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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. <br> 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. <br> 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.<b
 r> <br> 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. <br> <br> 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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