Abstract

A motion of fluid's 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. Both a single ideal fluid dynamics
(corresponds e.g. to oceanic waves dynamics) and a dynamics of
superfluid Helium 4 with two fluid components are considered. Both
systems share the same type of the non-canonical Hamiltonian
structure. A superfluid Helium case is shown to be completely
integrable for the zero gravity and surface tension limit with the
exact reduction to the Laplace growth equation which is completely
integrable through the connection to the dispersionless limit of the
integrable Toda hierarchy and existence of the infinite set of
complex pole solutions. A single fluid case with nonzero 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-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.