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SUMMARY:Irina Bobrova (HSE\, Moscow)
DTSTART:20200608T120000Z
DTEND:20200608T140000Z
DTSTAMP:20260423T005851Z
UID:GDEq/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GDEq/7/">On 
 the second Painlevé equation and its higher analogues</a>\nby Irina Bobro
 va (HSE\, Moscow) as part of Geometry of differential equations seminar\n\
 n\nAbstract\nSix Painlevé equations were obtained by Paul Painlevé and h
 is school during the classification of ODE's of the form $w'' = P (z\, w\,
  w')$\, where the function $P (z\, w\, w')$ is a polynomial in $w$ and $w'
 $ and is an analytic function of $z$. These equations are widely used in p
 hysics and have beautiful mathematical structures. My talk is devoted to t
 he second Painlevé equation.\n\nWe will discuss the integrability of this
  equation and introduce its Hamiltonian representation in terms of the Kaz
 uo Okamoto variables. On the other hand\, the PII equation is integrable i
 n the sense of the Lax pair and the isomonodromic representation\, that I 
 will present.\n\nThe Bäcklund transformation and the affine Weyl group ar
 e another interesting question. Using these symmetries\, we are able to co
 nstruct various rational solutions for the integer parameter PII equation.
 \n\nThe second Painlevé equation has one more important representation in
  terms of $\\sigma$-coordinates which are $log$-symplectic.\n\nThere are h
 igher analogues of the PII equation\, which we will obtain by self-similar
  reduction of the modified Korteveg-de Vries hierarchy.\n
LOCATION:https://researchseminars.org/talk/GDEq/7/
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