## Beyond the Painlevé Equations

At the turn of the 20th Century Painlevé and his school found a "special" set of second-order non-linear
differential equations (six in all, labelled I to VI). What was significant about these equations was that
certain solutions of these generalised very broad categories of special functions known at the time, such as the hypergeometric function.
In this way they were thought of as the non-linear "generalisations" of such special functions.
However their solutions were not limited to these alone and the most general solutions were thought to be "transcendental" in some sense.
The Painlevé Transcendents now occupy a chapter (Chapter 32) in the revised version of the old Natural Bureau of Standards handbook by Abramowitz and Stegun or
its online version as the NIST Digital Library of Mathematical Functions http://dlmf.nist.gov.

The Painlevé equations have been studied from many vantage points in the modern era:
through the *algebraic geometry* of their spaces of initial conditions **Sak_2001** using tools employed for the resolution of singularities;
the algebraic structure of their *symmetries* (also known as B\"acklund transformations) using finite and crystallographic *reflection groups* **Nou_2004**;
classical analytical approaches by imposing the *Painlevé property* on the solution to the differential equation {$ y(x) $} in the complex {$x$}-plane **Inc_1956**;
or the *monodromy preserving property* **IKSY_1991**, **FIKN_2006** of one of the pair of linear partial differential equations (known as a Lax pair) for which
the Painlevé equation is a compatibility condition.

Currently a program to enumerate all of the higher order analogues of the Painlevé equations, at least to the next level
of four-dimensional or four accessory parameters, has been initiated by H. Kawakami, A. Nakamura and H. Sakai
in the period 2012-15. The first phase of the task was achieved with the construction of isomonodromy deformation problems for
Fuchsian differential equations, extending the four singularity case corresponding to Painlevé VI, in **Sak_2010**
by the techniques of addition and middle convolution. This yielded the four master cases: the Garnier systems; the Fuji-Suzuki systems; the Sasano systems and the matrix Painlevé systems.
These four master cases were extended by constructing from them degeneration schemes of singularity confluence in **KNS_2012** and **KNS_2013**, and yields four families.
However such a classification treats only the unramified cases and only very recently have ramified cases been studied, with a partial list of results given in **Kaw_2015**.
A ramified case refers to a situation where the formal asymptotic solution as {$ x\to \infty $}, {$ y(x) \sim e^{P(x)} $} possesses fractional exponents for {$ P(x) $} rather than integer-valued ones.

Recently the author and P. Forrester studied an integrable equation arising from a problem in random matrix theory **WF_2016**. Of the four families described above
the only family relevant to the random matrix example, a certain higher order analogue of Painlevé III, is the Fuji-Suzuki family which have {$ 3\times 3 $} Lax pairs.
There are nine unramified and a further seven ramified cases in this degeneration scheme. However examining the details we find that it does not appear fit into this collection and
so it seems that the classification is still incomplete. It is the intention of the project to see how this example, and potentially others, can fit into the broad classification
and thus complete this task. This would suit a student who has a background in classical analysis and allied pure mathematics subjects such as group theory.

**FIKN_2006**
A.S. Fokas, A.R. Its, A.A. Kapaev, and V.Yu. Novokshenov.
*Painlevé transcendents. The Riemann-Hilbert Approach*, volume 128 of *Mathematical Surveys and Monographs*. American Mathematical Society, Providence, RI, 2006.

**Inc_1956**
E.L. Ince.
*Ordinary Differential Equations*, Dover Publications, New York, 1956.

**IKSY_1991**
K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida.
*From Gauss to Painlevé. A modern theory of special functions*. Aspects of Mathematics, E16. Friedr. Vieweg & Sohn, Braunschweig, 1991.

**Kaw_2015**
H. Kawakami.
*Four-dimensional Painlevé-type equations associated with ramified linear equations*. unpublished, 2015.

**KNS_2012**
H. Kawakami, A. Nakamura, and H. Sakai.
*Degeneration scheme of 4-dimensional Painlevé-type equations*. Sep 2012. arXiv:1209.3836.

**KNS_2013**
H. Kawakami, A. Nakamura, and H. Sakai.
*Toward a classification of four-dimensional Painlevé-type equations*.
In A. Dzhamay, K. Maruno, and V.U. Pierce, editors, *Algebraic and geometric aspects of integrable systems and random matrices*, volume 593 of *Contemporary Mathematics*, pg 143--161. Amer. Math. Soc., Providence, RI, 2013.
Joint Mathematics Meeting on Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, Boston, MA, JAN 06-07, 2012.

**Nou_2004**
M. Noumi.
*Painlevé equations through symmetry*, volume 223 of *Translations of Mathematical Monographs*. American Mathematical Society, Providence, RI, 2004.

**Sak_2001**
H. Sakai.
*Rational surfaces associated with affine root systems and geometry of the Painlevé equations*. Comm. Math. Phys., 220(1):165--229, 2001.

**Sak_2010**
H. Sakai.
*Isomonodromic deformation and 4-dimensional Painlevé type equations*. Technical report, 2010. unpublished.

**WF_2016**
N.S. Witte and P.J. Forrester.
*Singular values of products of Ginibre random matrices*. 2016.