# applied math seminar

### Event Description:

Title:

Conformal mapping, Hamiltonian methods and integrability for solving

the Euler equations

Abstract:

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.