  Models where the dark energy is a scalar field with a non-standard
Dirac-Born-Infeld (DBI) kinetic term are investigated. Scaling solutions are
studied and proven to be attractors. The corresponding shape of the brane
tension and of the potential is also determined and found to be, as in the
standard case, either exponentials or power-law of the DBI field. In these
scenarios, in contrast to the standard situation, the vacuum expectation value
of the field at small redshifts can be small in comparison to the Planck mass
which could be an advantage from the model building point of view. This
situation arises when the present-day value of the Lorentz factor is large,
this property being per se interesting. Serious shortcomings are also present
such as the fact that, for simple potentials, the equation of state appears to
be too far from the observational favored value -1. Another problem is that,
although simple stringy-inspired models precisely lead to the power-law shape
that has been shown to possess a tracking behavior, the power index turns out
to have the wrong sign. Possible solutions to these issues are discussed.
