This is a call for contributions to a special issue of Journal of Physics A: Mathematical and General
entitled `One hundred years of Painlevé VI, the Fuchs–Painlevé equation'. The motivation behind this
special issue is to celebrate the centenary of the discovery of this famous differential equation. The Editorial Board has invited P A Clarkson, N Joshi, M Mazzocco, F W Nijhoff and M Noumi to serve as Guest Editors for the issue.
The nonlinear ordinary differential equation, which is nowadays known as the Painlevé VI (PVI)
equation, is one of the most important differential equations in mathematical physics.
It was discovered 100 years ago by Richard Fuchs (son of the famous mathematician Lazarus Fuchs)
and reported for the first time in Comptes Rendus de l'Academie des Sciences Paris141 555–8 (1905). Gambier, in his seminal paper of 1906, included this equation
as the top equation in the list of what are now known as the six Painlevé transcendental equations.
The Painlevé list emerged from the work on the classification of all ordinary second-order
differential equations whose general solution are `uniform', in the sense that there are no
movable (i.e. as a function of the initial data) singularities (meaning branch points) worse than poles.
The latter is known as the Painlevé property.
As the top equation in the Painlevé list of transcendental equations, the importance of PVI can be
appreciated by recognizing that this is a universal differential equation,
which is the most general (in terms of number of free parameters) of the known second order ODEs defining
nonlinear special functions. As such, parallels can be drawn between the role played by PVI
transcendents in the nonlinear case and the hypergeometric functions at the linear level. In fact, the
monograph From Gauss to Painlevé by K Iwasaki, H Kimura, S Shimomura and M Yoshida
(Vieweg, 1991), draws very clearly the line stretching over more than 150 years of special function
theory in which PVI is placed as the key equation. In recent years these lines have been extended into
the discrete domain, i.e. the field of nonlinear ordinary difference equations, and discrete
analogues of PVI have opened entirely new fields of investigation.
The aim of the special issue, dedicated specifically to the PVI equations and its avatars rather
than to general Painlevé theory, is to consolidate the state-of-the-art knowledge of the properties
of this equation and to highlight modern developments (including generalizations of PVI, such as
the Garnier system, as well as discrete versions of the equation). The issue should be a repository
of high-quality original research papers as well as some invited topical reviews.
Scope of the special issue
The special issue is dedicated to the study of the Painlevé VI equation, its
solutions and properties, and to its generalizations—either in the direction of higher-order
differential equations associated with PVI (and related Garnier and Schlesinger systems), or in the
direction of difference analogues of the equation. The special issue will welcome contributions that
go into the analysis (including asymptotic theory) of Painlevé VI transcendents, the corresponding
monodromy theory, the representation theory aspects, the underlying algebraic geometry of the solution
manifolds, associated combinatorics and random matrix theory, as well as q-difference and discrete
versions of the equation, and last but not least applications in physics.
Papers dealing primarily with Painlevé equations other than PVI, or with general Painlevé theory, are not
encouraged as these would deflect the contents of the special issue from its
specific celebrational motivation.
Editorial policy
All contributions to the special issue will be refereed in accordance
with the refereeing policy of the journal. The Guest Editors will reserve the right to judge
whether a contribution fits the scope of the topic of the special issue.
Guidelines for preparation of contributions
•
We aim to publish the special issue in the first half of 2006, in order not to lose the connection
with the celebrational year 2005, marking the 100-year anniversary of the discovery of PVI.
To realize this, the deadline for contributed papers will be 31 January 2006.
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There is a page limit of 16 printed pages (approximately 9600 words) per contribution.
For submitted papers exceeding this length the Guest Editors reserve the right to request a
reduction in length. Further advice on document preparation can be found at www.iop.org/Journals/jphysa
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Contributions to the special issue should if possible be submitted electronically by web upload
at www.iop.org/Journals/jphysa, or by email to jphysa@iop.org, quoting `J. Phys. A Special Issue:
Painlevé VI'. Submissions should ideally be in standard LaTeX form; we are, however, able to
accept most formats including Microsoft Word. Please see the website for further information
on electronic submissions.
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Authors unable to submit electronically may send hard-copy contributions to: Publishing
Administrators, Journal of Physics A, Institute of Physics Publishing, Dirac House, Temple Back,
Bristol BS1 6BE, UK, enclosing electronic code on floppy disk if available and quoting
`J. Phys. A Special Issue: Painlevé VI'.
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All contributions should be accompanied by a read-me file or covering letter giving the postal
and email address for correspondence. The Publishing Office should be notified of any
subsequent change of address.
The special issue will be published in the paper and online version of the journal. The
corresponding author of each contribution will receive a complimentary copy of the issue.