Nonlinear Ocean Waves and the Inverse Scattering Transform (eBook)

Front Cover 1
Nonlinear Ocean Waves and the Inverse ScatteringTransform 1 ed. 4
Copyright Page 5
Title Page-II 6
Table of Contents 8
Preface 24
Part 1: Introduction: Nonlinear Waves 28
Chapter 1: Brief History and Overview of Nonlinear Water Waves 30
1.1. Linear and Nonlinear Fourier Analysis 30
1.2. The Nineteenth Century 33
1.2.1 Developments During the First Half of the Nineteenth Century 33
1.2.2 The Latter Half of the Nineteenth Century 35
1.3. The Twentieth Century 37
1.4. Physically Relevant Nonlinear Wave Equations 40
1.4.1 The Korteweg-deVries Equation 40
1.4.2 The Kadomtsev-Petviashvili Equation 42
1.4.3 The Nonlinear Schro¨ dinger Equation 44
1.4.4 Numerical Examples of Nonlinear Wave Dynamics 50
1.5. Laboratory and Oceanographic Applications of IST 51
1.5.1 Laboratory Investigations 53
1.5.2 Surface Waves in the Adriatic Sea 53
1.6. Hyperfast Numerical Modeling 54
Chapter 2: Nonlinear Water Wave Equations 60
2.1. Introduction 60
2.2. Linear Equations 61
2.3. The Euler Equations 62
2.4. Wave Motion in 2 + 1 Dimensions 63
2.4.1 The Zakharov Equation 63
2.4.2 The Davey-Stewartson Equations 64
2.4.3 The Davey-Stewartson Equations in Shallow Water 66
2.4.4 The Kadomtsev-Petviashvili Equation 66
2.4.5 The KP-Gardner Equation 67
2.4.6 The 2 + 1 Gardner Equation 67
2.4.7 The 2 + 1 Boussinesq Equation 67
2.5. Wave Motion in 1 + 1 Dimensions 67
2.5.1 The Zakharov Equation 67
2.5.2 The Nonlinear Schro¨ dinger Equation for Arbitrary Water Depth 68
2.5.3 The Deep-Water Nonlinear Schro¨ dinger Equation 70
2.5.4 The KdV Equation 70
2.5.5 The KdV Equation Plus Higher-Order Terms 70
2.6. Perspective in Terms of the Inverse Scattering Transform 72
2.7. Characterizing Nonlinearity 73
Chapter 3: The Infinite-Line Inverse Scattering Transform 76
3.1. Introduction 76
3.2. The Fourier Transform Solution to the Linearized KdV Equation 81
3.3. The Scattering Transform Solution to the KdV Equation 82
3.4. The Relationship Between the Fourier Transform and the Scattering Transform 85
3.5. Review of Assumptions Implicit in the Discrete, Finite Fourier Transform 88
3.6. Assumptions Leading to a Discrete Algorithm for the Direct Scattering Transform 91
Chapter 4: The Infinite-Line Hirota Method 96
4.1. Introduction 96
4.2. The Hirota Method 96
4.3. The Korteweg-deVries Equation 96
4.4. The Hirota Method for Solving the KP Equation 100
4.5. The Nonlinear Schrödinger Equation 101
4.6. The Modified KdV Equation 103
Part 2: Periodic Boundary Conditions 106
Chapter 5: Periodic Boundary Conditions 108
5.1. Introduction 108
5.2. Riemann Theta Functions as Ordinary Fourier Analysis 112
5.3. The Use of Generalized Fourier Series to Solve Nonlinear Wave Equations 114
5.3.1 Near-Shore, Shallow-Water Regions 114
5.3.2 Shallow- and Deep-Water Nonlinear Wave Dynamics for Narrow-Banded Wave Trains 116
5.4. Dynamical Applications of Theta Functions 117
5.5. Data Analysis and Data Assimilation 119
5.6. Hyperfast Modeling of Nonlinear Waves 120
Chapter 6: The Periodic Hirota Method 122
6.1. Introduction 122
6.2. The Hirota Method 122
6.3. The Burgers Equation 123
6.4. The Korteweg-de Vries Equation 125
6.5. The KP Equation 127
6.6. The Nonlinear Schrödinger Equation 131
6.7. The KdV-Burgers Equation 134
6.8. The Modified KdV Equation 135
