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Semiclassical Soliton Ensembles for the Focusing Nonlinear Schroedinger Equation (AM-154) : Annals of Mathematics Studies - Spyridon Kamvissis

Semiclassical Soliton Ensembles for the Focusing Nonlinear Schroedinger Equation (AM-154)

Annals of Mathematics Studies

Paperback

Published: 7th September 2003
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This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrodinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing the semiclassical behavior. In doing so they also aim to explain the observed gradient catastrophe for the underlying nonlinear elliptic partial differential equations, and to set forth a detailed, pointwise asymptotic description of the violent oscillations that emerge following the gradient catastrophe.


To achieve this, the authors have extended the reach of two powerful analytical techniques that have arisen through the asymptotic analysis of integrable systems: the Lax-Levermore-Venakides variational approach to singular limits in integrable systems, and Deift and Zhou's nonlinear Steepest-Descent/Stationary Phase method for the analysis of Riemann-Hilbert problems. In particular, they introduce a systematic procedure for handling certain Riemann-Hilbert problems with poles accumulating on curves in the plane. This book, which includes an appendix on the use of the Fredholm theory for Riemann-Hilbert problems in the Holder class, is intended for researchers and graduate students of applied mathematics and analysis, especially those with an interest in integrable systems, nonlinear waves, or complex analysis.

"Overall, this ... book [gives] a deep insight into the application of inverse scattering to equation... Most of the current books on solution theory tend to focus mainly on inverse scattering for the KdV equation, so a book that concentrates solely on the NLS equation is refreshing."--Peter Clarkson, Bulletin of the London Mathematical Society

List of Figures and Tables
Preface
Introduction and Overviewp. 1
Holomorphic Riemann-Hilbert Problems for Solitonsp. 13
Semiclassical Soliton Ensemblesp. 23
Formal WKB Formulae for Even, Bell-Shaped, Real-Valued Initial Conditionsp. 23
Asymptotic Properties of the Discrete WKB Spectrump. 26
The Satsuma-Yajima Semiclassical Soliton Ensemblep. 34
Asymptotic Analysis of the Inverse Problemp. 37
Introducing the Complex Phasep. 38
Representation as a Complex Single-Layer Potential. Passing to the Continum Limit. Conditions on the Complex Phase Leading to the Outer Model Problemp. 40
Exact Solution of the Outer Model Problemp. 51
Inner Approximationsp. 69
Estimating the Errorp. 106
Direct Construction of the Complex Phasep. 121
Postponing the Inequalities. General Considerationsp. 121
Imposing the Inequalities. Local and Global Continuation Theoryp. 138
Modulation Equationsp. 148
Symmetries of the Endpoint Equationsp. 159
The Genus-Zero Ansatzp. 163
Location of the Endpoints for General Datap. 163
Success of the Ansatz for General Data and Small Time. Rigorous Small-Time Asymptotics for Semiclassical Soliton Ensemblesp. 164
Larger-Time Analysis for Soliton Ensemblesp. 175
The Elliptic Modulation Equations and the Particular Solution of Akhmanov, Sukhorukov, and Khokhlov for the Satsuma-Yajima Initial Datap. 191
The Transition to Genus Twop. 195
Matching the Critical G = 0 Ansatz with a Degenerate G = 2 Ansatzp. 196
Perturbing the Degenerate G = 2 Ansatz. Opening the Band I[subscript l][superscript +] by Varying x near x[subscript crit]p. 200
Variational Theory of the Complex Phasep. 215
Conclusion and Outlookp. 223
Generalization for Nonquantum Values of hp. 223
Effect of Complex Singularities in p[superscript 0(n)p. 224
Uniformity of the Error near t = 0p. 225
Errors Incurred by Modifying the Initial Datap. 225
Analysis of the Max-Min Variational Problemp. 226
Initial Data with S(x) [is not equal to] 0p. 227
Final Remarksp. 228
Holder Theory of Local Riemann-Hilbert Problemsp. 229
Near-Identity Riemann-Hilbert Problems in L[superscript 2]p. 253
Bibliographyp. 255
Indexp. 259
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780691114828
ISBN-10: 069111482X
Series: Annals of Mathematics Studies
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 312
Published: 7th September 2003
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.32 x 15.55  x 1.83
Weight (kg): 0.45