  The problem of Turing instabilities for a reaction-diffusion system defined
on a complex Cartesian product networks is considered. To this end we operate
in the linear regime and expand the time dependent perturbation on a basis
formed by the tensor product of the eigenvectors of the discrete Laplacian
operators, associated to each of the individual networks that build the
Cartesian product. The dispersion relation which controls the onset of the
instability depends on a set of discrete wave- lenghts, the eigenvalues of the
aforementioned Laplacians. Patterns can develop on the Cartesian network, if
they are supported on at least one of its constituive sub-graphs. Multiplex
networks are also obtained under specific prescriptions. In this case, the
criteria for the instability reduce to compact explicit formulae. Numerical
simulations carried out for the Mimura-Murray reaction kinetics confirm the
adequacy of the proposed theory.
