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Riemann-Hilbert Problem (riemann-hilbert + problem)

Distribution by Scientific Domains
Mathematics and Statistics100%

Selected Abstracts

Integrable operators and canonical differential systems

MATHEMATISCHE NACHRICHTEN, Issue 1-2 2007
Lev Sakhnovich
Abstract In this article we consider a class of integrable operators and investigate its connections with the following theories: the spectral theory of the non-self-adjoint operators, the Riemann-Hilbert problem, the canonical differential systems, the random matrices theory and the limit values of the multiplicative integral. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Universality in the two-matrix model: a Riemann-Hilbert steepest-descent analysis

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 8 2009
Maurice Duits
The eigenvalue statistics of a pair (M1, M2) of n × n Hermitian matrices taken randomly with respect to the measure can be described in terms of two families of biorthogonal polynomials. In this paper we give a steepest-descent analysis of a 4 × 4 matrix-valued Riemann-Hilbert problem characterizing one of the families of biorthogonal polynomials in the special case W(y) = y4/4 and V an even polynomial. As a result, we obtain the limiting behavior of the correlation kernel associated to the eigenvalues of M1 (when averaged over M2) in the global and local regime as n , , in the one-cut regular case. A special feature in the analysis is the introduction of a vector equilibrium problem involving both an external field and an upper constraint. © 2008 Wiley Periodicals, Inc. [source]

Moderate deviations for longest increasing subsequences: The upper tail

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 12 2001
Matthias Löwe
We derive the upper-tail moderate deviations for the length of a longest increasing subsequence in a random permutation. This concerns the regime between the upper-tail large-deviation regime and the central limit regime. Our proof uses a formula to describe the relevant probabilities in terms of the solution of the rank 2 Riemann-Hilbert problem (RHP); this formula was invented by Baik, Deift, and Johansson [3] to find the central limit asymptotics of the same quantities. In contrast to the work of these authors, who apply a third-order (nonstandard) steepest-descent approximation at an inflection point of the transition matrix elements of the RHP, our approach is based on a (more classical) second-order (Gaussian) saddle point approximation at the stationary points of the transition function matrix elements. © 2001 John Wiley & Sons, Inc. [source]

Unified approach to KdV modulations

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2001
Gennady A. El
We develop a unified approach to integrating the Whitham modulation equations. Our approach is based on the formulation of the initial-value problem for the zero-dispersion KdV as the steepest descent for the scalar Riemann-Hilbert problem [6] and on the method of generating differentials for the KdV-Whitham hierarchy [9]. By assuming the hyperbolicity of the zero-dispersion limit for the KdV with general initial data, we bypass the inverse scattering transform and produce the symmetric system of algebraic equations describing motion of the modulation parameters plus the system of inequalities determining the number the oscillating phases at any fixed point on the (x, t)-plane. The resulting system effectively solves the zero-dispersion KdV with an arbitrary initial datum. © 2001 John Wiley & Sons, Inc. [source]

Long-time asymptotics of the nonlinear Schrödinger equation shock problem

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2007
Robert Buckingham
The long-time asymptotics of two colliding plane waves governed by the focusing nonlinear Schrödinger equation are analyzed via the inverse scattering method. We find three asymptotic regions in space-time: a region with the original wave modified by a phase perturbation, a residual region with a one-phase wave, and an intermediate transition region with a modulated two-phase wave. The leading-order terms for the three regions are computed with error estimates using the steepest-descent method for Riemann-Hilbert problems. The nondecaying initial data requires a new adaptation of this method. A new breaking mechanism involving a complex conjugate pair of branch points emerging from the real axis is observed between the residual and transition regions. Also, the effect of the collision is felt in the plane-wave state well beyond the shock front at large times. © 2007 Wiley Periodicals, Inc. [source]