Related papers: The extended $Z_N$-Toda hierarchy
To each partition function of cohomological field theory one can associate an Hamiltonian integrable hierarchy of topological type. The Givental group acts on such partition functions and consequently on the associated integrable…
A generalized Toda Lattice equation is considered. The associated linear problem (Lax representation) is found. For simple case N=3 the $\tau$-function Hirota form is presented that allows to construct an exast solutions of the equations of…
The universal Witham hierarchy is considered from the point of view of topological field theories. The $\tau$-function for this hierarchy is defined. It is proved that the algebraic orbits of Whitham hierarchy can be identified with various…
A higher dimensional analogue of the dispersionless KP hierarchy is introduced. In addition to the two-dimensional ``phase space'' variables $(k,x)$ of the dispersionless KP hierarchy, this hierarchy has extra spatial dimensions…
Using the matrix-resolvent method and a formula of the second-named author on the $n$-point function for a KP tau-function, we show that the tau-function of an arbitrary solution to the Toda lattice hierarchy is a KP tau-function. We then…
Properties of Hamiltonian symmetry flows on hyperbolic Euler-type Liouvillean equations E' are analyzed. Description of their Noether symmetries assigned to the integrals for these equations is obtained. The integrals provide Miura…
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…
We construct an integrable hierarchy in the form of Hirota quadratic equations (HQE) that governs the Gromov--Witten (GW) invariants of the Fano orbifold projective curve $\mathbb{P}^1_{a_1,a_2,a_3}$. The vertex operators in our…
Some new developments in constrained Lax integrable systems and their applications to physics are reviewed. After summarizing the tau function construction of the KP hierarchy and the basic concepts of the symmetry of nonlinear equations,…
A continuum limit of the Toda lattice field theory, called the SDiff(2) Toda equation, is shown to have a Lax formalism and an infinite hierarchy of higher flows. The Lax formalism is very similar to the case of the self-dual vacuum…
Using the free fermions technique and non-abelian bosonization rules we introduce the multi-component Pfaff-Toda hierarchy. The tau-function is defined as vacuum expectation value of a Clifford group element of the algebra of…
A $q$-analogue of the tau function of the modified KP hierarchy is defined by a change of independent variables. This tau function satisfies a system of bilinear $q$-difference equations. These bilinear equations are translated to the…
The multicomponent 2D Toda hierarchy is analyzed through a factorization problem associated to an infinite-dimensional group. A new set of discrete flows is considered and the corresponding Lax and Zakharov--Shabat equations are…
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…
The construction of the master T-operator recently suggested in Alexandrov et al. (arXiv:1112.3310) is applied to integrable vertex models and associated quantum spin chains with trigonometric R-matrices. The master T-operator is a…
An infinite class of fermionic flows of the N=(1|1) superconformal Toda lattice hierarchy is constructed and their algebraic structure is studied. We completely solve the semi-infinite N=(1|1) Toda lattice and chain hierarchies and derive…
We consider the Dubrovin--Frobenius manifold of rank $2$ whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces,…
We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical W-algebra; we compute the central charge…
Adler, Shiota and van Moerbeke obtained for the KP and Toda lattice hierarchies a formula which translates the action of the vertex operator on tau--functions to an action of a vertex operator of pseudo-differential operators on wave…
Extending the gauge-invariance principle for $\tau$ functions of the standard bilinear formalism to the supersymmetric case, we define ${\cal N}=1$ supersymmetric Hirota bilinear operators. Using them we bilinearize supersymmetric nonlinear…