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Conformal mapping, Hamiltonian methods and integrability of surface dynamics

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CAT - Complex analysis: techniques, applications and computations

A Hamiltonian formulation of the time dependent potential flow of ideal
incompressible fluid with a free surface is considered in two dimensional
(2D) geometry. It is well known that the dynamics of small to moderate
amplitudes of surface perturbations can be reformulated in terms of the
canonical Hamiltonian structure for the surface elevation and Dirichlet
boundary condition of the velocity potential. Arbitrary large
perturbations can be efficiently characterized through a time-dependent
conformal mapping of a fluid domain into the lower complex half-plane. We
reformulate the exact Eulerian dynamics through a non-canonical nonlocal
Hamiltonian system for the pair of 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 bracket. 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 singularities in the upper
complex half-plane of the conformal map and the complex velocity.
Analytical continuation through the branch cuts generically results in the
Riemann surface with infinite number of sheets. An infinite family of
solutions with moving poles are found on the Riemann surface. Residues of
poles are the constants of motion. These constants commute with each other
in the sense of underlying non-canonical Hamiltonian dynamics which
provides an argument in support of the conjecture of complete Hamiltonian
integrability of surface dynamics.

This talk is part of the Isaac Newton Institute Seminar Series series.

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