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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Painleve Equations - Nonlinear Special Functions
III - Peter Clarkson (University of Kent)
DTSTART;TZID=Europe/London:20190912T090000
DTEND;TZID=Europe/London:20190912T100000
UID:TALK129445AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/129445
DESCRIPTION:The six Painleve equations\, whose solutions are c
alled the Painleve transcendents\, were derived by
Painleve and his colleagues in the late 19th and
early 20th centuries in a classification of second
order ordinary differential equations whose solut
ions have no movable critical points.
In the
18th and 19th centuries\, the classical special fu
nctions such as Bessel\, Airy\, Legendre and hyper
geometric functions\, were recognized and develope
d in response to the problems of the day in electr
omagnetism\, acoustics\, hydrodynamics\, elasticit
y and many other areas.
Around the middle of
the 20th century\, as science and engineering cont
inued to expand in new directions\, a new class of
functions\, the Painleve functions\, started to a
ppear in applications. The list of problems now kn
own to be described by the Painleve equations is l
arge\, varied and expanding rapidly. The list incl
udes\, at one end\, the scattering of neutrons off
heavy nuclei\, and at the other\, the distributio
n of the zeros of the Riemann-zeta function on the
critical line Re(z) =1/2. Amongst many others\, t
here is random matrix theory\, the asymptotic theo
ry of orthogonal polynomials\, self-similar soluti
ons of integrable equations\, combinatorial proble
ms such as the longest increasing subsequence prob
lem\, tiling problems\, multivariate statistics in
the important asymptotic regime where the number
of variables and the number of samples are compara
ble and large\, and also random growth problems.
The Painleve equations possess a plethora
of interesting properties including a Hamiltonian
structure and associated isomonodromy problems\, w
hich express the Painleve equations as the compati
bility condition of two linear systems. Solutions
of the Painleve equations have some interesting as
ymptotics which are useful in applications. They p
ossess hierarchies of rational solutions and one-p
arameter families of solutions expressible in term
s of the classical special functions\, for special
values of the parameters. Further the Painleve eq
uations admit symmetries under affine Weyl groups
which are related to the associated Backlund trans
formations.
In these lectures I shall fi
rst review many of the remarkable properties which
the Painleve equations possess. In particular I w
ill discuss rational solutions of Painleve equatio
ns. Although the general solutions of the six Pain
leve equations are transcendental\, all except the
first Painleve equation possess rational solution
s for certain values of the parameters. These solu
tions are usually expressed in terms of logarithmi
c derivatives of special polynomials that are Wron
skians\, often of classical orthogonal polynomials
such as Hermite and Laguerre. It is also known th
at the roots of these special polynomials are high
ly symmetric in the complex plane. The polynomials
arise in applications such as random matrix theor
y\, vortex dynamics\, in supersymmetric quantum me
chanics\, as coefficients of recurrence relations
for semi-classical orthogonal polynomials and are
examples of exceptional orthogonal polynomials.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:INI IT
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