Construction of Exact Solutions in Two-Fields Models and the Crossing of the Cosmological Constant Barrier

A dark energy model with a phantom scalar field, an usual scalar field and the string field theory inspired polynomial potential has been constructed. A two-parameter set of exact solutions to the Friedmann equations has been found. We have constructed such string inspired potential that some exact solutions correspond to the state parameter $w_{DE}>-1$ at large time, whereas other ones correspond to $w_{DE}<-1$ at large time. We demonstrate that the superpotential method is very effective to seek new exact solutions. We also present a two-fields model with a polynomial potential and the state parameter, which crosses the cosmological barrier infinitely often.