  In this work, we derive an exact hydrodynamical description for the coupled,
charge and operator dynamics, in a quantum many-body system with U(1) symmetry.
Using an emergent symmetry in the complex Brownian SYK model with charge
conservation, we map the operator dynamics in the model to the imaginary-time
dynamics of an SU(4) spin-chain. We utilize the emergent SU(4) description to
demonstrate that the U(1) symmetry causes quantum-coherence to persist even
after disorder-averaging, in sharp contrast to models without symmetries. In
line with this property, we write down a 'restricted' Fokker-Planck equation
for the out-of-time ordered correlator (OTOC) in the large-$N$ limit, which
permits a classical probability description strictly in the incoherent sector
of the global operator-space. We then exploit this feature to describe the OTOC
in terms of a Fisher-Kolmogorov-Petrovsky-Piskun (FKPP)-equation which couples
the operator with the charge and is valid at all time-scales and for arbitrary
charge-density profiles. The coupled equations obtained belong to a class of
models also used to describe the population dynamics of bacteria embedded in a
diffusive media. We simulate them to explore operator-dynamics in a background
of non-uniform charge configuration, which reveals that the charge transport
can strongly affect dynamics of operators, including those that have no overlap
with the charge.
