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SUMMARY:Integral preserving discretization of 2D Toda lattice
DTSTART;VALUE=DATE-TIME:20230621T074500Z
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UID:indico-contribution-5-45@cern.ch
DESCRIPTION:Speakers: Dr. SMIRNOV\, Sergey (Lomonosov Moscow State Univers
 ity)\n2D-Toda lattices corresponding to the Cartan matrices of simple Lie 
 algebras are known to be Darboux integrable\, i.e. they admit complete fam
 ilies of essentially independent characteristic integrals. During the last
  three decades various discrete analogs of these systems were obtained. In
  2011 Habibullin proposed a systematic way to discretize 2D-Toda lattices.
  His approach was based on the \nidea to look for semi-discrete systems su
 ch that they have the same characteristic integrals as their continuous an
 alogs. Careful analysis of the systems corresponding to the Cartan matrice
 s of the rank 2 allowed Habibullin and his collaborators to introduce semi
 -discrete and purely discrete analogs of 2D-Toda lattices and to conjectur
 e that they are Darboux integrable for Cartan matrices \nof arbitrary rank
 . After that some partial results on Darboux integrability of these system
 s were obtained\, but the general claim remained unproved.\n\nWe prove tha
 t if function I is a y-integral of 2D-Toda lattice corresponding to some C
 artan matrix\, then this function is an n-integral of its semi-discrete an
 alog. This implies the existence of a complete family of n-integrals for e
 ach of these systems. We use the concept of characteristic algebra to prov
 e that these systems admit complete families of characteristic x-integrals
  as well.\n\nhttp://indico.fuw.edu.pl/contributionDisplay.py?contribId=45&
 sessionId=5&confId=67
LOCATION:Faculty of Physics\, University of Warsaw Lecture hall: 0.06
URL:http://indico.fuw.edu.pl/contributionDisplay.py?contribId=45&sessionId
 =5&confId=67
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SUMMARY:The theory of periodic anomalous (rogue) waves in continuous and d
 iscrete NLS type equations
DTSTART;VALUE=DATE-TIME:20230621T071500Z
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UID:indico-contribution-5-64@cern.ch
DESCRIPTION:Speakers: Prof. SANTINI\, Paolo (University of Roma "La Sapien
 za")\nModulation instability and nonlinearity are the main causes of the a
 ppearance of anomalous (rogue) waves (AWs) in several physical contexts. T
 he theory of periodic anomalous waves has been recently developed on the b
 asic Nonlinear Schrödinger (NLS) model in 1+1 dimensions\, adapting the f
 inite gap method to the Cauchy problem for periodic initial perturbations 
 of the homogeneous background solution of NLS [1]. This theory allows one 
 to express the solution of the Cauchy problem\, to leading order\, in term
 s of elementary functions of the unstable part of the initial data\, and h
 as already been tested in the nonlinear optics of a photorefractive crysta
 l [2]. Also a perturbation theory of AWs allowing one to study the leading
  order effects of small perturbations of the NLS equation on the dynamics 
 of AWs has been constructed [3]. In this lecture we develop a lattice theo
 ry of AWs using as basic model the Ablowitz-Ladik lattice\, integrable dis
 cretization of the NLS equation [4]\, [5].\n\n[1] P. G. Grinevich and P. M
 . Santini: ``The finite-gap method and the periodic NLS Cauchy problem of 
 anomalous waves for a finite number of unstable modes''\, Russian Math. Su
 rveys {\\bf 74:2} 211-263 (2019). DOI:10.1070/RM9863.\n[2] D. Pierangeli\,
  M. Flammini\, L. Zhang\, G. Marcucci\, A. J. Agranat\, P. G. Grinevich\, 
 P. M. Santini\, C. Conti\, and E. DelRe\, ``Observation of exact Fermi-Pas
 ta-Ulam-Tsingou recurrence and its exact dynamics''\, Phys. Rev. X {\\bf 8
 }\, 041017 (2018). doi.org/10.1103/ PhysRevX.8.041017. \n[3] F. Coppini\, 
 P. G. Grinevich and P. M. Santini: ``The effect of a small loss or gain in
  the periodic NLS anomalous wave dynamics. I''\,  Phys. Rev. E {\\bf 101}\
 , 032204 (2020). DOI: 10.1103/PhysRevE.101.032204. arXiv:1910.13176. \n[4]
  F. Coppini and P. M. Santini: ``Modulation instability\, periodic anomalo
 us wave recurrence\, and blow up in the Ablowitz - Ladik lattices''. Prepr
 int 2023. \n[5] F. Coppini and P. M. Santini: ``The effect of loss/gain an
 d hamiltonian perturbations of the Ablowitz - Ladik lattice on the recurre
 nce of periodic anomalous waves''. Preprint 2023.\n\nhttp://indico.fuw.edu
 .pl/contributionDisplay.py?contribId=64&sessionId=5&confId=67
LOCATION:Faculty of Physics\, University of Warsaw Lecture hall: 0.06
URL:http://indico.fuw.edu.pl/contributionDisplay.py?contribId=64&sessionId
 =5&confId=67
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BEGIN:VEVENT
SUMMARY:A Truss Structure with Mechanical Optimality\, Integrability and A
 rtisiticity
DTSTART;VALUE=DATE-TIME:20230621T081500Z
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DTSTAMP;VALUE=DATE-TIME:20260617T052635Z
UID:indico-contribution-5-16@cern.ch
DESCRIPTION:Speakers: Prof. KAJIWARA\, Kenji (Institute of Mathematics for
  Industry\, Kyushu University)\nWe report that a class of integrable discr
 ete holomorphic functions can generate planar truss structures with a cert
 ain mechanical optimality called the Michell structure\, well-known in the
  area of architecture.  Further\, the discrete planar curves formed by the
  edges are nothing but the discrete analogue of the logarithmic spiral whi
 ch is a special case of the discrete log-aesthetic curves. Discrete log-ae
 sthetic curves are integrable discrete analogue of the log-aesthetic curve
 s\, which is known as a class of planar curves with built-in aesthetic nat
 ure\, and those curves are invariant curves with respect to integrable def
 ormation of planar curves in similarity geometry.\n\nhttp://indico.fuw.edu
 .pl/contributionDisplay.py?contribId=16&sessionId=5&confId=67
LOCATION:Faculty of Physics\, University of Warsaw Lecture hall: 0.06
URL:http://indico.fuw.edu.pl/contributionDisplay.py?contribId=16&sessionId
 =5&confId=67
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