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

Overview

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 ...

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Overview

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.

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Editorial Reviews

From the Publisher
"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
Bulletin of the London Mathematical Society
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 - Peter Clarkson
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.
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Product Details

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
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