  One of the predictions of the standard CDM is that dark haloes have centrally
divergent density profiles. An extensive body of rotation curve observations of
dwarf and low surface brightness galaxies shows the dark haloes of those
systems to be characterized by soft constant density central cores. Several
physical processes have been proposed to produce soft cores in dark haloes,
each one with different scaling properties. With the aim of discriminating
among them we have examined the rotation curves of dark matter dominated dwarf
and low surface brightness galaxies and the inner mass profiles of two clusters
of galaxies lacking a central cD galaxy and with evidence of soft cores in the
centre. The core radii and central densities of these haloes scale in a well
defined manner with the depth of their potential wells, as measured through the
maximum circular velocity. As a result of our analysis we identify
self-interacting CDM as a viable solution to the core problem, where a
non-singular isothermal core is formed in the halo center surrounded by a
Navarro, Frenk, & White profile in the outer parts. We show that this
particular physical situation predicts core radii in agreement with
observations. Furthermore, using the observed scalings, we derive an expression
for the minimum cross section (\sigma) which has an explicit dependence with
the halo dispersion velocity (v). If m_x is the mass of the dark matter
particle: \sigma/m_x ~4 10^-25 (v/100 km s^-1)^-1 cm^2/Gev.
