# On Classical Integrability of the Hydrodynamics of Quantum Integrable Systems

## TL;DR

This paper investigates the integrability of the Bethe-Boltzmann kinetic equation in quantum integrable systems by constructing finite-dimensional approximations, revealing Hamiltonian structures and conserved quantities that suggest hydrodynamic integrability.

## Contribution

It introduces a finite-dimensional approximation framework for the Bethe-Boltzmann equation, deriving Hamiltonian structures and conserved charges that support its integrability in the hydrodynamic sense.

## Key findings

- Finite-dimensional systems exhibit non-local Poisson brackets and Hamiltonian densities.
- Conserved charges correspond to Casimir invariants of the hydrodynamic bracket.
- Evidence suggests the Bethe-Boltzmann equation is integrable in the hydrodynamic sense.

## 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.