  Deep HI observations of the outer parts of disc galaxies demonstrate the
frequent presence of extended, well-developed spiral arms far beyond the
optical radius. To understand the nature and the origin of such outer spiral
structure, we investigate the propagation in the outer gaseous disc of
large-scale spiral waves excited in the bright optical disc. Using
hydrodynamical simulations, we show that non-axisymmetric density waves,
penetrating in the gas through the outer Lindblad resonance, can exhibit
relatively regular spiral structures outside the bright optical stellar disc.
For low-amplitude structures, the results of numerical simulations match the
predictions of a simple WKB linear theory. The amplitude of spiral structure
increases rapidly with radius. Beyond $\approx 2$ optical radii, spirals become
nonlinear (the linear theory becomes quantitatively and qualitatively
inadequate) and unstable to Kelvin-Helmholtz instability. In numerical
simulations, in models for which gas is available very far out, spiral arms can
extend out to 25 disc scale-lengths. A comparison between the properties of the
models we have investigated and the observed properties of individual galaxies
may shed light into the problem of the amount and distribution of dark matter
in the outer halo.
