Exact Localized and Periodic Solutions of the Discrete Complex Ginzburg-Landau Equation

We study, analytically, the discrete complex cubic Ginzburg-Landau (dCCGL) equation. We derive the energy balance equation for the dCCGL and consider various limiting cases. We have found a set of exact solutions which includes as particular cases periodic solutions in terms of elliptic Jacobi functions, bright and dark soliton solutions, and constant magnitude solutions with phase shifts. We have also found the range of parameters where each exact solution exists. We discuss the common features of these solutions and solutions of the continuous complex Ginzburg-Landau model and solutions of Hamiltonian discrete systems and also their differences.