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SUMMARY:Anatolij Prykarpatski
DTSTART:20210324T162000Z
DTEND:20210324T180000Z
DTSTAMP:20260423T010004Z
UID:GDEq/32
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GDEq/32/">On
  integrability of some Riemann type hydrodynamical systems and Dubrovin in
 tegrability classification of perturbed Korteweg-de Vries type equations</
 a>\nby Anatolij Prykarpatski as part of Geometry of differential equations
  seminar\n\n\nAbstract\nIn our report we will stop on two closely related 
 to each other integrability theory aspects. The first one concerns the obt
 ained integrability results\, based on the gradient-holonomic integrabilit
 y scheme\, devised and applied by me jointly with Maxim Pavlov and collabo
 rators to a virtually new important Riemann type hierarchy $D_{t}^{N-1}u=z
 _{x}^{s}$\, $D_{t}z=0$\, where $s$\,&nbsp\;$N\\in N$ are arbitrary natural
  numbers\, and proposed in our work (M. Pavlov\, A. Prykarpatsky\, at al.\
 , <a href="https://arxiv.org/abs/1108.0878">arXiv:1108.0878</a>) as a nont
 rivial generalization of the infinite hierarchy of the Riemann type flows\
 , suggested before by M. Pavlov and D. Holm in the form of dynamical syste
 ms $D_{t}^{N}u=0$\, defined on a $2\\pi$-periodic functional manifold $M^{
 N}\\subset C^{\\infty}( R/2\\pi Z\; R^{N})$\, the vector $(u\,D_{t}u\,D_{t
 }²u\,...\,D_{t}^{N-1}u\,z)^{\\intercal}\\in M^{N}$\, the differentiations
  $D_{x}:=\\partial/\\partial x$\, $D_{t}:=\\partial/\\partial t+u\\partial
 /\\partial x$ satisfy as above the Lie-algebraic commutator relationship $
 [D_{x}\,D_{t}]=u_{x}D_{x}$ and t\\in R is an evolution parameter. The seco
 nd aspect of our report concerns the integrability results obtained by B. 
 Dubrovin jointly with Y. Zhang and collaborators\, devoted to classificati
 on of a special perturbation of the Korteweg-de Vries equation in the form
  $u_{t}=uu_{x}+\\epsilon^2[f_{31}(u)u_{xxx}+f_{32}(u)u_{xx}u_{x}+f_{33}(u)
 u_{x}^3]$\, where $f_{jk}(u)\,~j=3\,~k=1\,~3$\, are some smooth functions 
 and \\epsiln\\in R is a real parameter. We will deal with classification s
 cheme of evolution equations of a special type suspicious on being integra
 ble which was devised some years ago by untimely passed away Prof. Boris D
 ubrovin (19 March 2019) and developed with his collaborators\, mainly with
  Youjin Zhang. We have reanalyzed in detail their interesting results on i
 ntegrability classification of a suitably perturbed KdV type equation with
 in our gradient-holonomic integrability scheme\, devised many years ago an
 d developed by me jointly with Maxim Pavlov and collaborators\, and found 
 out that the Dubrovin's scheme has missed at least a one very interesting 
 integrable equation\, whose natural reduction became similar to the well-k
 nown Krichever-Novikov equation\, yet different from it. As a consequence 
 of the analysis\, we presented one can firmly claim that the Dubrovin-Zhan
 g integrability criterion inherits some important part of the mentioned ab
 ove gradient-holonomic integrability scheme properties\, coinciding with t
 he statement about the necessary existence of suitably ordered reduction e
 xpansions with coefficients to be strongly homogeneous differential polyno
 mials.\n\nJoint with Alex A. Balinsky\, Radoslaw Kycia and Yarema A. Pryka
 rpatsky.\n\nAlthough the talk will be in Russian\, the slides will be in E
 nglish and the discussion will be in both languages.\n
LOCATION:https://researchseminars.org/talk/GDEq/32/
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