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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…
The paper addresses a conjecture of Shapiro and Tater on the similarity between two sets of points in the complex plane; on one side is the values of $t\in \mathbb{C}$ for which the spectrum of the quartic anharmonic oscillator in the…
We establish Liouville type theorems for degenerate conformally invariant equations.
Equivalence is established between a special class of Painleve VI equations parametrized by a conformal dimension $\mu$, time dependent Euler top equations, isomonodromic deformations and three-dimensional Frobenius manifolds. The…
Starting with a Riccati equation solved by hypergeometric functions, some sequences of rational solutions to Painleve' VI are obtained.
We consider the linear least squares problem with linear equality constraints (LSE problem) formulated as $\min_{x\in\mathbb{R}^{n}}\|Ax-b\|_2 \ \mathrm{s.t.} \ Cx = d$. Although there are some classical methods available to solve this…
A new integrable nonautonomous nonlinear ordinary difference equation is presented which can be considered to be a discrete analogue of the Painleve V equation. Its derivation is based on the similarity reduction on the two-dimensional…
We find and study coupled Painlev\'e II systems in dimension 4, which can be obtained by a degeneration from the systems of type ${A_4}^{(1)}$. We compare these systems with other types of coupled Painlev\'e II systems from the viewpoint of…
Considering a certain interpolation problem, we derive a series of elliptic difference isomonodromic systems together with their Lax forms. These systems give a multivariate extension of the elliptic Painlev\'e equation.
An algebro-geometric setting for the study of the Painlev\'e VI equation is introduced. Hamiltonian form of the equation is realized on a twisted relative cotangent bundle to the universal elliptic curve with labelled points of order two.…
Necessary conditions are obtained for certain types of rational delay differential equations to admit a non-rational meromorphic solution of hyper-order less than one. The equations obtained include delay Painlev\'e equations and equations…
We study the analytic properties and the critical behavior of the elliptic representation of solutions of the Painlev\'e 6 equation. We solve the connection problem for elliptic representation in the generic case and in a non-generic case…
We develop regularity theory for degenerate elliptic equations with the degeneracy controlled by a weight. More precisely, we show local boundedness and continuity of weak solutions under the assumption of a weighted Orlicz-Sobolev and…
Model theoretic ranks of solutions to Painleve equations are calculated, and the type of the generic solution of the second Painleve equation is shown to be disintegrated, strengthening a theorem of Nagloo. A question of Hrushovski and…
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).
This is a continuation of the first author's development of the theory of elliptic differential operators with edge degeneracies. That first paper treated basic mapping theory, focusing on semi-Fredholm properties on weighted Sobolev and…
We focus on Fuchsian equations with four accessory parameters and three singular points. We see that the Fuchsian equations admit a "degeneration scheme" in some sense, which is expected to give rise to a degeneration scheme of discrete…
We consider solutions to degenerate anisotropic elliptic equations in order to study their regularity. In particular we establish second-order estimates and enclose regularity results for the stress field. All our results are new even in…
While over the last century or more considerable effort has been put into the problem of finding approximate solutions for wave equations in general, and quantum mechanical problems in particular, it appears that as yet relatively little…
Problem of asymptotic description for global solutions to the six Painleve equations was investigated. Elliptic anzatzes and appropriate modulation equations were written out.