Related papers: Pfaffian structures and certain solutions to BKP h…
The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such…
We consider the Schwarzian KP and Harry Dym hierarchies in the framework of the bilinear formalism which is well known for such integrable hierarchies as KP, modified KP, BKP, Toda lattice and other. We show that, similarly to the bilinear…
We give a natural definition of open Hurwitz numbers, where the weight of each ramified covering includes an integer parameter $N$ taken to the power that is equal to the number of boundary components of a Riemann surface with boundary…
Questions on random matrices and on non-intersecting Brownian motions have led to the study of moment matrices with regard to several weights. The purpose of this paper is to show that the determinants of such moment matrices satisfy, upon…
This paper is an extension of Parts I and Ia of a series about Nu-class generalized functions. We provide hand-generated algebraic expressions for integrals of single Matern-covariance functions, as well as for products of two…
We derive a bilinear expansion expressing elements of a lattice of KP $\tau$-functions, labelled by partitions, as a sum over products of pairs of elements of an associated lattice of BKP $\tau$-functions, labelled by strict partitions.…
In this paper we study the addition formulae of the KP, the mKP and the BKP hierarchies. We prove that the total hierarchies are equivalent to the simplest equations of their addition formulae. In the case of the KP and the mKP hierarchies…
We revisit dispersionless version of the multicomponent KP hierarchy considered previously by Takasaki and Takebe. In contrast to their study, we do not fix any distinguished component treating all of them on equal footing. We obtain…
For each partition p of an integer N \geq 2, consisting of r parts, an integrable hierarchy of Lax type Hamiltonian PDE has been constructed recently by some of us. In the present paper we show that any tau-function of the p-reduced…
In this short article, we will reconstruct the KP equation from Plucker relations and provide some generalizations on this topic. Additionally, in the final section, we define the discrete function $\tau$ in a similar manner, leading to the…
For one-matrix models with polynomial potentials, the explicit relationship between the partition function and the isomonodromic tau function for the 2x2 polynomial differential systems satisfied by the associated orthogonal polynomials is…
This paper is a brief review of recent results on the concept of ``generalized $\tau$-function'', defined as a generating function of all the matrix elements in a given highest-weight representation of a universal enveloping algebra ${\cal…
For an arbitrary solution to the Burgers--KdV hierarchy, we define the tau-tuple $(\tau_1,\tau_2)$ of the solution. We show that the product $\tau_1\tau_2$ admits Buryak's residue formula. Therefore, according to Alexandrov's theorem,…
We review various aspects of integrable hierarchies appearing in N=2 supersymmetric gauge theories. In particular, we show that the blowup function in Donaldson-Witten theory, up to a redefinition of the fast times, is a tau function for a…
Multivariate orthogonal polynomials in $D$ real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials,…
We use a Grassmannian framework to define multi-component tau functions as expectation values of certain multi-component Fermi operators satisfying simple bilinear commutation relations on Clifford algebra. The tau functions contain both…
This work investigates the intricate relationship between the q-boson model, a quantum integrable system, and classical integrable systems such as the Toda and KP hierarchies. Initially, we analyze scalar products of off-shell Bethe states…
Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable…
First we introduce the two tau-functions which appeared either as the $\tau$-function of the integrable hierarchy governing the Riemann mapping of Jordan curves or in conformal field theory and the universal Grassmannian. Then we discuss…
We study the relation between the periodic Benjamin-Ono equation with discrete Laplacian and the two dimensional Toda hierarchy. We introduce the tau-functions tau_pm(z) for the periodic Benjamin-Ono equation, construct two families of…