Related papers: Generalized Hirota Equations in Models of 2D Quant…
A recently introduced model of dually weighted planar graphs is solved in terms of an elliptic parametrization for some particular collection of planar graphs describing the 2D $R^2$ quantum gravity. Along with the cosmological constant…
Riemann theta functions are used to construct one-periodic and two-periodic wave solutions to a class of (2+1)-dimensional Hirota bilinear equations. The basis for the involved solution analysis is the Hirota bilinear formulation, and the…
Multiparametric quantum deformations of $gl(2)$ are studied through a complete classification of $gl(2)$ Lie bialgebra structures. From them, the non-relativistic limit leading to harmonic oscillator Lie bialgebras is implemented by means…
We study integrable non-degenerate Monge-Ampere equations of Hirota type in 4D and demonstrate that their symmetry algebras have a distinguished graded structure, uniquely determining the equations. This is used to deform these heavenly…
In this paper we consider the field equations for linearized gravity and other integer spin fields on the Kerr spacetime, and more generally on spacetimes of Petrov type D. We give a derivation, using the GHP formalism, of decoupled field…
Using the Hirota representation of dispersionless dKP and dToda hierarchies, we show that the chordal L\"owner equations and radial L\"owner equations respectively serve as consistency conditions for one variable reductions of these…
A general method based on the polynomial deformations of the Lie algebra sl(2,R) is proposed in order to exhibit the quasi-exactly solvability of specific Hamiltonians implied by quantum physical models. This method using the…
The symmetries of generic 2D dilaton models of gravity with (and without) matter are studied in some detail. It is shown that $\delta_2$, one of the symmetries of the matterless models, can be generalized to the case where matter fields of…
The hyper-CR Einstein-Weyl structures on $\R^3$ can be described in terms of the solutions to the dispersionless Hirota equation. In the present paper we show that simple geometric constructions on the associated twistor space lead to…
Using the free fermions technique and bosonization rules we introduce the multicomponent DKP hierarchy as a generating bilinear integral equation for the tau-function. A number of bilinear equations of the Hirota-Miwa type are obtained as…
We classify integrable Hamiltonian equations in 3D with the Hamiltonian operator d/dx, where the Hamiltonian density h(u, w) is a function of two variables: dependent variable u and the non-locality w such that w_x=u_y. Based on the method…
The discrete KP hierarchy is also known as the $(l-l')$--th modified KP hierarchy. Here in this paper, we consider the corresponding two--component generalization, called the two--component discrete KP (2dKP) hierarchy. Firstly, starting…
Tracy and Widom showed that the level spacing function of the Gaussian unitary ensemble is related to a particular solution of the fourth Painlev\'e equation. We reconsider this problem from the viewpoint of Hirota's bilinear method in…
Partly inspired by Sato's theory of the Kadomtsev-Petviashvili (KP) hierarchy, we start with a quite general hierarchy of linear ordinary differential equations in a space of matrices and derive from it a matrix Riccati hierarchy. The…
We demonstrate how the Einstein's equations for the $D$-dimensional spherical gravity can be written in the covariant vector-like form. These equations reveal easily the causal structure of curved spherically symmetric manifolds and may…
Starting from the ELSV formula, we derive a number of new equations on the generating functions for Hodge integrals over the moduli space of complex curves. This gives a new simple and uniform treatment of certain known results on Hodge…
We start with a Riemann-Hilbert problem (RHP) related to a BD.I-type symmetric spaces $SO(2r+1)/S(O(2r-2s +1)\otimes O(2s))$, $s\geq 1$. We consider two Riemann-Hilbert problems: the first formulated on the real axis $\mathbb{R}$ in the…
We construct a family of bilinear differential operators which satisfy certain gauge properties. These operators can be naturally associated with $q$-deformations of classical integrable hierarchies. In particular, we consider the case when…
Hirota's discrete KdV equation is a well-known integrable two-dimensional partial difference equation regarded as a discrete analogue of the KdV equation. In this paper, we show that a variation of Hirota's discrete KdV equation with an…
Bigraded Toda hierarchy $L_1^M(n)=L_2^N(n)$ is generalized to $L_1^M(n)=L_2^{N}(n)+\sum_{j\in \mathbb Z}\sum_{i=1}^{m}q^{(i)}_n\Lambda^jr^{(i)}_{n+1}$, which is the analogue of the famous constrained KP hierarchy $L^{k}=…