Riemann-Hilbert Methods Research Articles

A Riemann Hilbert Approach to the Study of the Generating Function Associated to the Pearcey Process

Using Riemann–Hilbert methods, we establish a Tracy–Widom like formula for the generating function of the occupancy numbers of the Pearcey process. This formula is linked to a coupled vector differential equation of order three. We also obtain a non linear coupled heat equation. Combining these two equations we obtain a PDE for the logarithm of the the generating function of the Pearcey process.

The Fourier extension method and discrete orthogonal polynomials on an arc of the circle

The Fourier extension method, also known as the Fourier continuation method, is a method for approximating non-periodic functions on an interval using truncated Fourier series with period larger than the interval on which the function is defined. When the function being approximated is known at only finitely many points, the approximation is constructed as a projection based on this discrete set of points. In this paper we address the issue of estimating the absolute error in the approximation. The error can be expressed in terms of a system of discrete orthogonal polynomials on an arc of the unit circle, and these polynomials are then evaluated asymptotically using Riemann–Hilbert methods.

Sharp upper bound for amplitudes of hyperelliptic solutions of the focusing nonlinear Schrödinger equation

Hyperelliptic or finite-gap solutions of the focusing nonlinear Schrödinger equation are quasiperiodic N-phase solutions whose time dependence linearizes on a real subtorus of the Jacobi variety of an invariant hyperelliptic Riemann surface. A new proof is given for the formula for an upper bound, attained for some initial values, on amplitudes of hyperelliptic solutions. The upper bound is equal to the sum of the imaginary parts of all the branch points of the invariant Riemann surface which lie in the upper half-plane. A similar formula is proven for bounded solutions of the defocusing nonlinear Schrödinger equation. The new proof generalizes the derivation of the two-phase formula (Wright 2016 Physica D 321–2 16–38), and the result is the same N-phase formula obtained using Riemann–Hilbert methods by Bertola and Tovbis (2017 Commun. Math. Phys. 354 525–47).

ON THE CONNECTION FORMULAS OF THE FOURTH PAINLEVÉ TRANSCENDENT

Using the method of "uniform asymptotics" proposed by Bassom, Clarkson, Law and McLeod [3], we provide a simpler and more rigorous proof of the connection formulas of the Clarkson–McLeod solution for the fourth Painlevé transcendent, which has been established earlier by Its and Kapaev [11] via the isomonodromy and Riemann–Hilbert methods.

Asymptotics of the Airy-Kernel Determinant

The authors use Riemann-Hilbert methods to compute the constant that arises in the asymptotic behavior of the Airy-kernel determinant of random matrix theory.

Asymptotic zero distribution of a class of hypergeometric polynomials

We prove that the zeros of 2 F 1 (−n, n+1/2; n+3/2; z asymptotically approach the section of the lemniscate {z : |z(1 − z)2&verbar = 4/27; Re(z) > 1/3} as n → ∞. In recent papers (cf. [8], [10]), Martínez-Finkelshtein and Kuijlaars and their co-authors have used Riemann-Hilbert methods to derive the asymptotic zero distribution of Jacobi polynomials P n (αn , βn ) when the limits A = lim n → ∞ αn /n and B = lim n → ∞ βn /n exist and lie in the interior of certain specified regions in the AB-plane. Our result corresponds to one of the transitional or boundary cases for Jacobi polynomials in the Kuijlaars Martínez-Finkelshtein classification.

Mixed Plane Problems in Linearized Solid Mechanics: Exact Solutions

The paper analyzes the exact solutions to mixed plane problems of linearized solid mechanics in cases of statics, dynamics, stability, and fracture. The exact solutions have a universal form for compressible and incompressible, elastic and plastic bodies and account for stresses and displacements expressed in terms of analytical functions of complex variables. To obtain these solutions, the use is made of complex variable theory, in particular, the Riemann–Hilbert methods and Keldysh–Sedov formula. When the initial (residual) stresses tend to zero, the exact solutions go over into the corresponding exact solutions of classical linear solid mechanics, which are based on the complex representations due to Muskhelishvili, Lekhnitskii, and Galin

Recursion Relations for Unitary Integrals, Combinatorics and the Toeplitz Lattice

In a discussion in spring 2001, Alexei Borodin showed us recursion relations for the Toeplitz determinants going with the symbols e t(z + z−1) and \!. Borodin obtained these relations using Riemann-Hilbert methods; see the recent work of Borodin B and Baik Baik. The nature of Borodin's recursion relations pointed towards the Toeplitz lattice and its Virasoro algebra, introduced by us in AvM1. In this paper, we take the Toeplitz lattice and Virasoro algebra approach for a fairly large class of symbols, leading to a systematic way of generating recursion relations. The latter are very naturally expressed in terms of the L-matrices appearing in the Toeplitz lattice equations. As a surprise, we find, compared to Borodin's, a different set of relations, except for the 3-step relations associated with the symbol e t(z + z−1) . The Painlevé analysis of the Toeplitz lattice enables us to show the ``singularity confinement'' for these recursion relations.

