Partial differential equations and solitary waves theory

Cover......Page 1
NONLINEAR PHYSICAL SCIENCE......Page 3
Partial Differential Equations and Solitary Waves Theory......Page 4
3642002501......Page 5
Preface......Page 8
Contents......Page 11
Part I - Partial Differential Equations......Page 20
1.1 Introduction......Page 22
1.2.1 Definition of a PDE......Page 23
1.2.2 Order of a PDE......Page 24
1.2.3 Linear and Nonlinear PDEs......Page 25
1.2.5 Some Nonlinear Partial Differential Equations......Page 26
1.2.7 Solution of a PDE......Page 28
1.2.8 Boundary Conditions......Page 30
1.2.10 Well-posed PDEs......Page 31
1.3 Classifications of a Second-order PDE......Page 33
References......Page 36
2.2 Adomian Decomposition Method......Page 38
2.3 The Noise Terms Phenomenon......Page 55
2.4 The Modified Decomposition Method......Page 60
2.5 The Variational Iteration Method......Page 66
2.6 Method of Characteristics......Page 73
2.7 Systems of Linear PDEs by Adomian Method......Page 78
2.8 Systems of Linear PDEs by Variational Iteration Method......Page 82
References......Page 87
3.1 Introduction......Page 88
3.2 The Adomian Decomposition Method......Page 89
3.2.1 Homogeneous Heat Equations......Page 92
3.2.2 Inhomogeneous Heat Equations......Page 99
3.3 The Variational Iteration Method......Page 102
3.3.1 Homogeneous Heat Equations......Page 103
3.3.2 Inhomogeneous Heat Equations......Page 106
3.4.1 Analysis of the Method......Page 108
3.4.2 Inhomogeneous Boundary Conditions......Page 118
3.4.3 Equations with Lateral Heat Loss......Page 121
References......Page 125
4.1 Introduction......Page 126
4.2.1 Two Dimensional Heat Flow......Page 127
4.2.2 Three Dimensional Heat Flow......Page 135
4.3.1 Two Dimensional Heat Flow......Page 143
4.3.2 Three Dimensional Heat Flow......Page 153
References......Page 159
5.1 Introduction......Page 162
5.2 Adomian Decomposition Method......Page 163
5.2.1 Homogeneous Wave Equations......Page 165
5.2.2 Inhomogeneous Wave Equations......Page 171
5.2.3 Wave Equation in an Infinite Domain......Page 176
5.3.1 Homogeneous Wave Equations......Page 181
5.3.2 Inhomogeneous Wave Equations......Page 187
5.3.3 Wave Equation in an Infinite Domain......Page 189
5.4.1 Analysis of the Method......Page 193
5.4.2 Inhomogeneous Boundary Conditions......Page 203
5.5 Wave Equation in an Infinite Domain: D’Alembert Solution......Page 209
References......Page 213
6.2 Adomian Decomposition Method......Page 214
6.2.1 Two Dimensional Wave Equation......Page 215
6.2.2 Three Dimensional Wave Equation......Page 229
6.3 Method of Separation of Variables......Page 239
6.3.1 Two Dimensional Wave Equation......Page 240
6.3.2 Three Dimensional Wave Equation......Page 250
References......Page 255
7.1 Introduction......Page 256
7.2.1 Two Dimensional Laplace’s Equation......Page 257
7.3 The Variational Iteration Method......Page 266
7.4.1 Laplace’s Equation in Two Dimensions......Page 270
7.4.2 Laplace’s Equation in Three Dimensions......Page 278
7.5 Laplace’s Equation in Polar Coordinates......Page 286
7.5.1 Laplace’s Equation for a Disc......Page 287
7.5.2 Laplace’s Equation for an Annulus......Page 294
References......Page 302
8.1 Introduction......Page 304
8.2 Adomian Decomposition Method......Page 306
8.2.1 Calculation of Adomian Polynomials......Page 307
8.2.2 Alternative Algorithm for Calculating Adomian Polynomials......Page 311
8.3 Nonlinear ODEs by Adomian Method......Page 320
8.4 Nonlinear ODEs by VIM......Page 331
8.5 Nonlinear PDEs by Adomian Method......Page 338
8.6 Nonlinear PDEs by VIM......Page 353
8.7 Nonlinear PDEs Systems by Adomian Method......Page 360
8.8 Systems of Nonlinear PDEs by VIM......Page 366
References......Page 370
9.1 Introduction......Page 372
9.2 The Nonlinear Advection Problem......Page 373
9.3 The Goursat Problem......Page 379
