Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation

By S. Kamvissis, K. Mclaughlin, Peter D. Miller. 2000

240 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e251619da724b8bf50883f1a4f554509f6dadf5e

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Citations in the corpus

On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation (2004, methodology): “The first analytic breakthrough [24], using initial data (1.”
Asymptotics of Hermite–Padé Rational Approximants for Two Analytic Functions with Separated Pairs of Branch Points (Case of Genus 0) (2007, methodology): “It has later been noticed [6, 11, 45] that the method also works for the non-Hermitian orthogonality in the complex plane with respect to complex valued weights.”
Long time asymptotic behavior of the focusing nonlinear Schrödinger equation (2016, unknown): “Such potentials have been extensively studied in the semi-classical limit [23,32,6] where the size of the discrete spectrum becomes asymptotically large.”
Padé approximants for functions with branch points — strong asymptotics of Nuttall–Stahl polynomials (2011, methodology): “It also has been noticed [7, 3, 29, 12] (see also recent paper [13]) that the method works for the non-Hermitian orthogonality in the complex plane with respect to complex-valued weights.”
Long‐Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions at Infinity and Asymptotic Stage of Modulational Instability (2015, background): “, see [56]) and remains true in the case of nonzero boundary conditions.”
The ∂ Steepest Descent Method and the Asymptotic Behavior of Polynomials Orthogonal on the Unit Circle with Fixed and Exponentially Varying Nonanalytic Weights (2004, methodology): “The contour selection principle was also encoded into a variational problem in [19].”
Dispersive shock waves and modulation theory (2016, methodology): “is shown in Fig. 18. Analysis of these semi-classical structures heavily relies on the integrability of the NLS equation and typically involves the Riemann-Hilbert problem formulation of the IST (see [143], [144] and references therein; also the next section for the discussion of various relations b”
Riemann-Hilbert Analysis for Laguerre Polynomials with Large Negative Parameter (2002, methodology): “…method for Riemann–Hilbert problems was augmented to handle cases in which the contour selection involves determining a set of nontrivial curves in the plane in [18], by Kamvissis, McLaughlin, and Miller, in the context of the semi-classical limit of the focusing nonlinear Schrödinger equation.”
On the Asymptotic Stability of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}$$\end{document}-Soliton Solutions (2016, methodology): “As claimed above and proven in Appendix A, the RHP for m(sol) can be solved exactly in closed form, but we will instead give the solution using the small norm theory of Riemann–Hilbert problems, see Appendix B [32], as this more naturally leads to the asymptotic form of the solution for t 1.”
Asymptotic Zero Behavior of Laguerre Polynomials with Negative Parameter (2002, methodology): “Riemann-Hilbert analysis in which a contour selection method was required appeared also in the recent papers [1, 3, 19].”
The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1] (2001, unknown): “See [8, 11] for an introduction, and [15, 19, 21, 27] for the latest developments.”
Strong asymptotics for Jacobi polynomials with varying nonstandard parameters (2003, methodology): “The optimal curves are trajectories of a quadratic differential and they were used for steepest descent analysis of Riemann-Hilbert problems first in [3] and later in [2, 19, 21, 22, 24].”
Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials (2009, methodology): “In a rather surprising twist, a completely different problem was reduced to the existence of an S-curve in [46], where the semiclassical solution of the focusing nonlinear Schrödinger equation was constructed using methods of the inverse scattering theory.”
The Steepest Descent Method for Orthogonal Polynomials on the Real Line with Varying Weights (2008, background): “Specific information relevant to the application of small-norm theory in Hölder spaces can be found in Appendix A of [11].”
Universality for the Focusing Nonlinear Schrödinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronquée Solution to Painlevé I (2010, background): “Within each region, the strong asymptotics of q(x, t, ε) can be expressed in terms of Riemann Theta-functions to within an error term of order O(ε) that is uniform on compact subsets of the region ([19] for the pure soliton case and [26] for the general case).”