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In a number of cases (e.g., in NGC 6946), deep HI and UV observations of the outer parts of spiral galaxies indicate the presence of well-developed spiral arms in the gas well outside the bright optical disk. Such prominent structures in regions where the disk is light and the stars are not the dominant visible component may be surprising. It is thus important to study what mechanisms may be at work to support the observed structures, which might also have some bearing on the problem of the amount and distribution of dark matter in disk galaxies. Here we follow the picture that the structures observed in the outer disk can be interpreted as the result of the density waves that, by outward transport of angular momentum, are expected to be associated with the excitation of the spiral modes that dominate the bright optical disk (Bertin & Amorisco, 2010, 512, 17). By means of 3D hydrodynamical simulations, we investigate the properties of nonaxisymmetric density wave trains in the outermost regions. The wave trains are taken to be generated in the inner disk; in the gas they can penetrate through the outer Lindblad resonance and propagate outwards. We study the case when one dominant mode is present, and then other situations when more than one mode participate in determining the large-scale spiral structure. We show that the amplitude of the perturbations increases with radius, forming prominent spiral patterns. Outside the bright optical stellar disk, the regular spiral pattern can thus be explained as the signature of the dominant modes generated in the inner disk. In the low-amplitude regime, the patterns are characterized by properties in general agreement with the predictions of the linear theory. At large radii, the density waves eventually become nonlinear. With the aim to compare results of our simulations with specific observed cases we also computed the synthetic HI data cubes of model gaseous disks.