Институт теоретической физики им. Л.Д. Ландау РАН

L.D. Landau Institute for Theoretical Physics RAS

L.D. Landau Institute for Theoretical Physics RAS

"Quantum Fluids, Quantum Field Theory, and Gravity"

Motion of complex singularities and integrability of fully nonlinear free surface dynamics of superfluid Helium vs. single ideal fluid

Date/Time: 09:40 19-Oct-2019

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 motion of complex singularities

outside of fluid, i.e. in the upper complex half-plane, for the analytical

continuation 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.

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 motion of complex singularities

outside of fluid, i.e. in the upper complex half-plane, for the analytical

continuation 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.

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