Related papers: Sur les Theoremes I et II de Painleve
A unique analytic continuation result is proved for solutions of a relatively general class of difference equations, using techniques of generalized Borel summability. This continuation allows for Painlev\'e property methods to be extended…
We study a second order ordinary differential equation corresponding to rotationally symmetric $p$-harmonic maps. We show unique continuation and Liouville's type theorems for positive solutions. We discuss the existence of bounded positive…
In this short note we give two examples of using the algebro-geometric theory of Painlev\'e equations to solve the Painlev\'e identification problem. The equations that we consider were recently obtained by M. van der Put and J. Top in…
In this paper we introduce a generalisation of the notion of holonomy for connections over a bundle map on a principal fibre bundle. We prove that, as in the standard theory on principal connections, the holonomy groups are Lie subgroups of…
Link between the Painleve property and the first integrals of nonlinear ordinary differential equations in polynomial form is discussed. The form of the first integrals of the nonlinear differential equations is shown to determine by the…
We study the underlying relationship between Painleve equations and infinite-dimensional integrable systems, such as the KP and UC hierarchies. We show that a certain reduction of these hierarchies by requiring homogeneity and periodicity…
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…
Locally variational systems of differential equations on smooth manifolds, having certain de Rham cohomology group trivial, automatically possess a global Lagrangian. This important result due to Takens is, how-ever, of sheaf-theoretic…
The Painlev\'e classification is one of the central problems in analytics theory of differential equations rooted in the XIX century. Although it saw many significant advances in analyzing certain classes of equations, the classification…
These course note first provide an introduction to secondary characteristic classes and differential cohomology. They continue with a presentation of a stable homotopy theoretic approach to the theory of differential extensions of…
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…
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…
The main subject of the paper is the so-called Discrete Painlev\'e-1 Equation (DP1). Solutions of the DP1 are classified under criterion of their behavior while argument tends to infinity. The appropriate theorems of existence are proved.
We show that several dichotomy theorems concerning the second level of the Borel hierarchy are special cases of the $\aleph_0$-dimensional generalization of the open graph dichotomy, which itself follows from the usual proof(s) of the…
The Painleve and weak Painleve conjectures have been used widely to identify new integrable nonlinear dynamical systems. The calculation of the integrals relies though on methods quite independent from Painlev\'e analysis. This paper…
Starting with univariate polynomial interpolation we arrive to a natural generalization of fundamental theorem of algebra for certain systems of multivariate algebraic equations.
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…
The notions of equivalence and strict equivalence for order one differential equations are introduced. The more explicit notion of strict equivalence is applied to examples and questions concerning autonomous equations and equations having…
We show that the local equivalence problem for second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also…
The classical Pythagoras theorem, binomial theorem, de Moivre's formula, and numerous other deductions are made using the uniqueness theorem for the initial value problems in linear ordinary differential equations.