相关论文: A fully noncommutative analog of the Painlev\'e IV…
We construct solutions of an infinite Toda system and an analogue of the Painlev'e II equation over noncommutative differential division rings in terms of quasideterminants of Hankel matrices.
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…
A q-difference analogue of the fourth Painlev\'e equation is proposed. Its symmetry structure and some particular solutions are investigated.
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…
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…
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…
We examine quantum extensions of the continuous Painlev\'e equations, expressed as systems of first-order differential equations for non-commuting objects. We focus on the Painlev\'e equations II, IV and V. From their auto-B\"acklund…
We give a new approach to the symmetries of the Painlev\'e equations $P_{V},P_{IV},P_{III}$ and $P_{II}$, respectively. Moreover, we make natural extensions to fourth-order analogues for each of the Painlev\'e equations $P_{V}$ and…
A study is presented of two-dimensional superintegrable systems separating in Cartesian coordinates and allowing an integral of motion that is a fourth order polynomial in the momenta. All quantum mechanical potentials that do not satisfy…
Generalizations of GL(n) abelian Toda and $\widetilde{GL}(n)$ abelian affine Toda field theories to the noncommutative plane are constructed. Our proposal relies on the noncommutative extension of a zero-curvature condition satisfied by…
We present certain general structures related to the solutions of Painlev\'e equation II and to the solutions of the differential equation satisfied by the corresponding Hamiltonian equations, together with the tau functions. By taking…
In a recent breakthrough Kanzieper showed that it is possible to obtain exact nonperturbative Random Matrix results from the replica limit of the corresponding Painlev\'e equation. In this article we analyze the replica limit of the Toda…
We give a survey of the connection between orthogonal polynomials, Toda lattices and related lattices, and Painlev\'e equations (discrete and continuous).
We use methods from dynamical systems to study the fourth Painleve equation PIV. Our starting point is the symmetric form of PIV, to which the Poincare compactification is applied. The motion on the sphere at infinity can be completely…
We study the asymptotic behaviour of solutions of the fourth Pain\-lev\'e equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalisation of phase space described by Okamoto. We show…
Painleve transcendents are usually considered as complex functions of a complex variable, but in applications it is often the real cases that are of interest. Under a reasonable assumption (concerning the behavior of a dynamical system…
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.
Novel hybrid Ermakov-Painlev\'{e} IV systems are introduced and an associated Ermakov invariant is used in establishing their integrability. B\"{a}cklund transformations are then employed to generate classes of exact solutions via the…
The Painlev\'e equations possess transcendental solutions $y(t)$ with special initial values that are symmetric under rotation or reflection in the complex $t$-plane. They correspond to monodromy problems that are explicitly solvable in…
A totally nonnegative matrix is a real-valued matrix whose minors are all nonnegative. In this paper, we concern with the totally nonnegative structure of the finite Toda lattice, a classical integrable system, which is expressed as a…