Constructive Solvability Conditions for the Riemann-Hilbert Problem

Sufficient and necessary conditions for the solvability of the Riemann-Hilbert problem are studied. These conditions consist in the possibility of constructing stable and semistable pairs (of bundles and connections) for a given monodromy. The obtained results make it possible to develop algorithms for testing the solvability conditions for the Riemann-Hilbert problem.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 643–655.Original Russian Text Copyright ©2005 by I. V. V’yugin.I dedicate this work to the blessed memory of my teacher Andrei Andreevich Bolibrukh     

Uniqueness Results for the Riemann-Hilbert Problem with a Vanishing Coefficient

We study the Riemann-Hilbert problem of finding φ, ψ ∈ H^p such that their nontangential boundary values satisfy the equation

Special Monodromy Groups and the Riemann-Hilbert Problem for the Riemann Equation

In this paper, we solve the Riemann-Hilbert problem for the Riemann equation and for the hypergeometric equation. We describe all possible representations of the monodromy of the Riemann equation. We show that if the monodromy of the Riemann equation belongs to SL(2, ℂ), then it can be realized not only by the Riemann equation, but also by the more special Riemann-Sturm-Liouville equation. For the hypergeometric equation, we construct a criterion for its monodromy group to belong to SL(2, ℤ).__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 753–767.Original Russian Text Copyright ©2005 by V. A. Poberezhnyi.     

Gauge Fixing for Logarithmic Connections Over Curves and the Riemann-Hilbert Problem

Let be a compact Riemann surface of genus g, X={x₁, ..., x_n} a finite set of points, and ¹(log X) be the sheaf of 1-forms,holomorphic over \X and generated near x_j by dz_j/z_j for a coordinatez_j centred at x_j.     

三类多双曲复方程组的Riemann-Hilbert边值问题

用函数论的方法研究了三类超定双曲型方程组的解,即三类多双曲复方程组在一般柱形域上的Riemann-Hilbert边值问题的提法,可解条件,解的表示,唯一性和存在性.     

二阶复椭圆Riemann—Hilbert边值问题

讨论了一般二阶非线性椭圆复方程的Riemann-Hilbert边值问题,首先给出Riemann-Hilbert问题及其适应性的概念,其次给出改进后的边值问题解的表述并证明了它的可解性,最后导出原Rremann-Hilbert边值问题的可解条件。     

On the Existence of a Solution in Weighted Sobolev Space to the Riemann-Hilbert Problem for an Elliptic System with Piecewise Continuous Boundary Data

Given a first order elliptic partial differential equation weconstruct a solution which solves a given Riemann–Hilbertboundary value problem whose coefficients have singularitiesof the first kind at a finite number of some prescribed isolatedpoints and are Holder–continuous outside those pointswhile the free term has a finite number of integrable powersingularities at some prescribed points. It is shown that thesolution belongs to some weighted Sobolev space W_1,p(D;), wherethe weight function = (z;D) is the distance of the variablepoint z from the boundary D raised to a certain power. The problemis solved by first reducing it to an analogous problem for holomorphicfunctions. The latter is then solved. 2000 Mathematics SubjectClassification 35F15, 35F30, 35J55, 35J65, 36J67.     

关于拟线性系统的一个Robin问题

In this paper, a Robin problem for quasi-linear system is considered. Under the appropriate assumptions, the existence of solution for the problem is proved and the asymptotic behavior of the solution is studied using the theory of differential inequalities.     

Problem of the Riemann-Hilbert type for a third-order linear elliptic system with a singular line in a half-plane

We find an integral representation of the solutions of a third-order elliptic equation and the corresponding inversion formula depending on the roots of the characteristic equation. We study the influence of the singular line on the solvability of boundary value problems and find a well-posed statement of boundary value problems of the Riemann-Hilbert type.     

Riemann-Hilbert Problem and Multiple High-order Poles Solutions of the Focusing mKdV Equation with Nonzero Boundary Conditions

The focusing modified Korteweg-de Vries（mKdV） equation with multiple high-order poles under the nonzero boundary conditions is first investigated via developing a Riemann-Hilbert（RH） approach. We begin with the asymptotic property, symmetry and analyticity of the Jost solutions, and successfully construct the RH problem of the focusing mKdV equation. We solve the RH problem when 1/S₁₁（k） has a single highorder pole and multiple high-order poles. Furthermore, we derive the soliton solutions of the focusing mKdV equation which corresponding with a single high-order pole and multiple high-order poles, respectively. Finally,the dynamics of one-and two-soliton solutions are graphically discussed.     

Riemann-Hilbert problems for poly-Hardy space on the unit ball

In this paper, we focus on a Riemann–Hilbert boundary value problem (BVP) with a constant coefficients for the poly-Hardy space on the real unit ball in higher dimensions. We first discuss the boundary behaviour of functions in the poly-Hardy class. Then we construct the Schwarz kernel and the higher order Schwarz operator to study Riemann–Hilbert BVPs over the unit ball for the poly-Hardy class. Finally, we obtain explicit integral expressions for their solutions. As a special case, monogenic signals as elements in the Hardy space over the unit sphere will be reconstructed in the case of boundary data given in terms of functions having values in a Clifford subalgebra. Such monogenic signals represent the generalization of analytic signals as elements of the Hardy space over the unit circle of the complex plane.     

On the Riemann-Hilbert transformations for a Galilean invariant system

Summary As an application of the Riemann-Hilbert (RH) problem to mathematical physics, the RH transformations are considered for a Galilean invariant nonlinear system. Algebraic RH transformation gives rise to new solutions from the old via a calculation in linear algebra. It is proved that the infinitesimal RH transformations form an infinite-dimensional Lie algebra without using a hierarchy of potentials.     

On the Riemann-Hilbert factorization problem for positive definite functions

We give several general theorems concerning positive definite solutions of Riemann-Hilbert problems on the real line. Furthermore, as an example, we apply our theory to the characteristic function of a class of Lévy processes and we find the distribution of their extrema at a given stopping time.     

Riemann-Hilbert problem for analytic families of representations

Translated from Matematicheskie Zametki, Vol. 50, No. 2, pp. 89–97, August, 1991.     

A Riemann-Hilbert Approach to the Chen-Lee-Liu Equation on the Half Line

In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation

$i{\partial _t}u + {\partial_{xx}u - i |u{|^2}{\partial _x}u = 0}$

on the half line (?∞, 0] with decaying initial value. Assuming that the solution u(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which are obtained from the initial data u₀(x) = u(x, 0) and the boundary data g₀(t) = u(0, t), g₁(t) = u_x(0, t), respectively. The spectral functions are not independent, but satisfy a so-called global relation.

     

一类Riemann-Hilbert边值逆问题

给出解析函数的一类R iem ann-H ilbert边值逆问题的数学提法,依据解析函数R iem ann-H ilbert边值问题的经典理论,讨论了此边值问题的可解性,给出了该边值问题的可解条件和解的表示式.     

A Controllability Problem for a System Governed by the Heat Equation

A control system described by the one-dimensional heat equationwith a boundary control is considered. The problem of findinga control which drives the system, in finite time T, to a constantfinal temperature distribution is discussed. A simple boundfor the norm of the control in terms of the final time T isderived, and a numerical example is presented.