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Уважаемые сотрудники ИТФ,<br>
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На заседании Ученого совета ИТФ в пятницу 8 июня будет заслушан
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<p class="speakers">A. I. Dyachenko, S. A. Dyachenko, <u>P. M.
Lushnikov</u> and V. E. Zakharov</p>
Dynamics of Poles in 2D Hydrodynamics with Free Surface: New
Constants of Motion<br>
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<p class="abstract tex">
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. </p>
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