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SUMMARY:Conformal mapping\, Hamiltonian methods and integrability of surfa
 ce dynamics - Pavel Lushnikov (University of New Mexico\; Landau Institute
  for Theoretical Physics)
DTSTART:20191001T130000Z
DTEND:20191001T140000Z
UID:TALK132925@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:A Hamiltonian formulation of the time dependent potential flow
  of ideal<br> incompressible fluid with a free surface is considered in tw
 o dimensional<br> (2D) geometry. It is well known that the dynamics of sma
 ll to moderate<br> amplitudes of surface perturbations can be reformulated
  in terms of the<br> canonical Hamiltonian structure for the surface eleva
 tion and Dirichlet<br> boundary condition of the velocity potential. Arbit
 rary large<br> perturbations can be efficiently characterized through a ti
 me-dependent<br> conformal mapping of a fluid domain into the lower comple
 x half-plane. We<br> reformulate the exact Eulerian dynamics through a non
 -canonical nonlocal<br> Hamiltonian system for the pair of new conformal v
 ariables. The<br> corresponding non-canonical Poisson bracket is non-degen
 erate\,  i.e. it<br> does not have any Casimir invariant. Any two function
 als of the conformal<br> mapping commute with respect to the Poisson brack
 et. We  also consider a<br> generalized hydrodynamics for two components o
 f superfluid Helium which<br> has the same non-canonical Hamiltonian struc
 ture. In both cases  the fluid<br> dynamics is fully characterized by the 
 complex singularities in the upper<br> complex half-plane of the conformal
  map and the complex velocity.<br> Analytical continuation through the bra
 nch cuts generically results in the<br> Riemann surface with infinite numb
 er of sheets. An infinite family of<br> solutions with moving poles are fo
 und on the Riemann surface. Residues of<br> poles are the constants of mot
 ion. These constants commute with each other<br> in the sense of underlyin
 g non-canonical Hamiltonian dynamics which<br> provides an argument in sup
 port of the conjecture of complete Hamiltonian<br> integrability of surfac
 e dynamics.
LOCATION:Seminar Room 2\, Newton Institute
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