6.9. The Boussinesq Equation 135
6.10. The 2 1 Boussinesq Equation 136
6.11. The 2 1 Gardner Equation 136
Part 3: Multidimensional Fourier Analysis 140
Chapter 7: Multidimensional Fourier Series 142
7.1. Introduction 142
7.2. Linear Fourier Series 142
7.3. Multidimensional or N-Dimensional Fourier Series 144
7.4. Conventional Multidimensional Fourier Series 145
7.5. Dynamical Multidimensional Fourier Series 147
7.6. Alternative Notations for Multidimensional Fourier Series 149
7.6.1 Baker’s Notation 150
7.6.2 Inverse Scattering Transform Notation 150
7.6.3 Relationship to Riemann Theta Functions 152
7.7. Simple Examples of Dynamical Multidimensional Fourier Series 153
7.8. General Rules for Dealing with Dynamical Multidimensional Fourier Series 156
7.9. Reductions of Multidimensional Fourier Series 157
7.10. Theta Functions Solve a Diffusion Equation 160
7.11. Multidimensional Fourier Series Solve Linear Wave Equations 162
7.12. Details for Two Degrees of Freedom 165
7.13. Converting Multidimensional Fourier Series to Ordinary Fourier Series 168
Chapter 8: Riemann Theta Functions 174
8.1. Introduction 174
8.2. Riemann Theta Functions 174
8.3. Simple Properties of Theta Functions 176
8.3.1 Symmetry of the Riemann Matrix 176
8.3.2 One-Dimensional Theta Functions: Connection to Classical Elliptic Functions 177
8.3.3 Multiple, Noninteracting Degrees of Freedom 178
8.3.4 A Theta Function Identity 179
8.3.5 Relationship of Generalized Fourier Series to Ordinary Fourier Series 181
8.3.6 Alternative Form for Theta Functions in Terms of Cosines 182
8.3.7 Partial Sums of Theta Functions 184
8.3.8 Examples of Simple Partial Theta Sums 187
8.4. Statistical Properties of Theta Function Parameters 191
8.5. Theta Functions as Ordinary Fourier Series 196
8.6. Perturbation Expansion of Theta Functions in Terms of an Interaction Parameter 200
8.7. N-Mode Interactions 202
8.8. Poisson Summation for Theta Functions 203
8.8.1 Gaussian Series for One-Degree-of-Freedom Theta Functions 203
8.8.2 The Infinite-Line Limit 207
8.8.3 Fourier and Gaussian Series for N-Dimensional Theta Functions 208
8.8.4 Gaussian Series for Theta Functions 209
8.8.5 One-Degree-of-Freedom Gaussian Series 209
8.8.6 Many-Degree-of-Freedom Gaussian Series 210
8.8.7 Comments on Numerical Analysis 212
8.8.8 Modular Transformations for Computing Theta Function Parameters 212
8.9. Solitons on the Infinite Line and on the Periodic Interval 215
8.10. N-Dimensional Theta Functions as a Sum of One-Degree-of-Freedom Thetas 216
8.11. N-Dimensional Partial Theta Sums over One-Degree-of-Freedom Theta Functions 218
Appendix I: Various Notations for Theta Functions 224
Exponential Forms 224
Cosine Forms 225
Appendix II: Partial Sums of Theta Functions 226
Exponential Forms 226
Cosine Forms 226
Appendix III: Fourier Series of Theta Functions at t = 0 227
Appendix IV: Fourier Series of Theta Functions at Time t 228
Chapter 9: Riemann Theta Functions as Ordinary Fourier Series 230
9.1. Introduction 230
9.2. Theoretical Considerations 232
9.3. A Numerical Example for the KdV Equation 235
Appendix: Theta Function Run with KP Program 242
Part 4: Nonlinear Shallow-Water Spectral Theory 244
Chapter 10: The Periodic Korteweg-DeVries Equation 246
10.1. Introduction 246
10.2. Linear Fourier Series Solution to the Linearized KdV Equation 246
