A motion of fluid's free surface is considered in two dimensional (2D) geometry. A timedependent conformal transformation maps a fluid domain into the lower complex halfplane of a new spatial variable. The fluid dynamics is fully characterized by the complex singularities in the upper complex halfplane 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 noncanonical 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 halfplane. Residues of poles are the constants of motion. These constants commute with each other in the sense of underlying noncanonical 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.
You are here
Motion of complex singularities and Hamiltonian integrability of surface dynamics
Research Group:
Pavel Lushnikov
Location:
A133
Schedule:
Wednesday, June 19, 2019  11:00
Abstract:
Openings
 Public Calls for Professors
 Temporary Professors/Researchers/Visiting Professors
 SISSA Mathematical Fellowships
 Marie SklodowskaCurie Grants
 Post Doctoral Fellowships
 PhD Scolarships
 SIS Fellowships
 Undergraduate Fellowships
 Postgraduate Fellowships
 MSc in Mathematics
 MSc in Data Science and Scientific Computing (DSSC)
 Professional Master Courses
Upcoming events

Martin Andler
The ups and downs of the development of mathematics in France since 1870
Tuesday, October 22, 2019  10:00

Stefano Scrobogna
Stabilization of the RayleighTaylor instability of an underlying fluid layer
Tuesday, October 22, 2019  11:00

Abed Bounemoura
Positive measure of KAM tori for finitely differentiable Hamiltonians
Thursday, October 24, 2019  16:00

Massimiliano Berti
Long time dynamics of water waves
Thursday, October 24, 2019  17:00
Today's Lectures

09:00 to 11:00

09:00 to 11:00

11:00 to 13:00

11:00 to 13:00

11:00 to 13:00

14:00 to 16:00

14:00 to 16:00

14:00 to 16:00

16:00 to 18:00
Recent publications

R. Scandone,Zero modes and lowenergy reso...

A. Bawane; S. Benvenuti; G. Bonelli; N. Muteeb; A. Tanzini,N=2 gauge theories on unorient...

R. Feola; F. Iandoli,Local wellposedness for quasi...

D. Riccobelli; D. Ambrosi,Activation of a muscle as a ma...