Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation:
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Princeton [u.a.]
Princeton Univ. Press
2003
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| Series: | Annals of mathematics studies
154 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 265 S. graph. Darst. |
| ISBN: | 069111482X 0691114838 |
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MARC
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| 100 | 1 | |a Kamvissis, Spyridon |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation |c Spyridon Kamvissis ; Kenneth D. T.-R. McLaughlin ; Peter D. Miller |
| 264 | 1 | |a Princeton [u.a.] |b Princeton Univ. Press |c 2003 | |
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Record in the Search Index
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| adam_text | Contents
List of Figures and Tables ix
Preface xi
Chapter 1. Introduction and Overview 1
1.1 Background 1
1.2 Approach and Summary of Results 5
1.3 Outline and Method 8
1.4 Special Notation 11
Chapter 2. Holomorphic Riemann-Hilbert Problems for Solitons 13
Chapter 3. Semiclassical Soliton Ensembles 23
3.1 Formal WKB Formulae for Even, Bell-Shaped, Real-Valued Initial Conditions 23
3.2 Asymptotic Properties of the Discrete WKB Spectrum 26
3.2.1 Asymptotic Behavior for k Fixed 28
3.2.2 Letting k Approach the Origin 29
3.2.3 Approximations Uniformly Valid for k near the Origin 31
3.2.4 Convergence Theorems for Discrete WKB Spectra 34
3.3 The Satsuma-Yajima Semiclassical Soliton Ensemble 34
Chapter 4. Asymptotic Analysis of the Inverse Problem 37
4.1 Introducing the Complex Phase 38
4.2 Representation as a Complex Single-Layer Potential. Passing to the
Continum Limit. Conditions on the Complex Phase Leading to the Outer
Model Problem 40
4.3 Exact Solution of the Outer Model Problem 51
4.3.1 Reduction to a Problem in Function Theory on Hyperelliptic Curves 52
4.3.2 Formulae for the Baker-Akhiezer Functions 57
4.3.3 Making the Formulae Concrete 61
4.3.4 Properties of the Semiclassical Solution of the Nonlinear Schrodinger
Equation 66
4.3.5 Genus Zero 67
4.3.6 The Outer Approximation for N (A.) 68
4.4 Inner Approximations 69
4.4.1 Local Analysis for k near the Endpoint X2* for k = 0,..., G/2 70
4.4.2 Local Analysis for k near the Endpoint kU-1 for k = 1,..., G/2 82
vi CONTENTS
4.4.3 Local Analysis for X near the Origin 86
4.4.4 Note Added: Exact Solution of Riemann-Hilbert Problem 4.4.5 100
4.5 Estimating the Error 106
4.5.1 Defining the Parametrix 106
4.5.2 Asymptotic Validity of the Parametrix 107
Chapter 5. Direct Construction of the Complex Phase 121
5.1 Postponing the Inequalities. General Considerations 121
5.1.1 Collapsing the Loop Contour C 121
5.1.2 The Scalar Boundary Value Problem for Genus G. Moment Conditions 124
5.1.3 Ensuring n(4 ) = 0 in the Bands. Vanishing Conditions 1 33
5.1.4 Determination of the Contour Bands. Measure Reality Conditions 134
5.1.5 Restoring the Loop Contour C 137
5.2 Imposing the Inequalities. Local and Global Continuation Theory 1 38
5.3 Modulation Equations 148
5.4 Symmetries of the Endpoint Equations 1 59
Chapter 6. The Genus-Zero Ansatz 163
6.1 Location of the Endpoints for General Data 163
6.2 Success of the Ansatz for General Data and Small Time. Rigorous Small-
Time Asymptotics for Semiclassical Soliton Ensembles 164
6.2.1 The Genus-Zero Ansatz for t = 0. Success of the Ansatz and Recov¬
ery of the Initial Data 164
6.2.2 Perturbation Theory for Small Time 171
6.3 Larger-Time Analysis for Soliton Ensembles 175
6.3.1 The Explicit Solution of the Analytic Cauchy Problem for the Genus-
Zero Whitham Equations along the Symmetry Axis x = 0 176
6.3.2 Determination of the Endpoint for the Satsuma-Yajima Ensemble
and General x and t 179
6.3.3 Numerical Determination of the ContourBand forthe Satsuma-Yajima
Ensemble 181
6.3.4 Seeking a Gap Contour on Which 9t( / (A)) 0. The Primary
Caustic for the Satsuma-Yajima Ensemble 183
6.4 The Elliptic Modulation Equations and the Particular Solution of Akhmanov,
Sukhorukov, and Khokhlov for the Satsuma-Yajima Initial Data 191
Chapter 7. The Transition to Genus Two 195