10.3. The Hyperelliptic Function Solution to KdV 247
10.4. The theta-Function Solution to the KdV Equation 248
10.5. Special Cases of Solutions to the KdV Equation to Using theta-Functions 251
10.5.1 One Degree of Freedom 252
10.5.2 On the Possibility of Multiple, Noninteracting Cnoidal Waves 257
10.5.3 The Linear Fourier Limit 258
10.5.4 The Soliton and the N-Soliton Limits 259
10.5.5 Physical Selection of the Basis Cycles 259
10.6. Exact and Approximate Solutions to the KdV Equation for Specific Cases 260
10.6.1 A Single Cnoidal Wave 260
10.6.2 Multiple, Noninteracting Cnoidal Waves 262
10.6.3 Cnoidal Waves with Interactions 263
10.6.4 Approximate Solutions to KdV for Partial Theta Sums 266
10.6.5 Linear Limit of KdV Solutions 269
10.6.6 Approximate Solutions to KdV for Specific Cases 269
10.6.7 The Single Cnoidal Wave Solution to the KdV Equation 275
10.6.8 The Ursell Number 277
10.6.9 The Cnoidal Wave as a Classical Elliptic Function and Its Ursell Number 277
10.6.10 An Example Problem with 10 Degrees of Freedom 280
10.6.11 Relationship of Cnoidal Wave Parameters to the Parameter q 280
10.6.12 Wave Amplitudes and Heights for Each Degree of Freedom of KdV 282
Chapter 11: The Periodic Kadomtsev-Petviashvili Equation 288
11.1. Introduction 288
11.2. Overview of Periodic Inverse Scattering 289
11.3. Computation of the Spectral Parameters in Terms of Schottky Uniformization 291
11.3.1 Linear Fractional Transformation 292
11.3.2 Theta Function Spectrum as Poincareacute Series of Schottky Parameters 293
11.4. The Nakamura-Boyd Approach for Determining the Riemann Spectrum 294
Part 5: Nonlinear Deep-Water Spectral Theory 298
Chapter 12: The Periodic Nonlinear Schrödinger Equation 300
12.1. Introduction 300
12.2. The Nonlinear Schrödinger Equation 300
12.2.1 The “Time” NLS Equation and Its Relation to Physical Experiments 301
12.2.2 A Scaled Form of the NLS Equation 302
12.2.3 Small-Amplitude Modulations of the NLS Equation 302
12.3. Representation of the IST Spectrum in the Lambda Plane 303
12.4. Overview of Modulation Theory for the NLS Equation 305
12.5. Analytical Formulas for Unstable Wave Packets 312
12.6. Periodic Spectral Theory for the NLS Equation 315
12.6.1 The Lax Pair 315
12.6.2 The Spectra Eigenvalue Problem and Floquet Analysis 316
12.7. Overview of the Spectrum and Hyperelliptic Functions 320
12.7.1 The IST Spectrum 320
12.7.2 Generating Solutions to the NLS Equation 322
12.7.3 Applications to the Cauchy Problem: Space and Time Series Analysis 322
12.7.4 The Main Spectrum 323
12.7.5 The Auxiliary Spectrum of the muj(x, 0) 323
12.7.6 The Auxiliary Spectrum of the Riemann Sheet Indices sigmaj 324
12.7.7 The Auxiliary Spectrum of the gammaj(x, 0) 324
Appendix—Interpretation of the Hyperelliptic FunctionSuperposition Law 325
Chapter 13: The Hilbert Transform 328
13.1. Introduction 328
13.2. The Hilbert Transform 331
13.2.1 Properties of the Hilbert Transform 332
13.2.2 Numerical Procedure for Determining the Hilbert Transform 336
13.2.3 Table of Simple Hilbert Transforms 336
13.3. Narrow-Banded Processes 336
13.4. Statistical Properties of Complex Time Series 339
13.5. Relations Between the Surface Elevation and the Complex Envelope Function 342
13.6. Fourier Representation of the Free Surface Elevation and the Complex Envelope Function 347
