相关论文: Noncommutative Toda Chains, Hankel Quasideterminan…

We study a fully noncommutative generalisation of the commutative fourth Painlev\'e equation that possesses solutions in terms of an infinite Toda system over an associative unital division ring equipped by a derivation.

We consider the Hankel determinant formula of the $\tau$ functions of the Toda equation. We present a relationship between the determinant formula and the auxiliary linear problem, which is characterized by a compact formula for the $\tau$…

In this paper, I construct the Darboux transformations for the non-commutative Toda solutions at $ n=1 $ with the help of linear systems whose compatibility condition yields zero curvature representation of associated systems of non-linear…

Two types of determinant representations of the rational solutions for the Painlev\'e II equation are discussed by using the bilinear formalism. One of them is a representation by the Devisme polynomials, and another one is a Hankel…

A noncommutative version of the semi-discrete Toda equation is considered. A Lax pair and its Darboux transformations and binary Darboux transformations are found and they are used to construct two families of quasideterminant solutions.

In this article non-abelian version of quantum Painlev\'e II equation is presented with Its quasideterminant solutions has been derived by using the Darboux transformations. This non-abelian quantum Painlev\'e II equation may be considered…

We consider the iso-spectral real manifolds of tridiagonal Hessenberg matrices with real eigenvalues. The manifolds are described by the iso-spectral flows of indefinite Toda lattice equations introduced by the authors [Physica, 91D (1996),…

We have derived a non-abelian analog for the two-dimensional discrete Toda lattice which possesses solutions in terms of quasideterminants and admits Lax pairs of different forms. Its connection with non-abelian analogs for several…

Two families of solutions of a generalized non-Abelian Toda lattice are considered. These solutions are expressed in terms of quasideterminants, constructed by means of Darboux and binary Darboux transformations. As an example of the…

The Yablonskii-Vorob'ev polynomials $y_{n}(t)$, which are defined by a second order bilinear differential-difference equation, provide rational solutions of the Toda lattice. They are also polynomial tau-functions for the rational solutions…

Determinant formulas for the general solutions of the Toda and discrete Toda equations are presented. Application to the $\tau$ functions for the Painlev\'e equations is also discussed.

In [1], a generalized type of Darboux transformations defined in terms of a twisted derivation was constructed in a unified form. Such twisted derivations include regular derivations, difference operators, superderivatives and…

We find explicit (multisoliton) solutions for nonabelian integrable systems such as periodic Toda field equations, Langmuir equations, and Schrodinger equations for functions with values in any associative algebra. The solution for…

The problem of construction of integrable boundary conditions for the discrete Toda chain is considered. The restricted chains for properly chosen closure conditions are reduced to the well known discrete Painlev\'e equations $dP_{III}$,…

In this paper we consider a Hankel determinant formula for generic solutions of the Painleve' II equation. We show that the generating functions for the entries of the Hankel determinants are related to the asymptotic solution at infinity…

The noncommutative analogues of the nonisospectral Toda and Lotka-Volterra lattices are proposed and studied by performing nonisopectral deformations on the matrix orthogonal polynomials and matrix symmetric orthogonal polynomials without…

The extension of Painlev\'e equations to noncommutative spaces has been considering extensively in the theory of integrable systems and it is also interesting to explore some remarkable aspects of these equations such as Painlev\'e…

We derive determinant representations and nonlinear differential equations for the scaled 2-point functions of the 2D Ising model on the cylinder. These equations generalize well-known results for the infinite lattice (Painlev\'e III…

We integrate nonabelian Toda field equations for root systems of types A, B, C, for functions with values in any associative algebra. The solution is expressed via quasideterminants. In the appendix we review some results concerning…

Consider the Laguerre polynomials and deform them by the introduction in the measure of an exponential singularity at zero. In [Chen Y., Its A., J. Approx. Theory 162 (2010), 270-297, arXiv:0808.3590] the authors proved that this…