  We show that several aspects of the low-temperature hydrodynamics of a
discrete Gross-Pitaevskii equation (GPE) can be understood by mapping it to a
nonlinear version of fluctuating hydrodynamics. This is achieved by first
writing the GPE in a hydrodynamic form of a continuity and an Euler equation.
Respecting conservation laws, dissipation and noise due to the system's chaos
are added, thus giving us a nonlinear stochastic field theory in general and
the Kardar-Parisi-Zhang (KPZ) equation in our particular case. This mapping to
KPZ is benchmarked against exact Hamiltonian numerics on discrete GPE by
investigating the non-zero temperature dynamical structure factor and its
scaling form and exponent. Given the ubiquity of the Gross-Pitaevskii equation
(a.k.a. nonlinear Schrodinger equation), ranging from nonlinear optics to cold
gases, we expect this remarkable mapping to the KPZ equation to be of paramount
importance and far reaching consequences.
