Extended tachyon field, Chaplygin gas and solvable k-essence cosmologies

We investigate a flat Friedmann-Robertson-Walker spacetime filled with k-essence and find the set of functions F which generate three different families of extended tachyon fields and Chaplygin gases. They lead to accelerated and superaccelerated expanding scenarios. For any function F, we find the first integral of the k-field equation when the k-field is driven by an inverse square potential or by a constant one. In the former, we obtain the general solution of the coupled Einstein-k-field equations for a linear function F. This model shares the same kinematics of the background geometry with the ordinary scalar field one driven by an exponential potential. However, they are dynamically different. For a constant potential, we introduce a k-field model that exhibits a transition from a power-law phase to a de Sitter stage, inducing a modified Chaplygin gas.