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 non-local 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.

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