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SUMMARY:Motion of complex singularities and Hamiltonian integrability of s
urface dynamics - Pavel Lushnikov (University of New Mexico\; Landau Insti
tute for Theoretical Physics)
DTSTART:20191210T113000Z
DTEND:20191210T123000Z
UID:TALK135514@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:A motion of fluid'\;s free surface is considered in two dim
ensional
(2D) geometry. A time-dependent conformal transformation maps
a
fluid domain into the lower complex half-plane of a new spatial
variable. The fluid dynamics is fully characterized by the complex
si
ngularities in the upper complex half-plane of the conformal map
and t
he complex velocity. Both a single ideal fluid dynamics
(corresponds e
.g. to oceanic waves dynamics) and a dynamics of
superfluid Helium 4 w
ith two fluid components are considered. Both
systems share the same t
ype of the non-canonical Hamiltonian
structure. A superfluid Helium ca
se is shown to be completely
integrable for the zero gravity and surfa
ce tension limit with the
exact reduction to the Laplace growth equati
on which is completely
integrable through the connection to the disper
sionless limit of the
integrable Toda hierarchy and existence of the i
nfinite set of
complex pole solutions. A single fluid case with nonzer
o gravity and
surface tension turns more complicated with the infinite
set of new
moving poles solutions found which are however unavoidably
coupled
with the emerging moving branch points in the upper half-plan
e.
Residues of poles are the constants of motion. These constants
commute with each other in the sense of underlying non-canonical
Hamil
tonian dynamics. It suggests that the existence of these extra
constan
ts of motion provides an argument in support of the
conjecture of comp
lete Hamiltonian integrability of 2D free surface
hydrodynamics.
LOCATION:Seminar Room 1\, Newton Institute
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