  We introduce a technique for the analysis of general spatially coupled
systems that are governed by scalar recursions. Such systems can be expressed
in variational form in terms of a potential functional. We show, under mild
conditions, that the potential functional is \emph{displacement convex} and
that the minimizers are given by the fixed points of the recursions.
Furthermore, we give the conditions on the system such that the minimizing
fixed point is unique up to translation along the spatial direction. The
condition matches those in \cite{KRU12} for the existence of spatial fixed
points. \emph{Displacement convexity} applies to a wide range of spatially
coupled recursions appearing in coding theory, compressive sensing, random
constraint satisfaction problems, as well as statistical mechanical models. We
illustrate it with applications to Low-Density Parity-Check and generalized
LDPC codes used for transmission on the binary erasure channel, or general
binary memoryless symmetric channels within the Gaussian reciprocal channel
approximation, as well as compressive sensing.
