Painleve Equation Research Articles

On the Riemann–Hilbert–Birkhoff Inverse Monodromy Problem Associated with the Third Painlevé Equation

A linear system of ordinary differential equations corresponding by the isomonodromy deformation method to the third Painleve equation is considered. The surjectivity of the monodromy map generated by this system is proven using the Riemann–Hilbert factorization method.

HYPERGEOMETRIC SOLUTION OF A CERTAIN POLYNOMIAL HAMILTONIAN SYSTEM OF ISOMONODROMY TYPE

In our previous work, a unified description as polynomial Hamiltonian systems was established for a broad class of the Schlesinger systems including the sixth Painleve equation and Garnier systems. The main purpose of this paper is to present particular solutions of this Hamiltonian system in terms of a certain generalization of Gauss' hypergeometric function. Key ingredients of the argument are the linear Pfaffian system derived from an integral representation of the hypergeometric function (with the aid of twisted de Rham theory) and Lax formalism of the Hamiltonian system.

Transcendence of solutions of q-Painlevé equation of type $${A_7^{(1)}}$$

In this paper we prove the transcendence of the solutions of the q-Painleve equation of type \({A_6^{(1)}}\). The q-Painleve equation of type \({A_6^{(1)}}\) is also called d-P II or q-P II and has a continuous limit to the Painleve equation of type II. Its symmetry is (A 1 + A 1)(1).

Hamiltonians associated with the third and fifth Painlevé equations

We obtain a Painleve-type differential equation for the simplest rational Hamiltonian associated with the fifth Painleve equation in the case γ ≠ 0, δ = 0. We prove the existence of Hamiltonians of a nonrational type associated with the fifth Painleve equation in the case γ ≠ 0, δ = 0. We obtain a generalization of the Garnier and Okamoto formulas for rational Hamiltonians associated with the third Painleve tequation.

The Painlevé equation of the second kind for the binary ionic transport in diffusion boundary layers near ion-exchange membranes at overlimiting current

Abstract In ion-exchange membrane systems such as electrodialysis, the overlimiting current phenomenon still remains a difficult topic to study due to the complicated solution method for the Nernst–Planck–Poisson model. In this study, the Nernst–Planck–Poisson equations were prepared to simulate the steady-state binary ionic transport in the diffusion boundary layer near an ion-exchange membrane. The system of differential equations was converted into the Painleve equation of the second kind in such a way that the converted model domain explicitly shows the transition point from the space-charge region to the electroneutral region in the diffusion boundary layer even before the differential equation is solved. Based on this property, mathematical expressions were proposed to estimate the limiting current density and the width of the space-charge region in the diffusion boundary layer near an ideally perm-selective ion-exchange membrane. The so-called Airy solution of the Painleve equation of the second kind was used to describe the ionic transport in the space-charge region. It was also found that the Airy function of the second kind with its derivative describes the behavior of the electric double layer developed from the ion-exchange membrane surface. In addition, a relatively simple numerical method, including a stability criterion, was used to solve the Painleve equation of the second kind to simulate the ionic transport in the diffusion boundary layer near an ion-exchange membrane.

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

We show that the Belavin-Polyakov-Zamolodchikov equation of the minimal model of conformal field theory with the central charge c = 1 for the Virasoro algebra is contained in a system of linear equations that generates the Schlesinger system with 2×2 tmatrices. This generalizes Suleimanov’s result on the Painleve equations. We consider the properties of the solutions, which are expressible in terms of the Riemann theta function.

Vortices and Polynomials

The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed. Classical polynomials, such as the Hermite polynomials, have roots that describe the equilibria of identical vortices on the line. Stationary and uniformly translating vortex configurations with vortices of the same strength but positive or negative orientation are given by the zeros of the Adler–Moser polynomials, which arise in the description of rational solutions of the Korteweg–de Vries equation. For quadrupole background flow, vortex configurations are given by the zeros of polynomials expressed as Wronskians of Hermite polynomials. Further, new solutions are found in this case using the special polynomials arising in the description of rational solutions of the fourth Painlevé equation.

Basic asymptotic expansions of solutions to the sixth Painlevé equation

Asymptotic expansions of solutions of three types are found for the sixth Painleve equation in its three singular points.

A Lax Formalism for the Elliptic Difference Painlevé Equation

A Lax formalism for the elliptic Painleve equation is presented. The construction is based on the geometry of the curves on P 1 ◊ P 1 and described in terms of the point configurations.

Universal character and q-difference Painlevé equations

The universal character is a polynomial attached to a pair of partitions and is a generalization of the Schur polynomial. In this paper, we introduce an integrable system of q-difference lattice equations satisfied by the universal character, and call it the lattice q-UC hierarchy. We regard it as generalizing both q-KP and q-UC hierarchies. Suitable similarity and periodic reductions of the hierarchy yield the q-difference Painleve equations of types \({(A_{2g+1}+A_1)^{(1)}(g \geq 1)}\) , \({D_5^{(1)}}\) , and \({E_6^{(1)}}\) . As its consequence, a class of algebraic solutions of the q-Painleve equations is rapidly obtained by means of the universal character. In particular, we demonstrate explicitly the reduction procedure for the case of type \({E_6^{(1)}}\) via the framework of τ based on the geometry of certain rational surfaces.

