  The phase-turbulent (PT) regime for the one dimensional complex
Ginzburg-Landau equation (CGLE) is carefully studied, in the limit of large
systems and long integration times, using an efficient new integration scheme.
Particular attention is paid to solutions with a non-zero phase gradient. For
fixed control parameters, solutions with conserved average phase gradient $\nu$
exist only for $|\nu|$ less than some upper limit. The transition from phase to
defect-turbulence happens when this limit becomes zero. A Lyapunov analysis
shows that the system becomes less and less chaotic for increasing values of
the phase gradient. For high values of the phase gradient a family of
non-chaotic solutions of the CGLE is found. These solutions consist of
spatially periodic or aperiodic waves travelling with constant velocity. They
typically have incommensurate velocities for phase and amplitude propagation,
showing thereby a novel type of quasiperiodic behavior. The main features of
these travelling wave solutions can be explained through a modified
Kuramoto-Sivashinsky equation that rules the phase dynamics of the CGLE in the
PT phase. The latter explains also the behavior of the maximal Lyapunov
exponents of chaotic solutions.
