Exact solutions of nonlocal nonlinear field equations in cosmology

We consider a method for seeking exact solutions of the equation of a nonlocal scalar field in a nonflat metric. In the Friedmann–Robertson–Walker metric, the proposed method can be used in the case of an arbitrary potential except linear and quadratic potentials, and it allows obtaining solutions in quadratures depending on two arbitrary parameters. We find exact solutions for an arbitrary cubic potential, which consideration is motivated by string field theory, and also for exponential, logarithmic, and power potentials. We show that the $k$-essence field can be added to the model to obtain exact solutions satisfying all the Einstein equations.