Related papers: Hirota Quadratic Equations for the Extended Toda H…
Modified Toda hierarchy is a two-component generalization of the 1st modified KP hierarchy, which has been widely applied to analyze constraints of the Toda hierarchy, including the B--Toda and C--Toda hierarchies. In this paper, we…
A $(q,t)$-deformation of the 2d Toda integrable hierarchy is introduced by enhancing the underlying symmetry algebra $\mathfrak{gl}(\infty)\simeq \text{q-W}_{1+\infty}$ to the quantum toroidal $\mathfrak{gl}(1)$ algebra. The…
Geometric interpretation of the Hirota equation is presented as equation describing the Laplace sequence of two-dimensional quadrilateral lattices. Different forms of the equation are given together with their geometric interpretation: (i)…
We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the…
A review of selected topics in Hirota's bilinear difference equation (HBDE) is given. This famous 3-dimensional difference equation is known to provide a canonical integrable discretization for most important types of soliton equations.…
This is a short review of the construction of quasi-periodic (algebraic-geometrical) solutions to hierarchies of nonlinear integrable equations. As is well known, the solutions are expressed through Riemann's theta-functions associated with…
Analytic-bilinear approach for construction and study of integrable hierarchies, in particular, the KP hierarchy is discussed. It is based on the generalized Hirota identity. This approach allows to represent generalized hierarchies of…
We discuss the origin of the associativity (WDVV) equations in the context of quasiclassical or Whitham hierarchies. The associativity equations are shown to be encoded in the dispersionless limit of the Hirota equations for KP and Toda…
The modified Toda (mToda) hierarchy is a two-component generalization of the 1-st modified KP (mKP) hierarchy, which connects the Toda hierarchy via Miura links and has two tau functions. Based on the fact that the mToda and 1-st mKP…
The ancestor Gromov--Witten invariants of a compact {\Kahler} manifold $X$ can be organized in a generating function called the total ancestor potential of $X$. In this paper, we construct Hirota Quadratic Equations (HQE shortly) for the…
Starting from a fairly explicit homogeneous realization of the toroidal Lie algebra $\mathcal{L}^{\tor}_{r+1}(\fsl_\ell)$ via a lattice vertex algebra, we derive an integrable hierarchy of Hirota bilinear equations. Moreover, we represent…
In this paper, we construct the bilinear identities for the wave functions of an extended Kadomtsev-Petviashvili (KP) hierarchy, which is the KP hierarchy with particular extended flows (2008, Phys. Lett. A, 372: 3819). By introducing an…
We introduce a new integrable hierarchy of nonlinear differential-difference equations which is a subhierarchy of the 2D Toda lattice defined by imposing a constraint to the Lax operators of the latter. The 2D Toda lattice with the…
We construct a certain reduction of the 2D Toda hierarchy and obtain a tau-symmetric Hamiltonian integrable hierarchy. This reduced integrable hierarchy controls the linear Hodge integrals in the way that one part of its flows yields the…
The quasiclassical solution to the extended Toda chain hierarchy, corresponding to the deformation of the simplest Seiberg-Witten theory by all descendants of the dual topological string model, is constructed explicitly in terms of the…
Analytic-bilinear approach for construction and study of integrable hierarchies is discussed. Generalized multicomponent KP and 2D Toda lattice hierarchies are considered. This approach allows to represent generalized hierarchies of…
It is shown that the Affine Toda models (AT) constitute a ``gauge fixed'' version of the Conformal Affine Toda model (CAT). This result enables one to map every solution of the AT models into an infinite number of solutions of the…
We give a derivation of dispersionless Hirota equations for the extended dispersionless Toda hierarchy. We show that the dispersionless Hirota equations are nothing but a direct consequence of the genus-zero topological recursion relation…
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}=…
In this paper, we firstly give the definition of dipersionless bigraded Toda hierarchy (dBTH) and introduce some Sato theory on dBTH. Then we define Orlov-Schulman's $\M_L$, $\M_R$ operator and give the additional Block symmetry of dBTH.…