Analysis Seminar

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Complex Poles, Branch Cuts and Integrability of 2D Surface Dynamics

Speaker: Pavel Lushnikov, University of New Mexico

Location: Warren Weaver Hall 1302

Date: Thursday, October 25, 2018, 11 a.m.

Synopsis:

Euler equations for potential flow of ideal incompressible fluid
with a free surface are 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. An infinite family of solutions with moving poles is
found. 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 integrability of 2D free
surface hydrodynamics.