  An analysis of traveling wave solutions of pure cross-diffusion systems,
i.e., systems that lack reaction and self-diffusion terms, is presented. Using
the qualitative theory of phase plane analysis the conditions for existence of
different types of wave solutions are formulated. In particular, it is shown
that family of wave trains is a generic phenomenon in pure cross-diffusion
systems. The results can be used for construction and analysis of different
mathematical models describing systems with directional movement.
