1.6 The Method of Separation of Variables......Page 35
1.7 Fourier Transforms and Initial-Boundary-Value Problems......Page 47
1.8 Applications of Multiple Fourier Transforms to Partial Differential Equations......Page 59
1.9 Laplace Transforms and Initial-Boundary-Value Problems......Page 64
1.10 Hankel Transforms and Initial-Boundary-Value Prohlems......Page 73
1.11 Green's Functions and Boundary-Value Problems......Page 82
1.12 Exercises......Page 93
2.2 Basic Concepts and Definitions......Page 107
2.3 Some Nonlinear Model Equations......Page 108
2.4 Variational Principles and the Euler-Lagrange Equations......Page 113
2.5 The Variational Principle for Nonlinear Klein-Gordon Equations......Page 118
2.6 The Variational Principle for Nonlinear Water Waves......Page 119
2.7 Exercises......Page 121
3.2 The Classification of First-Order Equations......Page 125
3.3 The Construction of a First-Order Equation......Page 126
3.4 The Geometrical Interpretation of a First-Order Equation......Page 130
3.5 The Method of Characteristics and General Solutions......Page 132
3.6 Exercises......Page 144
4.2 The Generalized Method of Characteristics......Page 148
4.3 Complete Integrals of Certain Special Nonlinear Equations......Page 152
4.4 Examples of Applications to Analytical Dynamics......Page 158
4.5 Applications to Nonlinear Optics......Page 164
4.6 Exercises......Page 169
5.2 Conservation Laws......Page 172
5.3 Discontinuous Solutions and Shock Waves......Page 185
5.4 Weak or Generalized Solutions......Page 187
5.5 Exercises......Page 194
6.2 Kinematic Waves......Page 197
6.3 Traffic Flow Problems......Page 201
6.4 Flood Waves in Long Rivers......Page 214
6.5 Chromatographic Models and Sediment Transport in Rivers......Page 216
6.6 Glacier Flow......Page 222
6.7 Roll Waves and Their Stability Analysis......Page 225
6.8 Simple Waves and Riemann's Invariants......Page 231
6.9 The Nonlinear Hyperbolic System and Riemann's Invariants......Page 251
6.10 Generalized Simple Waves and Generalized Riemanns Invariants......Page 262<br>6.11 Exercises......Page 266<br>7.2 Linear Dispersive Waves......Page 275<br>7.3 Initial-Value Problems and Asymptotic Solutions......Page 279<br>7.4 Nonlinear Dispersive Waves and Whithams Equations......Page 282
7.5 Whitham's Theory of Nonlinear Dispersive Waves......Page 285
7.6 Whitham's Averaged Variational Principle......Page 288
7.7 The Whitham Instability Analysis and Its Applications to Water Waves......Page 290
7.8 Exercises......Page 293
8.2 Burgers' Equation and the Plane Wave Solution......Page 295
8.3 Traveling Wave Solutions and Shock-Wave Structure......Page 298
8.4 The Cole-Hopf Transformation and the Exact Solution of the Burgers Equation......Page 301
8.5 The Asymptotic Behavior of the Exact Solution of the Burgers Equation......Page 306
8.6 The N-Wave Solution......Page 308
8.7 Burgers' Initial- and Boundary-Value Problem......Page 310
8.8 Fisher's Equation and Diffusion-Reaction Process......Page 313
8.9 Traveling Wave Solutions and Stability Analysis......Page 315
8.10 Perturbation Solutions of the Fisher Boundary-Value Problem......Page 319
8.11 Similarity Methods and Similarity Solutions of Diffusion Equations......Page 321
8.12 Nonlinear Reaction-Diffusion Equations......Page 331
8.13 A Brief Summary of Recent Work with References......Page 336
8.14 Exercises......Page 337
9.2 The History of the Soliton and Soliton Interactions......Page 342
9.3 The Boussinesq and Korteweg-de Vries Equations......Page 347
9.4 Solutions of the KdV Equation, Solitons and Cnoidal Waves......Page 358
9.5 The Lie Group Method and Similarity and Rational Solutions of the KdV Equation......Page 367
9.6 Conservation Laws and Nonlinear Transformations......Page 370
9.7 The Inverse Scattering Transform (IST) Method......Page 374
9.8 Bäcklund Transformations and the Nonlinear Superposition Principle......Page 397
9.9 The Lax Formulation, Its KdV Hierarchy, and the Zakharov and Shabat (ZS) Scheme......Page 402
9.10 The AKNS Method......Page 411
9.11 Exercises......Page 412
10.2 The One-Dimensional Linear Schrödinger Equation......Page 416
10.3 The Derivation of the Nonlinear Schrödinger (NLS) Equation and Solitary Waves......Page 418
10.4 Properties of the Solutions of the Nonlinear Schrödinger Equation......Page 423
10.5 Conservation Laws for the NLS Equation......Page 430
10.6 The Inverse Scattering Method for the Nonlinear Schrödinger Equation......Page 433
10.7 Examples of Physical Applications in Fluid Dynamics and Plasma Physics......Page 435
10.8 Applications to Nonlinear Optics......Page 449
10.9 Exercises......Page 460
11.2 The One-Dimensional Linear Klein-Gordon Equation......Page 463
11.3 The Two-Dimensional Linear Klein-Gordon Equation......Page 466
11.4 The Three-Dimensional Linear Klein-Gordon Equation......Page 468
11.5 The Nonlinear Klein-Gordon Equation and Averaging Techniques......Page 469
11.6 The Klcin-Gordon Equation and the Whitham Averaged Variational Principle......Page 477
11.7 The Sine-Gordon Equation, Soliton and Antisoliton Solutions......Page 480
11.8 The Solution of the Sine-Gordon Equation by Separation of Variables......Page 485
11.9 Backlund Transformations for the Sine-Gordon Equation......Page 494
11.10 The Solution of the Sine-Gordon Equation by the Inverse Scattering Method......Page 497
11.11 The Similarity Method for the Sine-Gordon Equation......Page 501
11.12 Nonlinear Optics and the Sine-Gordon Equation......Page 502
11.13 Exercises......Page 506
12.1 Introduction......Page 510
12.2 The Reductive Perturbation Method and Quasi-Linear Hyperbolic Systems......Page 511
12.3 Quasi-Linear Dissipative Systems......Page 515
12.4 Weakly Nonlinear Dispersive Systems and the Korteweg-de Vries Equation......Page 517
12.5 Strongly Nonlinear Dispersive Systems and the Nonlinear Schrödinger Equation......Page 530
12.6 The perturbation Method of Ostrovsky and Pelinosky......Page 536
12.7 The Method of Multiple Scales......Page 540
12.8 The Method of Multiple Scales for the Case of the Long-Wave Approximation......Page 546
1.12 Exercises......Page 549
2.7 Exercises......Page 554
3.6 Exercises......Page 555
4.6 Exercises......Page 558
5.5 Exercises......Page 559
6.11 Exercises......Page 560
8.14 Exercises......Page 564
11.13 Exercises......Page 565
Bibliography......Page 566
Index......Page 588