13.6.1 Fourier Representations 349
13.7. Initial Modulations for Certain Special Solutions of the NLS Equation 355
Part 6: Theoretical Computation of the Riemann Spectrum 358
Chapter 14: Algebraic-Geometric Loop Integrals 360
14.1. Introduction 360
14.2. The Theta-Function Solutions to the KdV Equation 360
14.2.1 Holomorphic Differentials 361
14.2.2 Phases of the Theta Functions 366
14.2.3 The Period Matrix 367
14.2.4 One Degree of Freedom 368
14.2.5 Notation for Classical Jacobian Integrals 369
14.2.6 Notation for to the Theta-Function Formulation 369
14.3. On the Possibility of ``Interactionless´´ Potentials for the Two Degree-of-Freedom Case 373
14.4. Numerical Computation of the Riemann Spectrum 375
Appendix: Summary of Formulas for the Loop Integrals of the KdV Equation 376
Chapter 15: Schottky Uniformization 380
15.1. Introduction 380
15.2. IST Spectral Domain 380
15.3. Linear Oscillation Basis 381
15.3.1 An Overview of Schottky Uniformization in the Oscillation Basis 381
15.3.2 The Schottky Circles and Parameters 382
15.3.3 Linear Fractional Transformations 384
15.3.4 Poincare´ Series Relating the IST E-plane to the Schottky z-plane 386
15.3.5 Poincareacute Series for the Period Matrix 387
15.3.6 Poincareacute Series for the wavenumbers and Frequencies 388
15.3.7 How to Sum the Poincareacute Series 388
15.3.8 One Degree of Freedom 390
15.3.9 Two Degrees of Freedom 392
Appendix I:. Schottky Uniformization in the Small-Amplitude Limit of the Oscillation Basis 397
Appendix II:. Schottky Uniformization in the Large-Amplitude Limit of the Soliton Basis 398
Appendix III:. Poincaré Series from the Holomorphic Differentials 399
Appendix IV:. One Degree-of-Freedom Schottky z-Plane to IST E-Plane Poincaré Series 404
Chapter 16: Nakamura-Boyd Approach 410
16.1. Introduction 410
16.2. The Hirota Direct Method for the KdV Equation with Periodic Boundary Conditions 411
16.3. Theta Functions with Characteristics 414
16.4. Solution of the KdV Equation for the Theta Function with Characteristics 415
16.5. Determination of Theta-Function Parameters 417
16.6. Linearized Form for Riemann Spectrum for the KdV Equation 419
16.7. Strategy for Determining Solutions of Nonlinear Equations 419
16.8. One Degree-of-Freedom Riemann Spectrum and Solution of the KdV Equation 422
16.9. Two Degree of Freedom Riemann Spectrum and Solution of the KdV Equation 427
16.10. N Degree of Freedom Riemann Spectrum and Solution of the KdV Equation 430
16.10.1 Form Number 1 431
16.10.2 Form Number 2 432
16.10.3 Form Number 3 433
16.11. Numerical Algorithm for Solving Nonlinear Equations 434
16.12. Solving Systems of Two-Dimensional Nonlinear Equations 436
Appendix: Theta Functions with Characteristics 443
Part 7: Nonlinear Numerical and Time Series Analysis Algorithms 448
Chapter 17: Automatic Algorithm for the Spectral Eigenvalue Problem for the KdV Equation 450
17.1. Introduction 450
17.2. Formulation of the Problem 450
17.3. Periodic IST for the KdV Equation in the mu-Function Representation 453
17.4. The Spectral Structure of Periodic IST 456
17.5. A Numerical Discretization 459
17.5.1 Formulation 459
17.5.2 Implementation of the Numerical Algorithm 461
17.5.3 Reconstruction of Hyperelliptic Functions and Periodic Solutions to the KdV Equation 462
17.6. Automatic Numerical IST Algorithm 463
