A matrix Riemann-Hilbert problem associated with the one-dimensional SchrBdinger equation is considered, and the existence and uniqueness of its solutions are studied. The solution of this Riemann-Hilbert problem yields the solution of the inverse scattering problem for a larger class of potentials than the usual Faddeev class. Some examples of explicit solutions of the Riemann-Hilbert problem ...

We describe methods for the derivation of strong asymptotics for the denominator polynomials and the remainder of Padé approximants for a Markov function with a complex and varying weight. Two approaches, both based on a Riemann-Hilbert problem, are presented. The first method uses a scalar Riemann-Hilbert boundary value problem on a two-sheeted Riemann surface, the second approach uses a matri...

We are solving the classical Riemann-Hilbert problem of rank N > 1 on the extended complex plane punctured in 2m + 2 points, for N × N quasi-permutation monodromy matrices. Our approach is based on the finite gap integration method applied to study the Riemann-Hilbert by Kitaev and Korotkin [1], Deift, Its, Kapaev and Zhou [2] and Korotkin, [3]. This permits us to solve the Riemann-Hilbert prob...

We use discrete analogs of Riemann–Hilbert problem’s methods to derive the discrete Bessel kernel which describes the poissonized Plancherel measures for symmetric groups. To do this we define discrete analogs of a Riemann–Hilbert problem and of an integrable integral operator and show that computing the resolvent of a discrete integrable operator can be reduced to solving a corresponding discr...

In the earlier study of Riemann surfaces, one was interested in the embeddability of Riemann surfaces in Euclidean spaces. Complex Analysists showed that any Riemann surface can be embedded in Euclidean 3-space. But recently, this problem has been conveyed to the problem in the Riemannian manifolds. In this paper we investigate the history and very recent results of problem in affordable and sy...

In this paper we study the Fuchsian Riemann-Hilbert (inverse monodromy) problem corresponding to Frobenius structures on Hurwitz spaces. We find a solution to this Riemann-Hilbert problem in terms of integrals of certain meromorphic differentials over a basis of an appropriate relative homology space over a Riemann surface, study the corresponding monodromy group and compute the monodromy matri...

In the application of the Deift-Zhou steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials, a model Riemann-Hilbert problem that appears in the multi-cut case is solved with the use of hyperelliptic theta functions. We present here an alternative approach which uses meromorphic differentials instead of theta functions to construct the solution of the model Riemann-Hi...

is called Fuchsian if the N ×N coefficient matrix A(λ) is a rational function of λ whose singularities are simple poles. Each Fuchsian system generates, via analytic continuation of its fundamental solution Ψ (λ) along closed curves, a representation of the fundamental group of the punctured Riemann sphere (punctured at the poles of A(λ)) in the group of N ×N invertible matrices. This represent...

A two-dimensional Riemann solver is proposed for the solution of hyperbolic systems of conservation laws in two dimensions of space. The solver approximates the solution of a so-called angular two-dimensional Riemann problem as the weighted sum of the solutions of one-dimensional Riemann problems. The weights are proportional to the aperture of the regions of constant state. The two-dimensional...

In this paper, we present a general iterative method for the solution of the Riemann problem for hyperbolic systems of PDEs. The method is based on the multiple shooting method for free boundary value problems. We demonstrate the method by solving one-dimensional Riemann problems for hyperelastic solid mechanics. Even for conditions representative of routine laboratory conditions and military b...