Late-time background constraints on linear and non-linear interacting dark energy after DESI DR2

In this study, we present observational constraints on a class of phenomenological interacting dark energy (IDE) models that admit analytical solutions for the Hubble parameter $H(z)$. We consider a set of five linear and three non-linear IDE scenarios, encompassing both interactions proportional to the dark matter and/or dark energy densities, as well as non-linear combinations of the two. For all eight IDE models, we find a better fit than $Λ$CDM from a $Δχ^2$ analysis for both combinations of datasets considered. When using the Akaike Information Criterion ($Δ$AIC), we find a similarly improved fit in all cases, except for one dataset combination in $Q=3Hδρ_{\rm de}$. Our analysis also shows a preference for sign-switching interactions, with energy transfer from dark energy to dark matter at low redshift, reversing direction at higher redshift. These results should be interpreted with caution, as the latter direction of energy transfer is accompanied by negative dark energy densities in the past, which may be unphysical. Models that do not allow sign-changing behaviour instead show a preference for energy flow from dark matter to dark energy, and hence negative dark energy densities. The only exceptions are $Q=3Hδρ_{\rm de}$ and $Q=3Hδ\left(\tfrac{ρ_{\rm de}^2}{ρ_{\rm dm}+ρ_{\rm de}}\right)$, which exhibit energy flow in the opposite direction. Furthermore, for all interactions considered, we find a phantom-divide crossing in the effective dark energy equation of state $w^{\rm eff}_{\rm de}$, with the dark energy density decreasing ($w^{\rm eff}_{\rm de}>-1$) at present and at low redshift, while increasing ($w^{\rm eff}_{\rm de}