Some exact solutions in K-essence theory isotropic cosmology

We use a simple form of the K-essence theory and apply it to the classic isotropic cosmological model and seek exact solutions. The particular form of the kinetic term that we choose is $K \left(\phi, X \right)= K_0(\phi)X^m +K_1$. The resulting field equations in the homogeneous and isotropic cosmology (FRW)is considered. Several exact solutions are obtained.