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applied math seminar

Event Type: 
Pavel Lushnikov, Department of Mathematics and Statistics, University of New Mexico
Event Date: 
Monday, March 26, 2018 -
3:30pm to 4:30pm
SMLC 356
General PublicFaculty/StaffStudentsAlumni/Friends

Event Description: 


Conformal mapping, Hamiltonian methods and integrability for solving
the Euler equations


A time-dependent conformal transformation is used to address 2D
hydrodynamics of ideal fluid with free surface. A free surface is
mapped into the real line with fluid domain mapped into the lower
complex half-plane. The fluid dynamics is fully characterized by the
motion of complex singularities in the upper complex half-plane of
the conformal map and the complex velocity. Dynamical equations are
equivalently reformulated as the noncanonical Hamiltonian system
with nonlocal Poisson bracket. Infinite number of commuting
integrals of motion are found in conformal variables suggesting full
integrability of Euler equations with free surface because these
integrals are not Casimir invariants. The initially flat surface
with the pole in the complex velocity turns over arbitrary small
time into the branch cut connecting two square root branch points.
Without gravity one of these branch points approaches the fluid
surface with the approximate exponential law corresponding to the
formation of the fluid jet. The addition of gravity results in
wavebreaking in the form of plunging of the jet into the water
surface. The use of the additional conformal transformation to
resolve the dynamics near branch points allows to analyze
wavebreaking in details. The formation of multiple Crapper capillary
solutions is observed during overturning of the wave contributing to
the turbulence of surface wave. Another possible way for the
wavebreaking is the slow increase of Stokes wave amplitude through
nonlinear interactions until the limiting Stokes wave forms with
subsequent wavebreaking. For non-limiting Stokes wave the only
singularity in the physical sheet of Riemann surface is the
square-root branch point located. The corresponding branch cut
defines the second sheet of the Riemann surface if one crosses the
branch cut. The infinite number of pairs of square root
singularities is found corresponding to infinite number of
non-physical sheets of Riemann surface. Each pair belongs to its own
non-physical sheet of Riemann surface. Increase of the steepness of
the Stokes wave means that all these singularities simultaneously
approach the real line from different sheets of Riemann surface and
merge together forming 2/3 power law singularity of the limiting
Stokes wave. It is conjectured that non-limiting Stokes wave at the
leading order consists of the infinite product of nested square root
singularities which form the infinite number of sheets of Riemann
surface. The conjecture is also supported by high precision
simulations, where a quad (32 digits) and a variable precision (up
to 200 digits) were used to reliably recover the structure of square
root branch cuts in multiple sheets of Riemann surface.