Backlund Transformations and Hierarchies of Exact Solutions for the Fourth Painleve Equation and their Application to Discrete Equations

We take a third-order approach to the fourth Painlev\'e equation and indicate the value of such an approach to other second-order ODEs in the Painlev\'e-Gambier list of 50.

In this work the supersymmetric technique is applied to the truncated oscillator to generate Hamiltonians ruled by second and third-order polynomial Heisenberg algebras, which are connected to the Painlev\'e IV and Painlev\'e V equations respectively. The aforementioned connection is exploited to produce particular solutions to both non-linear differential equations and the B\"acklund transformations relating them.

A q-difference analogue of the fourth Painlev\'e equation is proposed. Its symmetry structure and some particular solutions are investigated.

The fourth-order analog to the first Painlev\'{e} equation is studied. All power expansions for solutions of this equation near points $z=0$ and $z=\infty$ are found. The exponential additions to the expansion of solution near $z=\infty$ are computed. The obtained results confirm the hypothesis that the fourth-order analog of the first Painlev\'{e} equation determines new transcendental functions. By means of the methods of power geometry the basis of the plane lattice is als...

We consider the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions of the inhomogeneous Painlev\'e II equation $$ u"(x)=2u^3(x)+xu(x)-\alpha \qquad \textrm{for } \alpha \in \mathbb{R} \textrm{ and } |\alpha| > \frac{1}{2}. $$ These solutions are obtained from the classical Ablowitz-Segur and Hastings-McLeod solutions via the B\"acklund transformation, and satisfy the same asymptotic behaviors when $x \to \pm \infty$. For $|\alpha| > 1/2$, we show that the quasi-Ablowitz...

In this paper we discuss Airy solutions of the second Painlev\'e equation (\mbox{\rm P$_{\rm II}$}) and two related equations, the Painlev\'e XXXIV equation ($\mbox{\rm P}_{34}$) and the Jimbo-Miwa-Okamoto $\sigma$ form of \mbox{\rm P$_{\rm II}$}\ (\mbox{\rm S$_{\rm II}$}), are discussed. It is shown that solutions which depend only on the Airy function $\mathop{\rm Ai}\nolimits(z)$ have a completely difference structure to those which involve a linear combination of the Airy...

This paper is an addendum to earlier papers \cite{R1,R2} in which it was shown that the unstable separatrix solutions for Painlev\'e I and II are determined by $PT$-symmetric Hamiltonians. In this paper unstable separatrix solutions of the fourth Painlev\'e transcendent are studied numerically and analytically. For a fixed initial value, say $y(0)=1$, a discrete set of initial slopes $y'(0)=b_n$ give rise to separatrix solutions. Similarly, for a fixed initial slope, say $y'(...

We will classify all rational transformations which change the confluent hypergeometric equations to linear equations of the Painleve type from the first to the fifth. We show such rational transformations correspond to almost all of algebraic solutions of the Painleve equations from the first to fifth up to the Backlund transformations.

A Riemann-Hilbert problem for a $q$-difference Painlev\'e equation, known as $q\textrm{P}_{\textrm{IV}}$, is shown to be solvable. This yields a bijective correspondence between the transcendental solutions of $q\textrm{P}_{\textrm{IV}}$ and corresponding data on an associated $q$-monodromy surface. We also construct the moduli space of $q\textrm{P}_{\textrm{IV}}$ explicitly.

The Yablonskii-Vorob'ev polynomials $y_{n}(t)$, which are defined by a second order bilinear differential-difference equation, provide rational solutions of the Toda lattice. They are also polynomial tau-functions for the rational solutions of the second Painlev\'{e} equation ($P_{II}$). Here we define two-variable polynomials $Y_{n}(t,h)$ on a lattice with spacing $h$, by considering rational solutions of the discrete time Toda lattice as introduced by Suris. These polynomia...