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

Fluctuating hydrodynamics for a discrete Gross-Pitaevskii equation: mapping to Kardar-Parisi-Zhang universality class