Asymptotics of semiclassical soliton ensembles: rigorous justification of the WKB approximation

Rigorous pointwise asymptotics are established for semiclassical soliton ensembles (SSEs) of the focusing nonlinear Schrodinger equation using techniques of asymptotic analysis of matrix Riemann- Hilbert problems. The accumulation of poles in the eigenfunction is handled using a new method in which the residues are simultaneously interpolated at the poles by two distinct interpolants. The results justify the WKB approximation for the nonselfadjoint Zakharov-Shabat operator with real-analytic, bell-shaped, even potentials. The new technique introduced in this paper is applicable to other problems as well: (i) it can be used to provide a unified treatment by Riemann-Hilbert methods of the zero-dispersion limit of the Korteweg-de Vries equation with negative (soliton generating) initial as studied by Lax, Levermore, and Venakides, and (ii) it allows one to compute rigorous strong asymptotics for systems of discrete orthogonal polynomials. Many important problems in the theory of integrable systems and approximation theory can be recast as Riemann-Hilbert problems for a matrix-valued unknown. Via the connection with approximation theory, and specifically the theory of orthogonal polynomials, one can also study problems from the theory of random matrix ensembles and combinatorics. Roughly speaking, solving a Riemann-Hilbert problem amounts to reconstructing a sectionally meromorphic matrix from given homogeneous multiplicative at the boundary contours of the domains of meromorphy, from part data given at the prescribed singularities, and from a normalization condition. So, many asymptotic questions in integrable systems (e.g. long time behavior and singular perturbation theory) and approximation theory (e.g. behavior of orthogonal polynomials in the limit of large degree) amount to determining asymptotic properties of the solution matrix of a Riemann-Hilbert problem from given asymptotics of the jump conditions and principal part data. In recent years a collection of techniques has emerged for studying certain asymptotic problems of this sort. These techniques are analogous to familiar asymptotic methods for expanding oscillatory integrals, and we often refer to them as steepest-descent methods. The basic method first appeared in the work of Deift and Zhou (DZ93). The first applications were to Riemann-Hilbert problems without poles, in which the solution matrix is sectionally holomorphic. Later, some problems were studied in which there were a number of poles — a number held fixed in the limit of interest — in the solution matrix (see, for example, the paper (DKKZ96) on the long-time behavior of the Toda lattice with rarefaction initial data). The previous methods were extended to these more complicated problems through the device of making a local change of variable near each pole in some small domain containing the pole. The change of variable is chosen so that it has the effect of removing the pole at the cost of introducing an explicit jump on the boundary of the domain around the pole in which the transformation is made. The result is a Riemann-Hilbert problem for a sectionally holomorphic matrix, which can be solved asymptotically by pre-existing steepest-descent methods. Recovery of an approximation for the original sectionally meromorphic matrix unknown involves putting back the poles by reversing the explicit change of variables that was designed to get rid of them to begin with. Yet another category of Riemann-Hilbert problems consists of those problems where the number of poles is not fixed, but becomes large in the limit of interest, with the poles accumulating on some closed set F in the finite complex plane. A problem of this sort has been addressed (KMM00) by making an explicit transformation of the type described above in a single fixed domain G that contains the locus of accumulation F of all the poles. The transformation is chosen to get rid of all the poles at once. In order to specify it,

Exact relativistic treatment of stationary counterrotating dust disks: Boundary value problems and solutions

This is the first in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which describe counter-rotating disks of dust. These disks can serve as models for certain galaxies and accretion disks in astrophysics. We review the Newtonian theory for disks using Riemann-Hilbert methods which can be extended to some extent to the relativistic case where they lead to modular functions on Riemann surfaces. In the case of compact surfaces these are Korotkin's finite gap solutions which we will discuss in this paper. On the axis we establish for general genus relations between the metric functions and hence the multipoles which are enforced by the underlying hyperelliptic Riemann surface. Generalizing these results to the whole spacetime we are able in principle to study the classes of boundary value problems which can be solved on a given Riemann surface. We investigate the cases of genus 1 and 2 of the Riemann surface in detail and construct the explicit solution for a family of disks with constant angular velocity and constant relative energy density which was announced in a previous Physical Review Letter.

Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution

Using the isomonodromy and Riemann-Hilbert methods, we perform a rigorous global asymptotic analysis of the Clarkson-McLeod fourth Painleve transcendent. In particular, we prove some of the Clarkson-McLeod conjectures concerning the asymptotic behaviour, as , of the solutions of Painleve IV equation which decay as . The relevant exact connection formulae are also rigorously derived.