A variety of new complex waves representing solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity is investigated. Two different approaches are used, namely the generalized exponential function and the unified methods. Complex periodic, solitary, soliton, and elliptic wave solutions of phenomena that occur in nonlinear optics or in plasma physics are obtained. The physical meaning of the geometrical structures for some solutions is discussed for different choices of the free parameters. It is shown that the proposed methods lead to powerful mathematical tools for obtaining the exact traveling wave solutions of complex nonlinear evolution equations.
