Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation / Spyridon Kamvissis, Kenneth D.T-R. McLaughlin, Peter D. Miller

Author(s): Kamvissis, Spyridon
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Subjects: Schrödinger equation
Solitons
Formats: Print
Material Type: Books
Language: English
Audience: Unspecified
Published: Princeton, N.J. : Princeton University Press, 2003
Series: Annals of mathematics studies no. 154
LC Classification: Q, QC
Physical Description: xii, 265 p. : ill. ; 24 cm
Table of Contents: List of Figures and Tables
Preface
Ch. 1. Introduction and Overview 1
Ch. 2. Holomorphic Riemann-Hilbert Problems for Solitons 13
Ch. 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
3.3. The Satsuma-Yajima Semiclassical Soliton Ensemble 34
Ch. 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
4.4. Inner Approximations 69
4.5. Estimating the Error 106
Ch. 5. Direct Construction of the Complex Phase 121
5.1. Postponing the Inequalities. General Considerations 121
5.2. Imposing the Inequalities. Local and Global Continuation Theory 138
5.3. Modulation Equations 148
5.4. Symmetries of the Endpoint Equations 159
Ch. 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
6.3. Larger-Time Analysis for Soliton Ensembles 175
6.4. The Elliptic Modulation Equations and the Particular Solution of Akhmanov, Sukhorukov, and Khokhlov for the Satsuma-Yajima Initial Data 191
Ch. 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 I[subscript l][superscript +] by Varying x near x[subscript crit] 200
Ch. 8. Variational Theory of the Complex Phase 215
Ch. 9. Conclusion and Outlook 223
9.1. Generalization for Nonquantum Values of h 223
9.2. Effect of Complex Singularities in p[superscript 0(n) 224
9.3. Uniformity of the Error near t = 0 225
9.4. Errors Incurred by Modifying the Initial Data 225
9.5. Analysis of the Max-Min Variational Problem 226
9.6. Initial Data with S(x) [is not equal to] 0 227
9.7. Final Remarks 228
App. A. Holder Theory of Local Riemann-Hilbert Problems 229
App. B. Near-Identity Riemann-Hilbert Problems in L[superscript 2] 253
Bibliography 255
Index 259
Additional Authors: McLaughlin, K. T-R (Kenneth T-R), 1969-
Miller, Peter D
Notes: LCCN: 2003108056
ISBN: 0691114838
ISBN: 069111482X (PBK.)£27.95
Includes bibliographical references (p. [255]-258) and index
OCLC Number: 51780336
ISBN/ISSN: 0691114838
069111482X