The Painleve equations-nonlinear special functions.
Peter Clarkson
Institute of Mathematics and Statistics, University of Kent, UK
Tuesday, November 20, 2001 11:00-12:00,
Room 145, NIST North (820)
Gaithersburg
Tuesday, November 20, 2001 09:00-10:00,
Room 5000
Boulder
Abstract:
The six Painleve equations (PI-PVI) were first derived around
the turn of the century in an investigation by Painleve and his
colleagues in a study of nonlinear second-order ordinary
differential equations. There has been considerable interest in
Painleve equations over the last few years primarily due to the
fact that they arise as reductions of soliton equations solvable by
inverse scattering. Further, the Painleve equations are regarded
as completely integrable equations and possess solutions which can
be expressed in terms of the solutions of linear integral equations.
Although first discovered from strictly mathematical
considerations, the Painleve equations have appeared in
several important physical applications including statistical
mechanics, plasma physics, nonlinear waves, quantum gravity,
quantum field theory, general relativity, nonlinear optics and
fibre optics.
The Painleve equations may also be thought of as nonlinear analogues
of the classical special functions such as Bessel functions.
They possess hierarchies of rational solutions and one-parameter
families of solutions expressible in terms of the classical
special functions for speical values of the parameters. For example,
there exist solutions of PII-PVI that are expressed in terms of
Airy, Bessel, parabolic cylinder, Whittaker and hypergeometric
functions, respectively. Further the Painleve equations admit
symmetries under affine Weyl groups which are related to the
associated Backlund transformations.
In this talk I shall describe some of the remarkable
properties which the Painleve equations possess (including connection
formulae, Backlund transformations, associated discrete equations
and hierarchies of exact solutions) and some of their applications.
Contact: D. W. LozierNote: Visitors from outside NIST must contact
Robin Bickel; (301) 975-3668;
at least 24 hours in advance.
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