دانلود کتاب Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation (AM-154)

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Contents\nList of Figures and Tables\nPreface\nChapter 1. Introduction and Overview\n 1.1 Background\n 1.2 Approach and Summary of Results\n 1.3 Outline and Method\n 1.4 Special Notation\nChapter 2. Holomorphic Riemann-Hilbert Problems for Solitons\nChapter 3. Semiclassical Soliton Ensembles\n 3.1 Formal WKB Formulae for Even, Bell-Shaped, Real-Valued Initial Conditions\n 3.2 Asymptotic Properties of the Discrete WKB Spectrum\n 3.2.1 Asymptotic Behavior for λ Fixed\n 3.2.2 Letting λ Approach the Origin\n 3.2.3 Approximations Uniformly Valid for λ near the Origin\n 3.2.4 Convergence Theorems for Discrete WKB Spectra\n 3.3 The Satsuma-Yajima Semiclassical Soliton Ensemble\nChapter 4. Asymptotic Analysis of the Inverse Problem\n 4.1 Introducing the Complex Phase\n 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\n 4.3 Exact Solution of the Outer Model Problem\n 4.3.1 Reduction to a Problem in Function Theory on Hyperelliptic Curves\n 4.3.2 Formulae for the Baker-Akhiezer Functions\n 4.3.3 Making the Formulae Concrete\n 4.3.4 Properties of the Semiclassical Solution of the Nonlinear Schrödinger Equation\n 4.3.5 Genus Zero\n 4.3.6 The Outer Approximation for Ns (?)\n 4.4 Inner Approximations\n 4.4.1 Local Analysis for λ near the Endpoint λ2k for k = 0, . . . , G/2\n 4.4.2 Local Analysis for λ near the Endpoint λ2k−1 for k = 1, . . . , G/2\n 4.4.3 Local Analysis for λ near the Origin\n 4.4.4 Note Added: Exact Solution of Riemann-Hilbert Problem 4.4.5\n 4.5 Estimating the Error\n 4.5.1 Defining the Parametrix\n 4.5.2 Asymptotic Validity of the Parametrix\nChapter 5. Direct Construction of the Complex Phase\n 5.1 Postponing the Inequalities. General Considerations\n 5.1.1 Collapsing the Loop Contour C\n 5.1.2 The Scalar Boundary Value Problem for Genus G. Moment Conditions\n 5.1.3 Ensuring = 0 in the Bands. Vanishing Conditions\n 5.1.4 Determination of the Contour Bands. Measure Reality Conditions\n 5.1.5 Restoring the Loop Contour C\n 5.2 Imposing the Inequalities. Local and Global Continuation Theory\n 5.3 Modulation Equations\n 5.4 Symmetries of the Endpoint Equations\nChapter 6. The Genus-Zero Ansatz\n 6.1 Location of the Endpoints for General Data\n 6.2 Success of the Ansatz for General Data and Small Time. Rigorous Small-Time Asymptotics for Semiclassical Soliton Ensembles\n 6.2.1 The Genus-Zero Ansatz for t = 0. Success of the Ansatz and Recovery of the Initial Data\n 6.2.2 Perturbation Theory for Small Time\n 6.3 Larger-Time Analysis for Soliton Ensembles\n 6.3.1 The Explicit Solution of the Analytic Cauchy Problem for the Genus-Zero Whitham Equations along the Symmetry Axis x = 0\n 6.3.2 Determination of the Endpoint for the Satsuma-Yajima Ensemble and General x and t\n 6.3.3 Numerical Determination of the Contour Band for the Satsuma-Yajima Ensemble\n 6.3.4 Seeking a Gap Contour on Which (˜φσ (λ)) < 0. The Primary Caustic for the Satsuma-Yajima Ensemble\n 6.4 The Elliptic Modulation Equations and the Particular Solution of Akhmanov, Sukhorukov, and Khokhlov for the Satsuma-Yajima Initial Data\nChapter 7. The Transition to Genus Two\n 7.1 Matching the Critical G = 0 Ansatz with a Degenerate G = 2 Ansatz\n 7.2 Perturbing the Degenerate G = 2 Ansatz. Opening the Band I x near xcrit\nChapter 8. Variational Theory of the Complex Phase\nChapter 9. Conclusion and Outlook\n 9.1 Generalization for Nonquantum Values of
\n 9.2 Effect of Complex Singularities in ρ0(η)\n 9.3 Uniformity of the Error near t = 0\n 9.4 Errors Incurred by Modifying the Initial Data\n 9.5 Analysis of the Max-Min Variational Problem\n 9.6 Initial Data with S(x) ≢ 0\n 9.7 Final Remarks\nAppendix A. Hölder Theory of Local Riemann-Hilbert Problems\n A.1 Local Riemann-Hilbert Problems. Statement of Results\n A.2 Umbilical Riemann-Hilbert Problems\n A.3 Review of Hölder Results for Simple Contours\n A.4 Generalization for Umbilical Contours\n A.5 Fredhold Alternative for Umbilical Riemann-Hilbert Problems\n A.6 Applications to Local Riemann-Hilbert Problems\nAppendix B. Near-Identity Riemann-Hilbert Problems in L2\nBibliography\nIndex