  Traveling waves triggered by a phase slip in coupled map lattices are
studied. A local phase slip affects globally the system, which is in strong
contrast with kink propagation. Attractors with different velocities coexist,
and form quantized bands determined by the number of phase slips. The mechanism
and statistical and dynamical characters are studied with the use of spatial
asymmetry, basin volume ratio, Lyapunov spectra, and mutual information. If the
system size is not far from an integer multiple of the selected wavelength,
attractors are tori, while weak chaos remains otherwise, which induces chaotic
modulation of waves or a chaotic itinerancy of traveling states. In the
itinerancy, the residence time distribution obeys the power law distribution,
implying the existence of a long-ranged correlation. Supertransients before the
formation of traveling waves are noted in the high nonlinearity regime. In the
weaker nonlinearity regime corresponding to the frozen random pattern, we have
found fluctuation of domain sizes and Brownian-like motion of domains.
Propagation of chaotic domains by phase slips is also found. Relevance of our
discoveries to B\'{e}nard convection experiments and possible applications to
information processing are briefly discussed.
