Journal ArticleInternational Mathematics Research Notices · December 1, 2008
For a broad class of unitary ensembles of random matrices, we demonstrate the universal nature of the Janossy densities of eigenvalues near the spectral edge, providing a different formulation of the probability distributions of the limiting second, third, ...
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Journal ArticleMathematical Physics Analysis and Geometry · November 1, 2008
Orthogonal rational functions are characterized in terms of a family of matrix Riemann-Hilbert problems on ℝ, and a related family of energy minimisation problems is presented. Existence, uniqueness, and regularity properties of the equilibrium measures wh ...
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Journal ArticleConstructive Approximation · March 1, 2008
Let Λ ℝ denote the linear space over ℝ spanned by zk k ℤ. Define the (real) inner product Ċ,Ċ L : Λ ℝ× Λ ℝ ℝ, (f,g) ∫ℝ f(s)g(s) exp(- N V(s)) ds, N ℕ, where V satisfies: (i) V is real analytic on ℝ 0; ...
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Journal ArticleActa Applicandae Mathematicae · January 1, 2008
Let Λℝ denote the linear space over ℝ spanned by z k , k ∈ ℤ. Define the real inner product 〈 .,.〉 L : Λℝ×Λℝ→ℝ, (f,g)∫ℝ}f(s)g(s)exp (-{N}V(s)){d}s, N ∈, where V satisfies: (i) V is real anal ...
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Journal ArticleInternational Mathematics Research Notices · December 1, 2007
We consider the semiclassical limit for the focusing nonlinear (cubic) Schrödinger Equation (NLS) in the pure radiational case. We present a method of reconstructing the leading order terms of the solitonless initial data and of its evolution for a wide cl ...
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Journal ArticleJournal of Computational and Applied Mathematics · May 1, 2007
In the bulk scaling limit for the Gaussian Unitary Ensemble in random matrix theory, the probability that there are no eigenvalues in the interval (0, 2 s) is given by Ps = det (I - Ks), where Ks is the trace-class operator ...
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Journal ArticleInternational Mathematics Research Papers · October 18, 2006
Let Λℝ denote the linear space over ℝ spanned by zk, k ∈ ℤ. Define the real inner product (with varying exponential weights) 〈̇,̇〉ℒ : ΛRdbl; x ΛRdbl;. → ℝ, (f, g) ∫ℝ f(s)g(s) exp(-NV(s))ds, N ∈ ℕ, where ...
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Journal ArticleCommunications on Pure and Applied Mathematics · January 1, 2006
In a previous paper [13] we calculated the leading-order term q 0(x, t, ε) of the solution of q(x, t, ε), the focusing nonlinear (cubic) Schrödinger (NLS) equation in the semiclassical limit (ε → 0) for a certain one-parameter family of initial ...
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Journal ArticleCommunications on Pure and Applied Mathematics · July 1, 2004
We calculate the leading-order term of the solution of the focusing nonlinear (cubic) Schrödinger equation (NLS) in the semiclassical limit for a certain one-parameter family of initial conditions. This family contains both solitons and pure radiation. In ...
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Journal ArticlePhysical Review B Condensed Matter and Materials Physics · May 1, 2002
The Landau-Lifshitz equation for an anisotropic (easy-plane) ferromagnet is formulated as a Riemann-Hilbert problem on a Riemann surface of the spectral parameter. Exact multiple domain wall solutions can be obtained in a systematic and exhaustive manner b ...
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Journal ArticleJournal of Computational and Applied Mathematics · August 1, 2001
A few years ago the authors introduced a new approach to study asymptotic questions for orthogonal polynomials. In this paper we give an overview of our method and review the results which have been obtained in Deift et al. (Internat. Math. Res. Notices (1 ...
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Journal ArticleAdvances in Theoretical and Mathematical Physics · January 1, 2001
In this paper, we obtain optimal uniform lower tail estimates for the probability distribution of the properly scaled length of the longest up/right path of the last passage site percolation model considered by Johansson in [12]. The estimates are used to ...
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Journal ArticleCommunications on Pure and Applied Mathematics · January 1, 1999
We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx = e-Q(x)dx on the real line, where Q(x) = Σ2mk=0qkxk, q2m > 0, denotes a polynomial of even order with positive leading ...
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Journal ArticleCommunications on Pure and Applied Mathematics · January 1, 1999
We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e-nV(x)dx on the line as n → ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The princip ...
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Journal ArticlePhysical Review B Condensed Matter and Materials Physics · January 1, 1999
By using a stereographic projection of the unit sphere of magnetization vector onto a complex plane for the equations of motion, the effect of an external magnetic field for integrability of the system is discussed. The properties of the Jost solutions and ...
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Journal ArticleCommunications in Mathematical Physics · January 1, 1999
A new approach to the construction of isomonodromy deformations of 2 × 2 Fuchsian systems is presented. The method is based on a combination of the algebro-geometric scheme and Riemann-Hilbert approach of the theory of integrable systems. For a given numbe ...
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Journal ArticleProceedings of the National Academy of Sciences of the United States of America · January 1998
This paper extends the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou in a critical new way. We present, in particular, an algorithm, to obtain the support of the Riemann-Hilbert problem for leading asymptotics. Applying ...
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Journal ArticleCommunications on Pure and Applied Mathematics · January 1, 1996
In the Toda shock problem (see [7], [11], [8], and also [3]) one considers a driving particle moving with a fixed velocity 2a and impinging on a one-dimensional semi-infinite lattice of particles, initially equally spaced and at rest, and interacting with ...
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Journal ArticleInverse Problems · December 1, 1992
As rigorous methodology for studying the Riemann-Hilbert problems associated with certain integrable nonlinear ODEs was introduced in 1992 by Fokas and Zhou, and was used to investigate Painleve II and Painleve IV equations. Here the authors apply this met ...
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Journal ArticleCommunications in Mathematical Physics · March 1, 1992
We introduce a rigorous methodology for studying the Riemann-Hilbert problems associated with certain integrable nonlinear ordinary differential equations. For concreteness we investigate the Painlevé II and Painlevé IV equations. We show that the Cauchy p ...
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Journal ArticleCommunications in Mathematical Physics · March 1, 1990
For the direct-inverse scattering transform of the time dependent Schrödinger equation, rigorous results are obtained based on an opertor-triangular-factorization approach. By viewing the equation as a first order operator equation, similar results as for ...
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