9.4 The Klein-Gordon Equation......Page 389
9.4.1 Linear Klein-Gordon Equation......Page 390
9.4.2 Nonlinear Klein-Gordon Equation......Page 394
9.4.3 The Sine-Gordon Equation......Page 397
9.5 The Burgers Equation......Page 400
9.6 The Telegraph Equation......Page 407
9.7.1 The Linear Schrodinger Equation......Page 413
9.7.2 The Nonlinear Schrodinger Equation......Page 416
9.8 Korteweg-de Vries Equation......Page 420
9.9.1 Equations with Constant Coefficients......Page 424
9.9.2 Equations with Variable Coefficients......Page 427
References......Page 432
10.1 Introduction......Page 434
10.2.1 Perturbation Problems......Page 435
10.2.2 Nonperturbed Problems......Page 440
10.3 Partial Differential Equations......Page 446
10.4 The Padé Approximants......Page 449
10.5 Padé Approximants and Boundary Value Problems......Page 458
References......Page 474
11.1 Introduction......Page 476
11.2 Solitons......Page 478
11.2.1 The KdV Equation......Page 479
11.2.2 The Modified KdV Equation......Page 481
11.2.4 The Sine-Gordon Equation......Page 483
11.2.5 The Boussinesq Equation......Page 484
11.2.6 The Kadomtsev-Petviashvili Equation......Page 486
11.3 Compactons......Page 488
11.4 The Defocusing Branch of K(n,n)......Page 493
References......Page 494
Part II - Solitray Waves Theory......Page 496
12.1 Introduction......Page 498
12.2 Definitions......Page 499
12.2.1 Dispersion and Dissipation......Page 501
12.2.2 Types of Travelling Wave Solutions......Page 503
12.2.3 Nonanalytic Solitary Wave Solutions......Page 509
12.3.1 The Tanh-coth Method......Page 510
12.3.2 The Sine-cosine Method......Page 512
12.3.3 Hirota’s Bilinear Method......Page 513
12.4 Conservation Laws......Page 515
References......Page 521
13.1 Introduction......Page 522
13.2.1 Third-order KdV Equations......Page 524
13.3 The KdV Equation......Page 526
13.3.1 Using the Tanh-coth Method......Page 527
13.3.3 Multiple-soliton Solutions of the KdV Equation......Page 529
13.4 The Modified KdV Equation......Page 537
13.4.1 Using the Tanh-coth Method......Page 538
13.4.2 Using the Sine-cosine Method......Page 539
13.4.3 Multiple-soliton Solutions of the mKdV Equation......Page 540
13.5 Singular Soliton Solutions......Page 542
13.6.1 Using the Tanh-coth Method......Page 545
13.7 The Potential KdV Equation......Page 547
13.7.1 Using the Tanh-coth Method......Page 548
13.7.2 Multiple-soliton Solutions of the Potential KdV Equation......Page 550
13.8.1 The Kink Solution......Page 552
13.8.2 The Soliton Solution......Page 553
13.8.3 N-soliton Solutions of the Positive Gardner Equation......Page 554
13.8.4 Singular Soliton Solutions......Page 556
13.9 Generalized KdV Equation with Two Power Nonlinearities......Page 561
13.9.1 Using the Tanh Method......Page 562
13.10 Compactons: Solitons with Compact Support......Page 563
13.10.1 The K(n,n) Equation......Page 565
13.11 Variants of the K(n,n) Equation......Page 566
13.11.1 First Variant......Page 567
13.11.2 Second Variant......Page 568
13.11.3 Third Variant......Page 570
13.12.1 The Modified KdV Equation......Page 572
13.12.3 The Modified Equal Width Equation......Page 573
References......Page 574
14.1 Introduction......Page 576
14.2.1 Fifth-order KdV Equations......Page 577
14.2.2 Seventh-order KdV Equations......Page 580
14.3 Fifth-order KdV Equations......Page 581
14.3.1 Using the Tanh-coth Method......Page 582
14.3.2 The First Condition......Page 583
14.3.3 The Second Condition......Page 585
14.3.4 N-soliton Solutions of the Fifth-order KdV Equations......Page 586
14.4.1 Using the Tanh-coth Method......Page 595
14.4.2 N-soliton Solutions of the Seventh-order KdV Equations......Page 597
14.5 Ninth-order KdV Equations......Page 601
14.5.1 Using the Tanh-coth Method......Page 602
14.5.2 The Soliton Solutions......Page 603
14.6 Family of Higher-order mKdV Equations......Page 604
