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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Conformal mappings\, Riemann surface sheets and in
tegrability of surface dynamics - Pavel Lushnikov
(University of New Mexico)
DTSTART;TZID=Europe/London:20221027T110000
DTEND;TZID=Europe/London:20221027T120000
UID:TALK184322AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/184322
DESCRIPTION:A fully nonlinear surface dynamics of the time dep
endent potential flow of ideal incompressible flui
d with a free surface is considered in two dimensi
onal geometry. Arbitrary large surface waves can b
e efficientlycharacterized through a time-dependen
t conformal mapping of a fluid domain into the low
er complex half-plane. We reformulate the exact Eu
lerian dynamics through a non-canonical nonlocal H
amiltonian system for the pair of new conformal va
riables. The corresponding non-canonical Poisson b
racket is non-degenerate\, i.e. it does not have a
ny Casimir invariant. Any two functionals of the c
onformal mapping commute with respect to the Poiss
on bracket. We also consider a generalized hydrody
namics for two components of superfluid Helium whi
ch has the same non-canonical Hamiltonian structur
e. In both cases the fluid dynamics is fully chara
cterized by the complex singularities in the upper
complex half-plane of the conformal map and the c
omplex velocity. Analytical continuation through t
he branch cuts generically results in the Riemann
surface with infinite number of sheets including S
tokes wave\, An infinite family of solutions with
moving poles are found on the Riemann surface. Res
idues of poles are the constants of motion. These
constants commute with each other in the sense of
underlying non-canonical Hamiltonian dynamics whic
h provides an argument in support of the conjectur
e of complete Hamiltonian integrability of surface
dynamics. If we consider initial conditions with
short branch cuts then fluid dynamics is reduced t
o the complex Hopf equation for the complex veloci
ty coupled with the complex transport equation for
the conformal mapping. These equations are fully
integrable by characteristics producing the infini
te family of solutions\, including the pairs of mo
ving square root branch points. The solutions are
compared with the simulations of the full Eulerian
dynamics giving excellent agreement
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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