Middle Convolution and Heun's Equation

Heun's equation naturally appears as special cases of Fuchsian system of differential equations of rank two with four singularities by introducing the space of ini- tial conditions of the sixth Painleve equation. Middle convolutions of the Fuchsian system are related with an integral transformation of Heun's equation.

Bäcklund Transformations for First and Second Painlevé Hierarchies

We give Backlund transformations for first and second Painleve hierarchies. These Backlund transformations are generalization of known Backlund transformations of the first and second Painleve equations and they relate the considered hierarchies to new hierarchies of Painleve-type equations.

On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions

A crucial ingredient in the recent discovery by Ablowitz, Halburd, Herbst and Korhonen (2000, 2007) that a connection exists between discrete Painleve equations and (finite order) Nevanlinna theory is an estimate of the integrated average of log + |f(z + 1)/f(z)| on |z| = r. We obtained essentially the same estimate in our previous paper (2008) independent of Halburd et al. (2006). We continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences, quotients and logarithmic derivatives for finite order meromorphic functions. In addition to the potential applications of our new estimates in integrable systems, they are also of independent interest. In particular, our findings show that there are marked differences between the growth of meromorphic functions with Nevanlinna order less than and greater than one. We have established a difference analogue of the classical Wiman-Valiron type estimates for meromorphic functions with order less than one, which allow us to prove that all entire solutions of linear equations (with polynomial coefficients) of order less than one must have positive rational order of growth. We have also established that any entire solution to a first order algebraic equation (with polynomial coefficients) must have a positive order of growth, which is a difference analogue of a classical result of Polya.

Global asymptotics for polynomials orthogonal with exponential quartic weight

In this paper, we study the asymptotics of polynomials orthogonal with respect to the varying quartic weight ω(x) = e −nV (x) ,w hereV (x) = Vt(x) = x 4 4 + t 2 x 2 . We focus on the critical case t = −2, in the sense that for t −2, the support of the associated equilibrium measure is a single interval, while for t< −2, the support consists of two intervals. Globally uniform asymptotic expansions are obtained for z in three unbounded regions. These regions together cover the whole complex z-plane. In particular, in the region containing the origin, the expansion involves the Ψ function affiliated with the Hastings- McLeod solution of the second Painleve equation. Our approach is based on a modified version of the steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou (Ann. Math. 137 (1993), 295-370).

Local expansion of Painlevé VI transcendents around a fixed singularity

We study special solutions of the sixth Painleve equation with generic values of parameters whose linear monodromy can be calculated explicitly. We give two kinds of special solutions. One is the meromorphic solutions at a fixed singularity and the other is the solutions expressed in the Puiseux series at t=0. We give the respective character of these special solutions.

Asymptotic forms of solutions to the fourth Painlevé equation

We consider the fourth Painleve equation with nonzero values of its two complex parameters a and b . Asymptotic forms of power, complicated, exotic, and elliptic types for its solutions near the points x = 0 and x = ∞ are sought by methods of two- and three-dimensional power geometry. We obtain nine families of power asymptotic forms extended by asymptotic expansions and one family of elliptic asymptotic forms. Complicated and exotic asymptotic forms are absent.

Quantum Painlevé equations: from continuous to discrete and back

We present a cascade of quantum Painleve equations consisting in successive contiguity relations, whereupon starting form a continuous equations we obtain a discrete one, and continuous limits of the latter. We start from the quantum Painleve V and in the process derive the quantum form of continuous PIII which was missing in previous studies.

The Hermite–Krichever Ansatz for Fuchsian equations with applications to the sixth Painlevé equation and to finite-gap potentials

Several results including integral representation of solutions and Hermite– Krichever Ansatz on Heun’s equation are generalized to a certain class of Fuchsian differential equations, and they are applied to equations which are related with physics. We investigate linear differential equations that produce Painleve equation by monodromy preserving deformation and obtain solutions of the sixth Painleve equation which include Hitchin’s solution. The relationship with finite-gap potential is also discussed. We find new finite-gap potentials. Namely, we show that the potential which is written as the sum of the Treibich–Verdier potential and additional apparent singularities of exponents − 1 and 2 is finite-gap, which extends the result obtained previously by Treibich. We also investigate the eigenfunctions and their monodromy of the Schrodinger operator on our potential.

“Quantizations” of the second Painlevé equation and the problem of the equivalence of its L-A pairs

We show that the known Flaschka-Newell L-A pair for the second Painleve equation gives solutions to linear evolutionary equations similar to the quantum Schrodinger equations. Using the Fourier transform for distributions, we derive this pair from the classical Garnier pair.

A note on an open problem about the first Painlevé equation

In this note, we apply numerical analysis to the first Painleve equation, find the conditions for it to have oscillating solutions and therefore solve an open problem posed by Peter A. Clarkson.