相关论文: Heun equation and Painlev\'e equation

We consider the orbits of a discrete Painlev\'e equation over finite fields and show that the number of points in such orbits satisfy the Hasse bound. The orbits turn out to lie on algebraic curves, whose defining polynomials are given…

An ergodic study of Painleve VI is developed. The chaotic nature of its Poincare return map is established for almost all loops. The exponential growth of the numbers of periodic solutions is also shown. Principal ingredients of 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…

We develop a dynamical study of the sixth Painleve equation for all parameters generalizing an earlier work for generic parameters. Here the main focus of this paper is on non-generic parameters, for which the corresponding character…

We show that for every second order Fuchsian linear differential equation $E$ with $n$ singularities of which $n-3$ are apparent there exists a hypergeometric equation $H$ and a linear differential operator with polynomial coefficients…

We establish, for the first time, a Zaremba-Hopf-Oleinik type boundary point lemma for uniformly elliptic partial differential equations in double divergence form, also known as stationary Fokker-Planck-Kolmogorov equations. As an…

Some fundamental solutions of radial type for a class of iterated elliptic singular equations including the iterated Euler equation are given.

A multidimensional generalization of Melvin's solution for an arbitrary simple Lie algebra $\cal G$ is considered. The gravitational model in $D$ dimensions, $D \geq 4$, contains $n$ 2-forms and $l \geq n$ scalar fields, where $n$ is the…

We study a special anisotropic XYZ-model on a periodic chain of an odd length and conjecture exact expressions for certain components of the ground state eigenvectors. The results are written in terms of tau-functions associated with…

Starting from the second Painlev\'{e} equation, we obtain Painlev\'{e} type equations of higher order by using the singular point analysis.

We exhibit a remarkable connection between sixth equation of Painleve list and infinite families of explicitly uniformizable algebraic curves. Fuchsian equations, congruences for group transformations, differential calculus of functions and…

We investigate the question of finding discrete Lax pairs for the six discrete Painlev\'e equations (Pn). The choice we make is to discretize the pairs of Garnier, once converted to matricial form.

This paper first discusses irreducibility of a Painlev\'e equation $P$. We explain how the Painlev\'e property is helpful for the computation of special classical and algebraic solutions. As in a paper of Morales-Ruiz we associate an…

The sextic oscillator is discussed as a potential obtained from the bi-confluent Heun equation after a suitable variable transformation. Following earlier results, the solutions of this differential equation are expressed as a series…

We consider a class of quasi-linear anisotropic elliptic equations, possibly degenerate or singular, which are of interest in several applications such as computer vision and continuum mechanics. We prove a Hopf Lemma as well as local and…

We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the…

Conditions are given for the second-order linear differential equation P3 y" + P2 y'- P1 y = 0 to have polynomial solutions, where Pn is a polynomial of degree n. Several application of these results to Schroedinger's equation are…

Fuchsian differential equations $H_j$ of order $j=3,\dots,6$ with three singular points and one accessory parameter are presented. The shift operators for $H_6$ are studied. They lead to assign the accessory parameter of $H_6$ a cubic…

We examine the series expansions of the solutions of the confluent Heun equation in terms of three different sets of the Kummer confluent hypergeometric functions. The coefficients of the expansions in general obey three-term recurrence…

In this paper classical solutions of the degenerate fifth Painlev\'e equation are classified, which include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions. Solutions of the degenerate fifth…