  We investigate a k-essence field localized on the brane evolving linearly
with the cosmological time for arbitrary kinetic functions and consider the
atypical k-essence model for linear and nonlinear k-fields in
Friedmann-Robertson-Walker (FRW) cosmology. In the former case the k-field is
driven by an inverse quadratic polynomial potential and the solutions exhibit
several different behaviors which include expanding, contracting and bouncing
universes as well as a model with a finite time span, some of them ending in a
big crunch or a big rip. In the latter case we find the potential and show that
the atypical k-essence model is dynamically uncomplete. Particularly, by
selecting the extended tachyonic kinetic functions we analyze the high and low
energy limits of our model, obtaining the nearly power law solution. We
introduce a tachyon field with negative energy density and show that the
universe evolves between two singularities.