17.7. Example of the Analysis of a Many-Degree-of-Freedom Wave Train and Nonlinear Filtering 473
17.8. Summary and Conclusions 475
Chapter 18: The Spectral Eigenvalue Problem for the NLS Equation 478
18.1. Introduction 478
18.2. Numerical Algorithm 478
18.3. The NLS Spectrum 480
18.3.1 The Main Spectrum 480
18.3.2 The Auxiliary Spectrum of the muj(x,0) 481
18.3.3 The Auxiliary Spectrum of the Riemann Sheet Indices sigmaj 481
18.3.4 The Auxiliary Spectrum of the gammaj(x,0) 481
18.3.5 Spines in the Spectrum 482
18.4. Examples of Spectral Solutions of the NLS Equation 482
18.4.1 Plane Waves 482
18.4.2 Small Modulations 482
18.5. Summary 486
Chapter 19: Computation of Algebraic-Geometric Loop Integrals for the KdV Equation 488
19.1. Introduction 488
19.2. Convenient Transformations 488
19.2.1 First Transformation 489
19.2.2 Second Transformation 492
19.2.3 A Final Transformation 494
19.3. The Landen Transformation 495
19.4. Search for an AGM Method for the Loop Integrals 495
19.4.1 One Degree-of-Freedom Case 496
19.4.2 An Alternative Approach 498
19.4.3 Two Degree-of-Freedom Case 501
19.5. Improving Loop Integral Behavior 505
19.6. Constructing the Loop Integrals and Parameters of Periodic IST 512
Chapter 20: Simple, Brute-Force Computation of Theta Functions and Beyond 516
20.1. Introduction 516
20.2. Brute-Force Method 516
20.3. Vector Algorithm for the Theta Function 517
20.4. Theta Functions as Ordinary Fourier Series 519
20.5. A Memory-Bound Brute-Force Method 523
20.6. Poisson Series for Theta Functions 524
20.7. Decomposition of Space Series into Cnoidal Wave Modes 524
Chapter 21: The Discrete Riemann Theta Function 528
21.1. Introduction 528
21.2. Discrete Fourier Transform 528
21.3. The Multidimensional Fourier Transform 534
21.4. The Theta Function 535
21.5. The Discrete Theta Function 537
21.6. Determination of the Period Matrix and Phases from a Space/Time Series 542
21.7. General Procedure for Computing the Period Matrix and Phases from the Q's 548
21.8. Embedding the Discrete Theta Function 552
21.9. A Numerical Example for Extracting the Riemann Spectrum from the Q's 553
Chapter 22: Summing Riemann Theta Functions over the N-Ellipsoid 558
22.1. Introduction 558
22.2. Summing over the N-Sphere or Hypersphere 559
22.3. The Ellipse in Two Dimensions 564
22.4. Principal Axis Coordinates in Two Dimensions 564
22.5. Solving for the Coordinate m2 in Terms of m1 568
22.6. The Case for Three and N Degrees of Freedom 570
22.7. Summation Values for m1 573
22.8. Summary of Theta-Function Summation over Hyperellipsoid 576
22.9. Discussion of Convergence of Summation Method 579
22.10. Example Problem 580
Chapter 23: Determining the Riemann Spectrum from Data and Simulations 584
23.1. Introduction 584
23.2. Space Series Analysis 585
23.3. Time Series Analysis 587
23.4. Nonlinear Adiabatic Annealing 588
23.5. Outline of Nonlinear Adiabatic Annealing on a Riemann Surface 591
23.6. Establishing the Riemann Spectrum for the Cauchy Problem 595
23.7. Data Assimilation 596
Part 8: Theoretical and Experimental Problemsin Nonlinear Wave Physics 598
Chapter 24: Nonlinear Instability Analysis of Deep-Water Wave Trains 600
24.1. Introduction 600
24.2. Unstable Modes and Their IST Spectra 601
24.3. Properties of Unstable Modes 605
24.4. Formulas for Unstable Modes and Breathers 610
