Title: On Classical Integrability of the Hydrodynamics of Quantum Integrable Systems,
Abstract: Recently, a hydrodynamic description of local equilibrium dynamics in quantum
integrable systems was discovered. In the diffusionless limit, this is
equivalent to a certain "Bethe-Boltzmann" kinetic equation, which has the form
of an integro-differential conservation law in $(1+1)$D. The purpose of the
present work is to investigate the sense in which the Bethe-Boltzmann equation
defines an "integrable kinetic equation". To this end, we study a class of $N$
dimensional systems of evolution equations that arise naturally as
finite-dimensional approximations to the Bethe-Boltzmann equation. We obtain
non-local Poisson brackets and Hamiltonian densities for these equations and
derive an infinite family of first integrals, parameterized by $N$ functional
degrees of freedom. We find that the conserved charges arising from quantum
integrability map to Casimir invariants of the hydrodynamic bracket and their
group velocities map to Hamiltonian flows. Some results from the
finite-dimensional setting extend to the underlying integro-differential
equation, providing evidence for its integrability in the hydrodynamic sense.