Related papers: Painleve equations, vector fields, and ranks in di…
In this manuscript we make major progress classifying algebraic relations between solutions of Painlev\'e equations. Our main contribution is to establish the algebraic independence of solutions of various pairs of equations in the…
Polynomials related to rational solutions of Painleve' equations satisfy certain difference equations. Conditions are given to acertain that all solutions really are polynomials.
Starting from the second Painlev\'{e} equation, we obtain Painlev\'{e} type equations of higher order by using the singular point analysis.
For an arbitrary ordinary second order differential equation a test is constructed that checks if this equation is equivalent to Painleve I, II or Painleve III with three zero parameters equations under the substitutions of variables. If it…
Differential equations for the special polynomials associated with the rational solutions of the second Painleve hierarchy are introduced. It is shown rational solutions of the Korteveg - de Vries hierarchy can be found taking the…
We consider semiclassical orthogonal polynomials on the unit circle associated with a weight function that satisfy a Pearson-type differential equation involving two polynomials of degree at most three. Structure relations and difference…
The relation between the Painleve equations and the algebraic equations with the catastrophe theory point of view are considered. The asymptotic solutions with respect to the small parameter of the Painleve equations different types are…
We utilise a recent approach via the so-called re-scaling method to derive a unified and comprehensive theory of the solutions to Painleve's differential equations (I), (II) and (IV), with emphasis on the most elaborate equation (IV).
We will explain how some new algebraic solutions of the sixth Painleve equation arise from complex reflection groups, thereby extending some results of Hitchin and Dubrovin-Mazzocco for real reflection groups. The problem of finding…
It is demonstrated that a certain integral equation can be solved using the Painleve equation of third kind. Inversely, a special solution of this Painleve equation can be expressed as the ratio of two infinite series of spheroidal…
In this article, we study model-theoretic properties of algebraic differential equations of order $2$, defined over constant differential fields. In particular, we show that the set of solutions of a general differential equation of order…
Special polynomials associated with rational solutions of the second Painlev'e equation and other equations of its hierarchy are studied. A new method, which allows one to construct each family of polynomials is presented. The structure of…
Special polynomials associated with rational solutions of the second Painlev\'{e} equation and other members of its hierarchy are discussed. New approach, which allows one to construct each polynomial is presented. The structure of the…
Differential equations with the Painlev\'e property have been studied extensively due to their appearance in many branches of mathematics and their applicability in physics. Although a modern, differential algebraic treatment of the order…
We prove the meromorphy of solutions for a wide class of ordinary differential equations. These equations are given by invariant manifolds of non-linear partial differential equations integrable by the inverse scattering method. Some higher…
We note here, in answer to a question of Poizat, that the Morley and Lascar ranks need not coincide in differentially closed fields. We approach this through the (perhaps) more fundamental issue of the variation of Morley rank in families.
New special polynomials associated with the rational solutions of analogue to the Painleve hierarchies are introduced. The Hirota relations for these special polynomials are found. Differential - difference hierarchies for finding special…
We propose multidimensional versions of the Painlev\'e VI equation and its degenerations. These field theories are related to the isomonodromy problems of flat holomorphic infinite rank bundles over elliptic curves and take the form of…
We lecture on fundamental Painleve's early Theorems on first order ordinary differential equations with many examples. We end-up with two conjectures about the global analytic continuation of holonomy maps locally defined by Theorem II.
The Painlev\'{e} equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of ``classical" weights multiplied by suitable ``deformation factors", usually dependent on a…