24.5. Examples of Unstable Mode (Rogue Wave) Solutions of NLS 613
24.6. Summary and Discussion 617
Appendix. Overview of Periodic Theory for the NLS Equation with Theta Functions 618
Chapter 25: Internal Waves and Solitons 624
25.1. Introduction 624
25.2. The Andaman Sea Measurements 628
25.3. The Theory of the KdV Equation as a Simple Nonlinear Model for Long Internal Wave Motions 631
25.4. Background on KdV Theory and Solitons 637
25.5. Nonlinear Fourier Analysis of Soliton Wave Trains 640
25.6. Nonlinear Spectral Analysis of Andaman Sea Data 641
25.7. Extending the KdV Model to Higher Order 647
Chapter 26: Underwater Acoustic Wave Propagation 650
26.1. Introduction 650
26.2. The Parabolic Equation 652
26.3. Solving the Parabolic Equation with Fourier Series 654
26.4. Solving the Parabolic Equation Analytically 657
26.5. The Functions F(r,z) and G(r,z) as Ordinary Fourier Series Solution of the PE in Terms of Matrix Equation
26.6. Solving the Parabolic Equation in Terms of Multidimensional Fourier Series 665
26.7. Rewriting the Theta Functions in Alternative Forms 667
26.8. Applying Boundary Conditions to the Theta Functions 674
26.9. One Degree-of-Freedom Case 683
26.10. Linear Limit of the Theta-Function Formulation 684
26.11. Implementation of Multidimensional Fourier Methods in Acoustics 686
26.12. Physical Interpretation of the Exact Solution of the PE 690
26.13. Solving the PE for a Given Source Function 691
26.14. Range-Independent Problem 694
26.15. Determination of the Environment from Measurements 695
26.16. Coherent Modes in the Acoustic Field 698
26.17. Shadow Zone Analysis 701
26.18. Application to Unmanned, Untethered, Submersible Vehicles 705
26.19. Application to Communications, Imaging, and Encryption 706
Appendix: Products of Fourier Series 709
Chapter 27: Planar Vortex Dynamics 712
27.1. Introduction 712
27.2. Derivation of the Poisson Equation for Vortex Dynamics in the Plane 715
27.3. Poisson Equation for Schrödinger Dynamics in the Plane 718
27.4. Specific Cases of the Poisson Equation for Vortex Dynamics in the Plane 718
27.5. Geophysical Fluid Dynamics 719
27.5.1 Linearization of the Potential Vorticity Equation 720
27.5.2 The KdV Equation as Derived from the Potential Vorticity Equation 721
27.6. The Poisson Equation for the Davey-Stewartson Equations 723
27.7. Nonlinear Separation of Variables for the Schrödinger Equation 723
27.8. Vortex Solutions of the sinh-Poisson Equation Using Soliton Methods 725
27.9. Vortex and Wave Solutions of the sinh-Poisson Equation Using Algebraic Geometry 728
Chapter 28: Nonlinear Fourier Analysis and Filtering of Ocean Waves 740
28.1. Introduction 740
28.2. Preliminary Considerations 741
28.3. Sine Waves and Linear Fourier Analysis 744
28.4. Cnoidal Waves and Nonlinear Fourier Analysis 745
28.5. Theoretical Background for Data Analysis Procedures 747
28.5.1 Cnoidal Wave Decomposition Theorem for theta-Functions 747
28.5.2 Nonlinear Filtering with theta-Functions 749
28.6. Physical Considerations and Applicability of the Nonlinear Fourier Approach 752
28.6.1 Properties of the Nonlinear Fourier Approach 752
28.6.2 Preliminary Tests of the Time Series 753
28.6.3 The Use of Periodic Boundary Conditions 754
28.7. Nonlinear Fourier Analysis of the Data 755
28.7.1 Applicability of the Nonlinear Fourier Approach 756
