Related papers: Integrable structures of specialized hypergeometri…
For systems of evolutionary partial differential equations the tau-structure is an important notion which originated from the deep relation between integrable systems and quantum field theories. We show that, under a certain non-degeneracy…
We consider solvable matrix models. We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals (Gross-Witten integral and integrals over complex matrices) using the notion of tau function of matrix argument. In this case one…
We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system $\tau$, arising from the universal family of Calabi-Yau hypersurfaces $Y_a$ in a $G$-variety $X$ of…
This paper addresses the issue of integrable structure in a modified melting crystal model of topological string theory on the resolved conifold. The partition function can be expressed as the vacuum expectation value of an operator on the…
In this paper, we construct a new integrable equation which is a generalization of $q$-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of the generalized $q$-Toda…
Recent work of Foda and his group on a connection between classical integrable hierarchies (the KP and 2D Toda hierarchies) and some quantum integrable systems (the 6-vertex model with DWBC, the finite XXZ chain of spin 1/2, the phase model…
Previous results on quasi-classical limit of the KP hierarchy and its W-infinity symmetries are extended to the Toda hierarchy. The Planck constant $\hbar$ now emerges as the spacing unit of difference operators in the Lax formalism. Basic…
A hierarchy of $\mathbb{Z}_2^2$-graded integrable equations is constructed using the loop extension of the $\mathbb{Z}_2^2$-graded Lie superalgebra $\mathfrak{osp}(1|2)$. This hierarchy includes $\mathbb{Z}_2^2$-graded extensions of the…
A pedagogical presentation of integrable models with special reference to the Toda lattice hierarchy has been attempted. The example of the KdV equation has been studied in detail, beginning with the infinite conserved quantities and going…
The squared eigenfunction symmetry for the Toda lattice hierarchy is explicitly constructed in the form of the Kronecker product of the vector eigenfunction and the vector adjoint eigenfunction, which can be viewed as the generating…
Multicurrent correlators associated to KP $\tau$-functions of hypergeometric type are used as generating functions for weighted Hurwitz numbers. These are expressed as formal Taylor series and used to compute generic, simple, rational and…
For integrable inhomogeneous supersymmetric spin chains (generalized graded magnets) constructed employing Y(gl(N|M))-invariant R-matrices in finite-dimensional representations we introduce the master T-operator which is a sort of…
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…
The monodromy map for a rank-two system of differential equations with three Fuchsian singularities is classically solved by the Kummer formul\ae\ for Gauss' hypergeometric functions. We define the tau-function of such a system as the…
This paper explores integrable structures of a generalized melting crystal model that has two $q$-parameters $q_1,q_2$. This model, like the ordinary one with a single $q$-parameter, is formulated as a model of random plane partitions (or,…
The Hlawka Zeta Function is a Dirichlet series defined geometrically which provides an integral representation of the number of lattice points contained in the dilation $tD$ for some star shaped region $D\subset \mathbb{R}^{2}$ and some…
Construction and classification of 2D superintegrable systems (i.e. systems admitting, in addition to two global integrals of motion guaranteeing the Liouville integrability, the third global and independent one) defined on 2D spaces of…
We provide a determinantal formula for tau-functions of the KP hierarchy in terms of rectangular, constant matrices $A$, $B$ and $C$ satisfying a rank one condition. This result is shown to generalize and unify many previous results of…
Starting from a so-called flat exact semisimple bihamiltonian structures of hydrodynamic type, we arrive at a Frobenius manifold structure and a tau structure for the associated principal hierarchy. We then classify the deformations of the…
Consequences of the Toda equations arising from the conjectural matrix model for the Riemann sphere are investigated. The Toda equations determine the Gromov-Witten descendent potential (including all genera) of the Riemann sphere from the…