Dark energy in some integrable and nonintegrable FRW cosmological models

One of the greatest challenges in cosmology today is to determine the nature of dark energy, the sourse of the observed present acceleration of the Universe. Besides the vacuum energy, various dark energy models have been suggested. The Friedmann - Robertson - Walker (FRW) spacetime plays an important role in modern cosmology. In particular, the most popular models of dark energy work in the FRW spacetime. In this work, a new class of integrable FRW cosmological models is presented. These models induced by the well-known Painlev$\acute{e}$ equations. Some nonintegrable FRW models are also considered. These last models are constructed with the help of Pinney, Schr$\ddot{o}$dinger and hypergeometric equations. Scalar field description and two-dimensional generalizations of some cosmological models are presented. Finally some integrable and nonintegrable $F(R)$ and $F(G)$ gravity models are constructed.