We examine the role of the gravitational instability in an isothermal, self-gravitating layer threaded by magnetic fields on the formation of filaments and dense cores. Using a numerical simulation, we follow the non-linear evolution of a perturbed equilibrium layer. The linear evolution of such a layer is described in the analytic work of Nagai et al. We find that filaments and dense cores form simultaneously. Depending on the initial magnetic field, the resulting filaments form either a spiderweb-like network (for weak magnetic fields) or a network of parallel filaments aligned perpendicular to the magnetic field lines (for strong magnetic fields). Although the filaments are radially collapsing, the density profile of their central region (up to the thermal scale height) can be approximated by a hydrodynamical equilibrium density structure. Thus, the magnetic field does not play a significant role in setting the density distribution of the filaments. The density distribution outside of the central region deviates from the equilibrium. The radial column density distribution is then flatter than the expected power law of r -4 and similar to filament profiles observed with Herschel. Our results do not explain the near constant filament width of ~0.1pc. However, our model does not include turbulent motions. It is expected that the accretion-driven amplification of these turbulent motions provides additional support within the filaments against gravitational collapse. Finally, we interpret the filamentary network of the massive star forming complex G14.225-0.506 in terms of the gravitational instability model and find that the properties of the complex are consistent with being formed out of an unstable layer threaded by a strong, parallel magnetic field.
