相关论文: A Four-parametric Rational Solution to Painleve' V…

In this paper, we analyze the asymptotic behaviour of the poles of certain rational solutions of the fifth Painlev\'e equation. These solutions are constructed by relating the corresponding tau function to a Hankel determinant of a certain…

The third Painlev\'e equation in its generic form, often referred to as Painlev\'e-III($D_6$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x}\right)^2-\frac{1}{x}\frac{{\rm d}u}{{\rm…

We show that the Painleve equations P3-P5 can be derived (in a unified way) from a periodic sequence of Darboux transformations for a Schrodinger problem with quadratic eigenvalue dependency. The general problem naturally divides into three…

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…

This article deals with the asymptotic behavior of fourth order differential equation where the coefficients are perturbations of linear constant coefficient equation. We introduce a change of variable and deduce that the new variable…

We consider a $q$-Painlev\'e IV equation which is the $A_4^{(1)}$-surface type in the Sakai's classification. We find three distinct types of classical solutions with determinantal structures whose elements are basic hypergeometric…

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…

We classify rational solutions of a specific type of the set theoretical version of the pentagon equation. That is, we find all quadrirational maps $R:(x,y)\mapsto (u(x,y),v(x,y)),$ where $u, v$ are two rational functions on two arguments,…

In this paper rational solutions of the fifth Painlev\'e equation are discussed. There are two classes of rational solutions of the fifth Painlev\'e equation, one expressed in terms of the generalised Laguerre polynomials, which are the…

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…

In this paper, we show that the B\"acklund transformations of Painlev\'e equations come from birational maps of rational surfaces constructed by Okamoto as the spaces of initial conditions. The simultaneous resolutions of rational double…

We present a general scheme to derive higher-order members of the Painleve VI (PVI) hierarchy of ODE's as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the PVI equation…

A $\tau$ function formalism for Sakai's elliptic Painlev'e equation is presented. This establishes the equivalence between the two formulations by Sakai and by Ohta-Ramani-Grammaticos. We also give a simple geometric description of the…

A proposed solution to the Riemann Hypothesis

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…

This paper gives parametric solutions to quartic equations of the type,(4-3-3),(4-4-4),(4-5-5) and (4-6-6), According to Lander, Parkin, and Selfridge (2) conjecture, there are non-trivial solutions of the quartic…

The $2\times 2$ Schlesinger system for the case of four regular singularities is equivalent to the Painlev\'e VI equation. The Painlev\'e VI equation can in turn be rewritten in the symmetric form of Okamoto's equation; the dependent…

Among the reductions of the resonant three-wave interaction system to six-dimensional differential systems, one of them has been specifically mentioned as being linked to the generic sixth Painleve' equation P6. We derive this link…

In this paper, we study Riccati solutions of Painlev\'e equations from a view point of geometry of Okamoto-Painlev\'e pairs $(S,Y)$. After establishing the correspondence between (rational) nodal curves on $S-Y$ and Riccati solutions, we…

Painleve equations belong to the class y'' + a_1 {y'}^3 + 3 a_2 {y'}^2 + 3 a_3 y' + a_4 = 0, where a_i=a_i(x,y). This class of equations is invariant under the general point transformation x=Phi(X,Y), y=Psi(X,Y) and it is therefore very…