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The gravitational instability of cosmological pancakes composed of ordinary gaseous matter and collisionless dark matter in an Einstein-de Sitter universe is investigated by numerical simulations using the Adaptive Smoothed Particle Hydrodynamics (ASPH) method and a Particle-Mesh (PM) gravity solver. Pancakes are generic structures which form as the result of the nonlinear growth of cosmological density flucutations in a wide range of models involving Gaussian random noise initial conditions. The gravitational instability of such structures may lead to fragmentation and the subsequent formation of bound substructures, important in the formation of galaxies, as well as the production of vorticity. Consistent with the assumption that at least $90 \%$ of the matter in the universe is composed of dark nonbaryonic matter which can be treated as a gas of collisionless particles, we consider both purely collisionless pancakes as well as collisionless-matter-dominated pancakes. We present numerical results in 2D of pancake instability and fragmentation due to symmetric perturbations (density mode) as well as antisymmetric perturbations (bending mode) of the pancake. We also consider the effects of an initial, non-zero velocity dispersion for the collisionless matter on the growth of instability.
