DBI-type tachyons for inverse cosh potential

We consider classical and quantum dynamics of a tachyonic system described by a DBI type Lagrangian and inverse cosh potential. This investigation is partially motivated by the string theory and D-brane dynamics, but mostly by their application in cosmological inflation. A formalism for describing dynamics of spatially homogenous tachyon scalar field with this kind of potentials is developed. Classical actions and corresponding quantum propagators in the Feynman path integral approach, both on real and nonarchimedean spaces, are calculated. Possibilities for a quantum adelic generalization of these models are noticed. Cosmological applications are pointed out and discussed.