The classical Stefan problem involves the motion of boundaries during phase transition, but this process can be greatly complicated by the presence of a fluid flow. Here, we consider a body undergoing material loss due to either dissolution (from molecular diffusion), melting (from thermodynamic phase change), or erosion (from fluid-mechanical stresses) in a fast-flowing fluid. In each case, th...

We consider polynomials orthogonal on [0,∞) with respect to Laguerre-type weights w(x) = xe, where α > −1 and where Q denotes a polynomial with positive leading coefficient. The main purpose of this paper is to determine Plancherel-Rotach type asymptotics in the entire complex plane for the orthonormal polynomials with respect to w, as well as asymptotics of the corresponding recurrence coeffic...

We show that renormalization in quantum field theory is a special instance of a general mathematical procedure of multiplicative extraction of finite values based on the Riemann–Hilbert problem. Given a loop γ(z), |z| = 1 of elements of a complex Lie group G the general procedure is given by evaluation of γ+(z) at z = 0 after performing the Birkhoff decomposition γ(z) = γ−(z) γ+(z) where γ±(z) ...

where we x a volume element on to achieve the last isomorphism. The section (3) vanishes precisely at the complex points 2 . Such a point is called elliptic if @F ^ @F is transversal to the zero section in (4) and the intersection number int (@F^ @F; 0detF T 10M) is one, see [K], and general elliptic is de ned below in section 1. Our main result implies the following Corollary 2. Let F : 2 ,! (...

In this paper we consider 2+1-dimensional gravity coupled to N point-particles. We introduce a gauge in which the z-and ¯ z-components of the dreibein field become holomorphic and anti-holomorphic respectively. As a result we can restrict ourselves to the complex plane. Next we show that solving the dreibein-field: e a z (z) is equivalent to solving the Riemann-Hilbert problem for the group SO(...

The inverse scattering transformation for a novel nonlocal Lakshmanan–Porsezian–Daniel (LPD) equation with rapidly decaying initial data is studied in the framework of Riemann–Hilbert problem. Firstly, integrable LPD corresponding to $$3\times 3$$ Lax pair proposed. Secondly, process left-right matrix Riemann–Hilbert(RH) problem constructed. analytical properties and symmetry relations Jost fun...

A GBDT version of the Bäcklund–Darboux transformation is constructed for a non-isospectral canonical system, which plays essential role in the theory of random matrix models. The corresponding Riemann–Hilbert problem is treated and some explicit formulas are obtained. A related inverse problem is formulated and solved.

We implement the unified transform method to the initial-boundary value (IBV) problem of the Sasa-Satsuma equation on the half line. In addition to presenting the basic Riemann-Hilbert formalism, which linearizes this IBV problem, we also analyse the associated general Dirichlet to Neumann map using the so-called global relation.

Correlation functions of the XXZ spin chain in the critical regime is studied at zerotemperature. They are exactly represented in the Fredholm determinant form and are related with an operator-valued Riemann–Hilbert problem. Analyzing this problem we prove that a two-point correlation function consisting of sufficiently separated spin operators is expressed by power-functions of the distance be...