A generalized cubic Volterra lattice hierarchy and its integrable couplings system

In terms of properties of the known loop algebra and difference operators, a new algebraic system χ is constructed. By using the algebraic system χ, a discrete matrix spectral problem is introduced and a hierarchy of nonlinear lattice equations is derived. From the hierarchy the celebrated cubic Volterra lattice equation is engendered. We call the hierarchy a generalized cubic Volterra hierarchy. Then an extended algebraic system of χ is presented, from which the integrable couplings system of the generalized cubic Volterra lattice are obtained.