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SUMMARY:Conformal mappings\, Riemann surface sheets and integrability of s
 urface dynamics - Pavel Lushnikov (University of New Mexico)
DTSTART:20221027T100000Z
DTEND:20221027T110000Z
UID:TALK184322@talks.cam.ac.uk
DESCRIPTION:A fully nonlinear surface dynamics of the time dependent poten
 tial flow of ideal incompressible fluid with a free surface is considered 
 in two dimensional geometry. Arbitrary large surface waves can be efficien
 tlycharacterized through a time-dependent conformal mapping of a fluid dom
 ain into the lower complex half-plane. We reformulate the exact Eulerian d
 ynamics through a non-canonical nonlocal Hamiltonian system for the pair o
 f new conformal variables. The corresponding non-canonical Poisson bracket
  is non-degenerate\, i.e. it does not have any Casimir invariant. Any two 
 functionals of the conformal mapping commute with respect to the Poisson b
 racket. We also consider a generalized hydrodynamics for two components of
  superfluid Helium which has the same non-canonical Hamiltonian structure.
  In both cases the fluid dynamics is fully characterized by the complex si
 ngularities in the upper complex half-plane of the conformal map and the c
 omplex velocity. Analytical continuation through the branch cuts generical
 ly results in the Riemann surface with infinite number of sheets including
  Stokes wave\, An infinite family of solutions with moving poles are found
  on the Riemann surface. Residues of poles are the constants of motion. Th
 ese constants commute with each other in the sense of underlying non-canon
 ical Hamiltonian dynamics which provides an argument in support of the con
 jecture of complete Hamiltonian integrability of surface dynamics. If we c
 onsider initial conditions with short branch cuts then fluid dynamics is r
 educed to the complex Hopf equation for the complex velocity coupled with 
 the complex transport equation for the conformal mapping. These equations 
 are fully integrable by characteristics producing the infinite family of s
 olutions\, including the pairs of moving square root branch points. The so
 lutions are compared with the simulations of the full Eulerian dynamics gi
 ving excellent agreement
LOCATION:Seminar Room 1\, Newton Institute
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