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ix | |
| Preface |
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xxi | |
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1 Derivation of Nonlinear Wave Equations |
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1 | (14) |
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1.1 The Nonlinear Schrodinger Equation for Weakly Nonlinear Wave Packets |
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1 | (4) |
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1.2 The Generalized Nonlinear Schrodinger Equation for Light Beam Propagation |
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5 | (4) |
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1.3 Other Nonlinear Wave Equations in Physical Systems |
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9 | (6) |
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2 Integrable Theory for the Nonlinear Schrodinger Equation |
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15 | (64) |
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2.1 Inverse Scattering Transform Method |
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16 | (16) |
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2.1.1 Riemann-Hilbert Formulation |
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17 | (6) |
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2.1.2 Solution of the Riemann-Hilbert Problem |
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23 | (6) |
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2.1.3 Time Evolution of Scattering Data |
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29 | (2) |
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2.1.4 Long-Time Behavior of the Solution |
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31 | (1) |
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32 | (5) |
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2.3 Infinite Number of Conservation Laws |
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37 | (1) |
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2.4 Discrete Eigenvalues in the Zakharov-Shabat System |
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38 | (12) |
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2.4.1 Eigenvalue Formulae for Special Initial Conditions |
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39 | (4) |
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2.4.2 General Criteria for Discrete Eigenvalues |
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43 | (2) |
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2.4.3 Numerical Computations of Eigenvalues |
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45 | (5) |
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2.5 Closure of Zakharov-Shabat Eigenstates |
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50 | (5) |
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2.6 Squared Eigenfunctions of the Zakharov-Shabat System |
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55 | (10) |
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2.6.1 Variations of Scattering Data via Variation of Potential |
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56 | (3) |
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2.6.2 Variation of Potential via Variations of Scattering Data |
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59 | (3) |
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2.6.3 Inner Products and Closure Relation of Squared Eigenfunctions |
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62 | (1) |
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2.6.4 Extension to the General Case |
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63 | (2) |
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2.7 Squared Eigenfunctions and the Linearization Operator |
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65 | (3) |
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2.8 Recursion Operator and the AKNS Hierarchy |
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68 | (3) |
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2.9 Squared Eigenfunctions and the Recursion Operator |
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71 | (2) |
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2.10 Amplitude-Changing and Self-Collapsing Solitons in the AKNS Hierarchy |
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73 | (6) |
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3 Theories for Integrable Equations with Higher-Order Scattering Operators |
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79 | (40) |
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3.1 Integrable Hierarchy for a Higher-Order Scattering Operator |
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80 | (2) |
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3.2 Various Reductions of the Hierarchy |
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82 | (2) |
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3.3 Riemann-Hilbert Problem for the Hierarchy |
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84 | (5) |
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3.4 Time Evolution of Scattering Data |
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89 | (1) |
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90 | (1) |
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3.6 Infinite Number of Conservation Laws |
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91 | (2) |
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3.7 Closure of Eigenstates in the Higher-Order Scattering Operator |
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93 | (2) |
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3.8 Squared Eigenfunctions of the Higher-Order Scattering Operator |
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95 | (10) |
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3.8.1 Squared Eigenfunctions for Generic Potentials |
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96 | (6) |
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3.8.2 Squared Eigenfunctions under Potential Reductions |
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102 | (3) |
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3.9 Squared Eigenfunctions, the Linearization Operator, and the Recursion Operator |
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105 | (4) |
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3.10 Solutions in the Manakov System |
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109 | (3) |
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3.11 Solutions in a Coupled Focusing-Defocusing NLS System |
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112 | (2) |
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3.12 Solutions in the Sasa-Satsuma Equation |
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114 | (5) |
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4 Soliton Perturbation Theories and Applications |
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119 | (44) |
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4.1 Direct Soliton Perturbation Theory for the NLS Equation |
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120 | (13) |
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4.1.1 Eigenfunctions and Adjoint Eigenfunctions of the Linearization Operator |
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122 | (5) |
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4.1.2 Solution for the Perturbed Soliton |
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127 | (3) |
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4.1.3 Evolution of a Perturbed Soliton in the NLS Equation |
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130 | (3) |
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4.2 Higher-Order Effects on Optical Solitons |
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133 | (8) |
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135 | (4) |
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4.2.2 Self-Steepening Effect |
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139 | (1) |
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4.2.3 Third-Order Dispersion Effect |
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140 | (1) |
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4.3 Weak Interactions of NLS Solitons |
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141 | (9) |
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4.4 Soliton Perturbation Theory for the Complex Modified KdV Equation |
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150 | (13) |
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5 Theories for Nonintegrable Equations |
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163 | (106) |
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5.1 Solitary Waves in Nonintegrable Equations |
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163 | (2) |