14.6.1 N-soliton Solutions for Fifth-order mKdV Equation......Page 605
14.6.2 Singular Soliton Solutions for Fifth-order mKdV Equation......Page 606
14.6.3 N-soliton Solutions for the Seventh-order mKdV Equation......Page 608
14.7 Complex Solution for the Seventh-order mKdV Equations......Page 610
14.8 The Hirota-Satsuma Equations......Page 611
14.8.1 Using the Tanh-coth Method......Page 612
14.8.2 N-soliton Solutions of the Hirota-Satsuma System......Page 613
14.8.3 N-soliton Solutions by an Alternative Method......Page 615
14.9 Generalized Short Wave Equation......Page 616
References......Page 621
15.1 Introduction......Page 624
15.2 The Complex Modified KdV Equation......Page 625
15.2.1 Using the Sine-cosine Method......Page 626
15.2.2 Using the Tanh-coth Method......Page 627
15.3.1 Using the Sine-cosine Method......Page 631
15.3.2 Using the Tanh-coth Method......Page 632
15.4.1 Using the Sine-cosine Method......Page 634
15.4.2 Using the Tanh-coth Method......Page 635
15.5 The Kawahara and the Modified Kawahara Equations......Page 636
15.5.1 The Kawahara Equation......Page 637
15.5.2 The Modified Kawahara Equation......Page 638
15.6 The Kadomtsev-Petviashvili (KP) Equation......Page 639
15.6.1 Using the Tanh-coth Method......Page 640
15.6.2 Multiple-soliton Solutions of the KP Equation......Page 641
15.7 The Zakharov-Kuznetsov (ZK) Equation......Page 645
15.8 The Benjamin-Ono Equation......Page 648
15.9 The KdV-Burgers Equation......Page 649
15.10.1 The Sech Method......Page 651
15.11.1 The Sech Method......Page 653
References......Page 656
16.1 Introduction......Page 658
16.2.1 Using the Tanh-coth Method......Page 660
16.2.2 Multiple-soliton Solutions of the Boussinesq Equation......Page 662
16.3 The Improved Boussinesq Equation......Page 665
16.4 The Klein-Gordon Equation......Page 667
16.5 The Liouville Equation......Page 668
16.6.1 Using the Tanh-coth Method......Page 670
16.6.2 Using the Bäcklund Transformation......Page 673
16.6.3 Multiple-soliton Solutions for Sine-Gordon Equation......Page 674
16.7 The Sinh-Gordon Equation......Page 676
16.8 The Dodd-Bullough-Mikhailov Equation......Page 677
16.9 The Tzitzeica-Dodd-Bullough Equation......Page 678
16.10 The Zhiber-Shabat Equation......Page 680
References......Page 681
17.1 Introduction......Page 684
17.2 The Burgers Equation......Page 685
17.2.1 Using the Tanh-coth Method......Page 686
17.2.2 Using the Cole-Hopf Transformation......Page 687
17.3 The Fisher Equation......Page 689
17.4 The Huxley Equation......Page 690
17.6 The Burgers-Huxley Equation......Page 692
17.7 The FitzHugh-Nagumo Equation......Page 694
17.8 Parabolic Equation with Exponential Nonlinearity......Page 695
17.9 The Coupled Burgers Equation......Page 697
17.10 The Kuramoto-Sivashinsky (KS) Equation......Page 699
References......Page 700
18.1 Introduction......Page 702
18.2.1 Using the Tanh-coth Method......Page 705
18.2.2 Using an Exponential Algorithm......Page 707
18.3 Schrodinger Equation of Cubic Nonlinearity......Page 708
18.4 Schrodinger Equation with Power Law Nonlinearity......Page 709
18.5 The Ginzburg-Landau Equation......Page 711
18.5.1 The Cubic Ginzburg-Landau Equation......Page 712
18.5.2 The Generalized Cubic Ginzburg-Landau Equation......Page 713
18.5.3 The Generalized Quintic Ginzburg-Landau Equation......Page 714
References......Page 715
A.1 Fundamental Forms......Page 718
A.3 Inverse Trigonometric Forms......Page 719
A.5 Hyperbolic Forms......Page 720
A.6 Other Forms......Page 721
B.2 Trigonometric Functions......Page 722
B.5 Inverse Hyperbolic Functions......Page 723
C Exact Solutions of Burgers’ Equation......Page 724
D.2 Trigonometric Functions......Page 726
D.4 Logarithmic Functions......Page 728
E.2 The Gamma function Γ(x)......Page 730
F.1 Numerical Series......Page 731
F.2 Trigonometric Series......Page 732
Answers......Page 734
Index......Page 758