Construction of exact solutions in two-field cosmological models

We construct a dark energy model with a phantom scalar field, a standard scalar field, and a polynomial potential inspired by string field theory. We find a two-parameter set of exact solutions of the Friedmann equations. We find a potential satisfying the conditions obtained from the string theory and such that at large times, some of the exact solutions correspond to the state parameter $w_{\mathrm{DE}}>-1$ while the others correspond to $w_{\mathrm{DE}}<-1$. We demonstrate that the superpotential method is very effective for seeking new exact solutions.