the truncated Wigner approximation are also very popular tools to evaluate corrections beyond the Bogoliubov approximation in both conservative cold-atom [@castin_TWA; @polkovnikov_phase_space] and lossy optical systems [@gardiner_quantum_noise; @Iacopo_QFL]. For the latter, this technique has been applied in various contexts, including the study of condensation and superfluid properties [@iacopo_spontaneous; @wouters2009stochastic; @gladilin2014spatial; @he2015scaling], dynamical phase transitions [@sieberer2013dynamical; @dagvadorj_2015_nonequilibrium; @fossfeig_optical_bistability], and even genuine quantum effects such as the dynamical Casimir emission [@koghee2014dynamical] and Hawking radiation [@Hawking; @Hawking2]. In contrast to these successes, we will show in this work that the truncated Wigner approximation, when naively adopted to study Beliaev-Landau scattering, may dramatically overestimate the corrections to Bogoliubov theory and even lead to unphysical results.

As an alternative approach we discuss how a truncated hierarchy of correlations can serve as a consistent expansion beyond the Bogoliubov approximation [@hartmann_2013_correlator; @kastner_BBGKY; @wim_hierarchy]. In particular, we will illustrate that the truncation of the hierarchy at the third-order correlation functions, i.e. one order beyond the Bogoliubov approximation, is sufficient to incorporate the corrections attributed to Beliaev-Landau processes, provided an adequate truncation scheme is employed.

The structure of our paper is as follows. In Sec. \[sec:model\] we present the 1D driven-dissipative Bose-Hubbard model and in Sec. \[sec:beliaev\] we illustrate 