7.1 Matching the Critical G = 0 Ansatz with a Degenerate G = 2 Ansatz 196
7.2 Perturbing the Degenerate G = 2 Ansatz. Opening the Band /,+ by Varying
x near xcrit 200
Chapter 8. Variational Theory of the Complex Phase 215
Chapter 9. Conclusion and Outlook 223
9.1 Generalization for Nonquantum Values of h 223
9.2 Effect of Complex Singularities in p°(ri) 224
9.3 Uniformity of the Error near / = 0 225
9.4 Errors Incurred by Modifying the Initial Data 225
9.5 Analysis of the Max-Min Variational Problem 226
CONTENTS vii
9.6 Initial Data with S(x) # 0 227
9.7 Final Remarks 228
Appendix A. Holder Theory of Local Riemann-Hilbert Problems 229
A.I Local Riemann-Hilbert Problems. Statement of Results 229
A.2 Umbilical Riemann-Hilbert Problems 233
A.3 Review of Holder Results for Simple Contours 237
A.4 Generalization for Umbilical Contours 239
A.5 Fredhold Alternative for Umbilical Riemann-Hilbert Problems 242
A.6 Applications to Local Riemann-Hilbert Problems 248
Appendix B. Near-Identity Riemann-Hilbert Problems in L2 253
Bibliography 255
Index 259
|
| any_adam_object | 1 |
| author | Kamvissis, Spyridon MacLaughlin, Kenneth T-R Miller, Peter D. |
| author_facet | Kamvissis, Spyridon MacLaughlin, Kenneth T-R Miller, Peter D. |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
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| spelling | Kamvissis, Spyridon Verfasser aut Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation Spyridon Kamvissis ; Kenneth D. T.-R. McLaughlin ; Peter D. Miller Princeton [u.a.] Princeton Univ. Press 2003 XII, 265 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Annals of mathematics studies 154 Schrödinger, Équation de Schrödingervergelijking gtt Solitons gtt Schrödinger equation Schrödinger-Gleichung (DE-588)4053332-3 gnd rswk-swf Nichtlineare Schrödinger-Gleichung (DE-588)4278277-6 gnd rswk-swf Soliton (DE-588)4135213-0 gnd rswk-swf Nichtlineare Schrödinger-Gleichung (DE-588)4278277-6 s Soliton (DE-588)4135213-0 s DE-604 Schrödinger-Gleichung (DE-588)4053332-3 s MacLaughlin, Kenneth T-R Verfasser aut Miller, Peter D. Verfasser aut Annals of mathematics studies 154 (DE-604)BV000000991 154 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010485260&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kamvissis, Spyridon MacLaughlin, Kenneth T-R Miller, Peter D. Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation Annals of mathematics studies Schrödinger, Équation de Schrödingervergelijking gtt Solitons gtt Schrödinger equation Schrödinger-Gleichung (DE-588)4053332-3 gnd Nichtlineare Schrödinger-Gleichung (DE-588)4278277-6 gnd Soliton (DE-588)4135213-0 gnd |
| subject_GND | (DE-588)4053332-3 (DE-588)4278277-6 (DE-588)4135213-0 |
| title | Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation |
| title_auth | Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation |
| title_exact_search | Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation |
| title_full | Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation Spyridon Kamvissis ; Kenneth D. T.-R. McLaughlin ; Peter D. Miller |
| title_fullStr | Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation Spyridon Kamvissis ; Kenneth D. T.-R. McLaughlin ; Peter D. Miller |
| title_full_unstemmed | Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation Spyridon Kamvissis ; Kenneth D. T.-R. McLaughlin ; Peter D. Miller |
| title_short | Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation |
| title_sort | semiclassical soliton ensembles for the focusing nonlinear schrodinger equation |
| topic | Schrödinger, Équation de Schrödingervergelijking gtt Solitons gtt Schrödinger equation Schrödinger-Gleichung (DE-588)4053332-3 gnd Nichtlineare Schrödinger-Gleichung (DE-588)4278277-6 gnd Soliton (DE-588)4135213-0 gnd |
| topic_facet | Schrödinger, Équation de Schrödingervergelijking Solitons Schrödinger equation Schrödinger-Gleichung Nichtlineare Schrödinger-Gleichung Soliton |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010485260&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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