UID:
edocfu_9958352590702883
Format:
1 online resource (312 pages) :
,
illustrations.
Edition:
Course Book.
Edition:
Electronic reproduction. Princeton, N.J. : Princeton University Press, 2003. Mode of access: World Wide Web.
Edition:
System requirements: Web browser.
Edition:
Access may be restricted to users at subscribing institutions.
ISBN:
9781400837182
Series Statement:
Annals of Mathematics Studies, 154
Content:
This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrödinger 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 Hölder 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.
Note:
Frontmatter --
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Contents --
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Figures and Tables --
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Preface --
,
Chapter 1. Introduction and Overview --
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Chapter 2. Holomorphic Riemann-Hilbert Problems for Solitons --
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Chapter 3. Semiclassical Soliton Ensembles --
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Chapter 4. Asymptotic Analysis of the Inverse Problem --
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Chapter 5. Direct Construction of the Complex Phase --
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Chapter 6. The Genus - Zero Ansatz --
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Chapter 7. The Transition to Genus Two --
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Chapter 8. Variational Theory of the Complex Phase --
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Chapter 9. Conclusion and Outlook --
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Appendix A. H¨older Theory of Local Riemann-Hilbert Problems --
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Appendix B. Near-Identity Riemann-Hilbert Problems in L --
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Bibliography --
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Index.
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In English.
Language:
English
DOI:
10.1515/9781400837182
URL:
https://doi.org/10.1515/9781400837182
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