Related papers: On tau functions for orthogonal polynomials and ma…

Let $K$ be a non-polar compact subset of $\mathbb{R}$ and $\mu_K$ denote the equilibrium measure of $K$. Furthermore, let $P_n\left(\cdot, \mu_K\right)$ be the $n$-th monic orthogonal polynomial for $\mu_K$. It is shown that…

A general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh-Weyl $m$-function is proved. This result is applied to scattering…

We find a local $(d+1) \times (d+1)$ Riemann-Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of the Gaussian Orthogonal Ensemble of random matrices with a potential function of degree $d$.…

[Note: important Corrigendum now available at arXiv:1601.04790] The isomonodromic tau function defined by Jimbo-Miwa-Ueno vanishes on the Malgrange's divisor of generalized monodromy data for which a vector bundle is nontrivial, or, which…

In this paper we express tau functions for the Korteweg de Vries (KdV) equation, as Laplace transforms of iterated Skorohod integrals. Our main tool is the notion of Fredholm determinant of an integral operator. Our result extends the paper…

We establish a new perturbation theory for orthogonal polynomials using a Riemann--Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the orthogonal polynomials with…

$Q$-systems and $T$-systems are systems of integrable difference equations that have recently attracted much attention, and have wide applications in representation theory and statistical mechanics. We show that certain $\tau$-functions,…

We consider the scattering problem for the nonlinear Schr\"{o}dinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator $A(s)$ appearing in commutator…

We prove that the polynomial form of the scattering equations is a Macaulay H-basis. We demonstrate that this H-basis facilitates integrand reduction and global residue computations in a way very similar to using a Gr\"obner basis, but…

For an arbitrary solution to the KdV hierarchy, the generating series of logarithmic derivatives of the tau-function of the solution can be expressed by the basic matrix resolvent via algebraic manipulations. Based on this we develop in…

In this paper, we analyze the asymptotic behaviour of the poles of certain rational solutions of the fifth Painlev\'e equation. These solutions are constructed by relating the corresponding tau function to a Hankel determinant of a certain…

We establish a deep connection between the Prandtl equations linearised around a quadratic shear flow, confluent hypergeometric functions of the first kind, and the Schr\"odinger operator. Our first result concerns an ODE and a spectral…

We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a close-to-equilibrium framework and show that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution enjoys a…

We study the Cauchy problem for the generalized elliptic and non-elliptic derivative nonlinear Schrodinger equations, the existence of the scattering operators and the global well posedness of solutions with small data in Besov spaces and…

The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with…

We consider the Itzykson-Zuber-Eynard-Mehta two-matrix model and prove that the partition function is an isomonodromic tau function in a sense that generalizes Jimbo-Miwa-Ueno's. In order to achieve the generalization we need to define a…

We consider the random matrix model with external source, in case where the potential V(x) is an even polynomial and the external source has two eigenvalues a, -a of equal multiplicity. We show that the limiting mean eigenvalue distribution…

We present the q-deformed multivariate hypergeometric functions related to Schur polynomials as tau-functions of the KP and of the two-dimensional Toda lattice hierarchies. The variables of the hypergeometric functions are the higher times…

The Hankel determinant representations for the partition function and boundary correlation functions of the six-vertex model with domain wall boundary conditions are investigated by the methods of orthogonal polynomial theory. For specific…

The scattering of a wave obeying Helmholtz equation by an elliptic obstacle can be described exactly using series of Mathieu functions. This situation is relevant in optics, quantum mechanics and fluid dynamics. We focus on the case when…