  Recently, Quastel and Remenik \cite{QRKP} [arXiv:1908.10353] found a
remarkable relation between some solutions of the finite time
Kardar-Parisi-Zhang (KPZ) equation and the Kadomtsev-Petviashvili (KP)
equation. Using this relation we obtain the large deviations at large time and
at short time for the KPZ equation with droplet initial conditions, and at
short time with half-Brownian initial conditions. It is consistent with
previous results and allows to obtain sub-leading corrections, as well as
results at intermediate time. In addition, we find that the appropriate
generating function associated to the full Brownian initial condition also
satisfies the KP equation. Finally, generating functions for some linear
statistics of the Airy point process are also found to satisfy the KP property,
and consequences are discussed.