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5.2 Linearization Spectrum of Solitary Waves |
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165 | (3) |
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5.3 Vakhitov-Kolokolov Stability Criterion and Its Generalization |
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168 | (12) |
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5.3.1 Vakhitov-Kolokolov Stability Criterion |
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169 | (6) |
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5.3.2 Generalization of Vakhitov-Kolokolov Criterion |
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175 | (5) |
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5.4 Stability Switching at a Power Extremum |
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180 | (3) |
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5.5 Nonlocal Waves and the Exponential Asymptotics Technique |
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183 | (10) |
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5.6 Embedded Solitons and Their Dynamics |
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193 | (17) |
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5.6.1 Isolated Embedded Solitons and Their Semistability |
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194 | (12) |
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5.6.2 Continuous Families of Embedded Solitons |
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206 | (4) |
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5.7 Fractal Scattering in Collisions of Solitary Waves |
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210 | (15) |
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5.7.1 PDE Simulation Results for Coupled NLS Equations |
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212 | (2) |
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5.7.2 A Reduced ODE Model and Its Analysis |
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214 | (11) |
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5.8 Fractal Scattering in Weak Interactions of Solitary Waves |
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225 | (31) |
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5.8.1 PDE Simulation Results for Generalized NLS Equations |
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227 | (3) |
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5.8.2 An Asymptotic ODE Model |
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230 | (7) |
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5.8.3 A Universal Separatrix Map |
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237 | (12) |
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5.8.4 Fractal in the Separatrix Map |
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249 | (5) |
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5.8.5 Physical Mechanism for Fractal Scatterings in Weak Interactions |
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254 | (2) |
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5.9 Transverse Instability of Solitary Waves |
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256 | (9) |
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5.9.1 Instability of Long Transverse Waves |
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258 | (2) |
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5.9.2 Instability of Short Transverse Waves |
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260 | (4) |
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5.9.3 Experimental Demonstrations of Transverse Instabilities |
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264 | (1) |
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5.10 Wave Collapse in the Two-Dimensional NLS Equation |
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265 | (4) |
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6 Nonlinear Wave Phenomena in Periodic Media |
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269 | (58) |
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6.1 One-Dimensional Gap Solitons Bifurcated from Bloch Bands and Their Stability |
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271 | (12) |
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6.1.1 Bloch Bands and Bandgaps |
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271 | (1) |
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6.1.2 Envelope Equations of Bloch Waves |
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272 | (3) |
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6.1.3 Locations of Envelope Solutions |
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275 | (2) |
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6.1.4 Families of Gap Solitons Bifurcated from Band Edges |
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277 | (2) |
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6.1.5 Stability of Gap Solitons Bifurcated from Band Edges |
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279 | (4) |
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6.2 One-Dimensional Gap Solitons Not Bifurcated from Bloch Bands |
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283 | (6) |
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6.3 Two-Dimensional Gap Solitons Bifurcated from Bloch Bands |
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289 | (8) |
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6.3.1 2D Bloch Bands and Bandgaps |
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289 | (2) |
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6.3.2 Envelope Equations of 2D Bloch Waves |
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291 | (3) |
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6.3.3 Families of 2D Gap Solitons Bifurcated from Band Edges |
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294 | (3) |
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6.4 Stability of 2D Gap Solitons Bifurcated from Bloch Bands |
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297 | (17) |
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6.4.1 Analytical Calculations of Eigenvalue Bifurcations near Band Edges |
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298 | (12) |
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6.4.2 Numerical Stability Results |
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310 | (4) |
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6.5 Two-Dimensional Gap Solitons Not Bifurcated from Bloch Bands |
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314 | (3) |
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317 | (10) |
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7 Numerical Methods for Nonlinear Wave Equations |
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327 | (78) |
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7.1 Numerical Methods for Evolution Simulations |
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328 | (30) |
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7.1.1 Pseudospectral Method |
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328 | (4) |
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7.1.2 Split-Step Method---Accuracy and Numerical Stability |
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332 | (21) |
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7.1.3 Integrating-Factor Method and Its Numerical Stability |
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353 | (5) |
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7.2 Numerical Methods for Computations of Solitary Waves |
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358 | (31) |
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7.2.1 Petviashvili Method |
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359 | (7) |
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7.2.2 Accelerated Imaginary-Time Evolution Method |
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366 | (8) |
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7.2.3 Squared-Operator Iteration Methods |
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374 | (7) |
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7.2.4 Newton Conjugate-Gradient Methods |
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381 | (8) |
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7.3 Numerical Methods for Linear-Stability Eigenvalues of Solitary Waves |
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389 | (16) |
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7.3.1 Fourier Collocation Method for the Whole Spectrum |
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390 | (7) |
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7.3.2 Newton Conjugate-Gradient Method for Individual Eigenvalues |
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397 | (8) |
| Bibliography |
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405 | (22) |
| Index |
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427 | |
