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Although the solutions of Painlev\'e equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions, there do exist rational solutions for specialized values of the equation parameters. A very…
We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalently $A^{(1)}_{N-1}$ invariant Painlev\'e equations. This construction…
This paper focuses on the construction of rational solutions for the $A_{2n}$ Painlev\'e system, also called the Noumi-Yamada system, which are considered the higher order generalizations of PIV. In this even case, we introduce a method to…
Polynomials related to rational solutions of Painleve' equations satisfy certain difference equations. Conditions are given to acertain that all solutions really are polynomials.
We provide a complete classification and an explicit representation of rational solutions to the fourth Painlev\'e equation PIV and its higher order generalizations known as the $A_{2n}$-Painlev\'e or Noumi-Yamada systems. The construction…
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).
The problem of Painleve classification of ordinary differential equations lasting since the end of XIX century saw significant advances for the limited equation order, however not that much for the equations of higher orders. In this work…
In this study, a recursive solution technique in conjunction with generalized integrating factors is presented and applied to address first and second order linear differential equations. This approach demonstrates practical utility in…
The method of this paper is my original creation. A new method for solving linear differential equations is proposed in this paper. The important conclusion of this paper is that arbitrary order linear ordinary differential equations with…
We give an explicit determinant formula for a class of rational solutions of the Painlev\'e V equation in terms of the universal characters.
We study a one-dimensional ordinary differential equation modelling optical conveyor belts, showing in particular cases of physical interest that periodic solutions exist. Moreover, under rather general assumptions it is proved that the set…
We will consider four hierarchies of higher order analogues of the fourth (P4) and fifth (P5) Painleve equations. The necessary and sufficient conditions for having rational solutions will be presented. Further we well consider two more…
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…
We give a method to determine an explicit solution to a system of two inhomogeneous linear recursive sequences of higher order. Our approach can be used efficiently in solving certain combinatorial problems. We finish the paper by…
A variational method is presented for directly finding the bifurcation point of nonlinear equations as the saddle-node point of the extended nonlinear Rayleigh quotient. The method is applied for solving an open problem on the existence of…
Starting from the second Painlev\'{e} equation, we obtain Painlev\'{e} type equations of higher order by using the singular point analysis.
In this paper, we give an algorithm for finding general rational solutions of a given first-order ODE with parametric coefficients that occur rationally. We present an analysis, complete modulo Hilbert's irreducibility problem, of the…
In this article we will obtain real and complex solutions to the Painleve IV equation through supersymmetric quantum mechanics. Then we will classify them into real solution hierarchies and also the complex solution hierarchies, which are…
In this Letter we identify special systems of (an arbitrary number) N of first-order Ordinary Differential Equations with homogeneous polynomials of arbitrary degree M on their right-hand sides, which feature very simple explicit solutions;…
The aim of this paper is to give two new algorithms, which are elimination free, to find polynomial and rational solutions for a given holonomic system associated to a set of linear differential operators in the Weyl algebra D = k<x_1, ...,…