  We consider the dynamics of message passing for spatially coupled codes and,
in particular, the set of density evolution equations that tracks the profile
of decoding errors along the spatial direction of coupling. It is known that,
for suitable boundary conditions and after a transient phase, the error profile
exhibits a "solitonic behavior". Namely, a uniquely-shaped wavelike solution
develops, that propagates with constant velocity. Under this assumption we
derive an analytical formula for the velocity in the framework of a continuum
limit of the spatially coupled system. The general formalism is developed for
spatially coupled low-density parity-check codes on general binary memoryless
symmetric channels which form the main system of interest in this work. We
apply the formula for special channels and illustrate that it matches the
direct numerical evaluation of the velocity for a wide range of noise values. A
possible application of the velocity formula to the evaluation of finite size
scaling law parameters is also discussed. We conduct a similar analysis for
general scalar systems and illustrate the findings with applications to
compressive sensing and generalized low-density parity-check codes on the
binary erasure or binary symmetric channels.
