  We establish a new self-consistent model of coupling between the cosmic dark
energy and dark matter in the framework of the rheological approach, which is
based on the representation of the equations of state in terms of integral
operators of the Volterra-type. We elaborate the so-called four-kernel model,
in the framework of which both the dark energy and dark matter pressures are
presented by two integrals containing the energy densities of the dark energy
and dark matter. For the Volterra operators, the kernels of which are
associated with the effects of fading memory, the corresponding isotropic
homogeneous cosmological model is shown to be exactly integrable. We consider
the classification of the model exact solutions, based on the analysis of roots
of the characteristic polynomial associated with the key equation of the
presented model. The scalars of the pressure and energy-density of the dark
energy and dark matter, the Hubble function and acceleration parameter are
presented explicitly as the functions of the dimensionless scale factor. The
scale factor as the function of the cosmological time is found in quadratures
and is described analytically, qualitatively and numerically. Asymptotic
analysis allowed us to classify the models with respect to behavior typical for
the Big Rip, Little Rip and Pseudo Rip (de Sitter type). Two intriguing exact
cosmological solutions are discussed, which describe the super-exponential
expansion and the symmetric bounce. New solutions are presented, which
correspond to the quasi-periodic behavior of the state functions of the dark
fluid and of the geometric characteristics of the Universe.
