Dynamics of complex singularities and integrability of 2D surface motion

Dynamics of complex singularities and integrability of 2D surface motion

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Pavel Lushnikov, UNM
Fine Hall 322

A potential flow of ideal incompressible fluid with a free surface
is considered in two dimensional (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 singularities in the upper complex
half-plane of the conformal map and the complex velocity. Analytical
continuation through the branch cuts results in the Riemann surface
with infinite number of sheets. An infinite family of solutions with
moving poles is found on the Riemann surface. These poles are
coupled with the emerging moving branch points in the upper
half-plane. Residues of poles are the constants of motion. These
constants commute with each other in the sense of underlying
non-canonical Hamiltonian dynamics. It suggests that the existence
of these extra constants of motion provides an argument in support
of the conjecture of complete Hamiltonian integrability of 2D free
surface hydrodynamics.