Related papers: Bispectrality for the quantum open Toda chain
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…
We show that the k-double Schur functions defined by the authors, and the quantum double Schubert polynomials studied by Kirillov and Maeno and by Ciocan-Fontanine and Fulton, can be obtained from each other by an explicit rational…
The extended flow equations of the multi-component Toda hierarchy are constructed. We give the Hirota bilinear equations and tau function of this new extended multi-component Toda hierarchy(EMTH). Because of logarithmic terms, some extended…
A way to obtain the series solutions of the 1 + 2 dimensional continuous Toda chain is presented.
We construct generalized additional symmetries of a two-component BKP hierarchy defined by two pseudo-differential Lax operators. These additional symmetry flows form a Block type algebra with some modified(or additional) terms because of a…
This paper proposes a method for identifying and classifying integrable nonlinear equations with three independent variables, one of which is discrete and the other two are continuous. A characteristic property of this class of equations,…
In this paper, we give finite dimensional exponential solutions of the bigraded Toda Hierarchy(BTH). As an specific example of exponential solutions of the BTH, we consider a regular solution for the $(1,2)$-BTH with $3\times 3$-sized Lax…
Sequences of bivariate orthogonal polynomials written as vector polynomials of increasing size satisfy a couple of three term relations with matrix coefficients. In this work, introducing a time-dependent parameter, we analyse a Lax-type…
We consider the relation of the multi-component 2D Toda hierarchy with matrix orthogonal and biorthogonal polynomials. The multi-graded Hankel reduction of this hierarchy is considered and the corresponding generalized matrix orthogonal…
At the turn of the century, Etingof and Sevostyanov independently constructed a family of quantum integrable systems, quantizing the open Toda chain associated to a simple Lie group $G$. The elements of this family are parameterized by…
We establish the pluri-Lagrangian structure for families of B\"acklund transformations of relativistic Toda-type systems. The key idea is a novel embedding of these discrete-time (one-dimensional) systems into certain two-dimensional…
The quantum double construction of a $q$-deformed boson algebra possessing a Hopf algebra structure is carried out explicitly. The $R$-matrix thus obtained is compared with the existing literature.
We consider a hierarchy of the natural type Hamiltonian systems of $n$ degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of $2\times 2$…
We investigate an N-state spin model called quantum relativistic Toda chain and based on the unitary finite dimensional representations of the Weyl algebra with q being N-th primitive root of unity. Parameters of the finite dimensional…
Results on the finite nonperiodic Toda lattice are extended to some generalizations of the system: The relativistic Toda lattice, the generalized Toda lattice associated with simple Lie groups and the full Kostant-Toda lattice. The areas…
On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear systems is obtained, and the integration scheme for such equations is proposed.
For some integrable systems, such as the open Toda molecule, the spectral curve of the Lax representation becomes the graph $C = \{(\lambda,z) \mid z = A(\lambda)\}$ of a function $A(\lambda)$. Those integrable systems provide an…
Differential-difference integrable exponential type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras $A_2$, $B_2$, $C_2$, $G_2$ the complete sets…
The reciprocal link between the reduced Ostrovsky equation and the $A_2^{(2)}$ two-dimensional Toda system is used to construct the $N$-soliton solution of the reduced Ostrovsky equation. The $N$-soliton solution of the reduced Ostrovsky…
The relativistic Toda lattice equation is decomposed into three Toda systems, the Toda lattice itself, B\"acklund transformation of Toda lattice and discrete time Toda lattice. It is shown that the solutions of the equation are given in…