[Landau ITP Seminars] Friday 08.06.2018

[Serge Krashakov]

Уважаемые сотрудники ИТФ,

На заседании Ученого совета ИТФ в пятницу 8 июня будет заслушан доклад:

A. I. Dyachenko, S. A. Dyachenko, _P. M. Lushnikov_ and V. E. Zakharov

Dynamics of Poles in 2D Hydrodynamics with Free Surface: New Constants 
of Motion

We consider Euler equations for potential flow of ideal incompressible 
fluid with a free surface and infinite depth in two dimensional 
geometry. We admit a presence of gravity forces and surface tension. A 
time-dependent conformal mapping z(w,t) of the lower complex half-plane 
of the variable w into the area filled with fluid is performed with the 
real line of w mapped into the free fluid's surface. We study the 
dynamics of singularities of both z(w,t) and the complex fluid potential 
Pi(w,t) in the upper complex half-plane of w. We show the existence of 
solutions with an arbitrary finite number N of simple complex poles in 
z_w(w,t) and Pi_w(w,t) which are the derivatives of z(w,t) and Pi(w,t) 
over w. These poles are often coupled with branch points located at 
other points of the upper half-plane of w. We find that the residues of 
the simple poles of z_w(w,t) are new, previously unknown constants of 
motion, provided surface tension is zero. All these constants of motion 
commute with each other in the sense of underlying Hamiltonian dynamics. 
In absence of both gravity and surface tension, the residues of simple 
poles of Pi_w(w,t) are also the constants of motion. For nonzero gravity 
and zero surface tension, the residues of poles of any order of 
Pi_w(w,t) are the trivial linear functions of time. Nonzero surface 
tension allows residues of poles of even order to be compatible with the 
fluid dynamics. We also found solutions with N higher order poles. In 
all above cases the number of independent real integrals of motion is 4N 
for zero gravity and 4N-1 for nonzero gravity. We suggest that the 
existence of these nontrivial constants of motion provides an argument 
in support of the conjecture of complete integrability of free surface 
hydrodynamics in deep water.

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