of the absence of a spectral gap and the convexity of the Bogoliubov dispersion of conservative continuum systems, criterium (\[eq:E\_cons\]) in continuum models can only be satisfied in two or more spatial dimensions, while in 1D sytems phonons can only decay through higher-order scattering processes [@ristivojevic_2016_decay]. However, there exist specifically engineered 1D optical lattices with a nonconvex (but gapless) spectrum, such that energy and momentum conservation can be simultaneously satisfied [@optical_lattice_griffin; @optical_lattice_tetsuro].

The situation is different in driven-dissipative systems, where the Bogoliubov spectrum is typically gapped when the drive is below resonance. In a continuous 1D setup, the presence of a finite spectral gap in combination with a convex excitation spectrum always allows for third-order scattering process that satisfy (\[eq:E\_cons\]). When the spectrum is not convex, as is the case in a lattice model, the situation is somehow more complicated. The subtle interplay between the spectral gap and the degree of nonconvexity determines whether resonant third-order scattering channels are present.

The allowed wavevectors $k$ and $q$ that exactly satisfy the energy and momentum conservation condition (\[eq:E\_cons\]) in one-dimension are indicated in Fig. \[fig:bel\_decay\]b). Importantly, maximal and minimal values can be deduced for $k$ and $q$ from the contours, which set 