28.7.2 Analysis of the Data 757
28.7.3 Nonlinear Filtering 761
28.8. Summary and Discussion 770
Chapter 29: Laboratory Experiments of Rogue Waves 772
29.1. Introduction 772
29.2. Linear Fourier Analysis and the Nonlinear Schrödinger Equation 774
29.3. Nonlinear Fourier Analysis for the Nonlinear Schrödinger Equation 776
29.4. Marintek Wave Tank 777
29.5. Deterministic Wave Trains as Time Series 778
29.6. Random Wave Trains 790
29.6.1 Characteristics of Random Wave Trains Using IST for NLS 790
29.6.2 Measured Random Wave Trains 794
29.6.3 Nonlinear Spectral Analysis of the Random Wave Trains 794
29.7. Summary and Discussion 803
Chapter 30: Nonlinearity in Duck Pier Data 806
30.1. Introduction 806
30.2. The Ursell Number 807
30.2.1 Cnoidal Waves and the Spectral Ursell Number 808
30.3. Estimates of the Ursell Number from Duck Pier Data 812
30.4. Analysis of Duck Pier Data 814
Chapter 31: Harmonic Generation in Shallow-Water Waves 822
31.1. Introduction 822
31.2. Nonlinear Fourier Analysis 823
31.3. Nonlinear Spectral Decomposition 823
31.4. Harmonic Generation in Shallow Water 824
31.5. Periodic Inverse Scattering Theory 825
31.6. Classical Harmonic Generation and FPU Recurrence in a Simple Model Simulation 825
31.7. Search for Harmonic Generation in Laboratory Data 836
31.8. Summary and Discussion 842
Part 9: Nonlinear Hyperfast Numerical Modeling 846
Chapter 32: Hyperfast Modeling of Shallow-Water Waves: The KdV and KP Equations 848
32.1. Introduction 848
32.2. Overview of the Literature 849
32.3. The Inverse Scattering Transform for Periodic Boundary Conditions 851
32.3.1 The KdV Equation 852
32.3.2 The Kadomtsev-Petviashvili Equation 854
32.4. Properties of Riemann Theta Functions and Partial Theta Summations 858
32.4.1 The KdV Equation 858
32.4.2 The KP Equation 861
32.5. Computation of the Spectral Parameters in Terms of Schottky Uniformization 864
32.5.1 Linear Fractional Transformation 865
32.5.2 Theta Function Spectrum as Poincareacute Series of Schottky Parameters 866
32.6. Leading Order Computation of KP Spectra Using Schottky Variables 868
32.7. The Method of Nakamura and Boyd 870
32.8. The Exact Solution of the Time Evolution of the Fourier Components for the KP Equation 872
32.9. Numerical Procedures for Computing the Riemann Spectrum from Poincaré Series 874
32.10. Numerical Procedures for Computing the Riemann Theta Function 875
32.11. Numerical Procedures for Computing Hyperfast Solutions of the KP Equation 876
32.12. Numerical Example for KP Evolution 877
Chapter 33: Modeling the 2 + 1 Gardner Equation 884
33.1. Introduction 884
33.2. The 2 + 1 Gardner Equation and Its Properties 884
33.3. The Lax Pair and Hirota Bilinear Form 886
33.4. The Extended KP Equation in Physical Units 889
33.5. Physical Behavior of the Extended KP Equation 890
Chapter 34: Modeling the Davey-Stewartson (DS) Equations 894
34.1. Introduction 894
34.2. The Physical Form of the Davey-Stewartson Equations 894
34.3. The Normalized Form of the Davey-Stewartson Equations 897
34.4. The Hirota Bilinear Forms 899
34.4.1 Davey-Stewartson I—Surface Tension Dominates 900
34.4.2 Davey-Stewartson II—Oceanic Waves in Shallow Water with Negligible Surface Tension 900
34.5. Numerical Examples 901
References 904
International Geophysics Series 924
Index 930
Color Plates 946