  Discrete solitons in the Ablowitz-Ladik (AL) and discrete nonlinear
Schr\"odinger (DNLS) equations with damping and strong rapid drive are
investigated. The averaged equations have the forms of the parametric AL and
DNLS equations. A new type of parametric bright discrete soliton and cnoidal
waves are found and the stability properties are analyzed. The analytical
predictions of the perturbed inverse scattering transform are confirmed by the
numerical simulations of the AL and DNLS equations with rapidly varying drive
and damping.
