  Recent isolated galactic simulations show that the morphology of galactic
discs in modified gravity differs from that of the standard dark matter model.
In this study, we focused on the vertical structure of galactic discs and
compared the bending instability in the vertical direction for both paradigms.
To achieve this, we utilized high-resolution N-body simulations to construct
two models in a specific nonlocal gravity theory (NLG) and the standard dark
matter model and compared their stability against the bending perturbations.
Our numerical results demonstrate that the outer regions of the disc are more
susceptible to the instability in NLG, whereas the disc embedded in the dark
matter halo is more unstable in the central regions. We then interpret these
results based on the dispersion relation of the bending waves. To do so, we
presented an analytical study to derive the dispersion relation in NLG. Our
numerical results align with the predictions of our analytical models.
Consequently, we conclude that the analysis of bending instability in galactic
discs offers an explanation for the distinct vertical structures observed in
simulated galactic discs under these two theories. These findings represent a
significant step towards distinguishing between the modified gravity and dark
matter models.
