| Summary: | We consider free surface dynamics of a two-dimensional incompressible fluid
with odd viscosity. The odd viscosity is a peculiar part of the viscosity
tensor which does not result in dissipation and is allowed when parity symmetry
is broken. For the case of incompressible fluids, the odd viscosity manifests
itself through the free surface (no stress) boundary conditions. We first find
the free surface wave solutions of hydrodynamics in the linear approximation
and study the dispersion of such waves. As expected, the surface waves are
chiral and even exist in the absence of gravity and vanishing shear viscosity.
In this limit, we derive effective nonlinear Hamiltonian equations for the
surface dynamics, generalizing the linear solutions to the weakly nonlinear
case. Within the small surface angle approximation, the equation of motion
leads to a new class of non-linear chiral dynamics governed by what we dub the
{\it chiral} Burgers equation. The chiral Burgers equation is identical to the
complex Burgers equation with imaginary viscosity and an additional analyticity
requirement that enforces chirality. We present several exact solutions of the
chiral Burgers equation. For generic multiple pole initial conditions, the
system evolves to the formation of singularities in a finite time similar to
the case of an ideal fluid without odd viscosity. We also obtain a periodic
solution to the chiral Burgers corresponding to the non-linear generalization
of small amplitude linear waves.
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