  Domain-wall solutions in four-dimensional supersymmetric field theories with
distinct discrete vacuum states lead to the spontaneous breaking of
supersymmetry, either completely or partially. We consider in detail the case
when the domain walls are the BPS-saturated states, and 1/2 of supersymmetry is
preserved. Several useful criteria that relate the preservation of 1/2 of
supersymmetry on the domain walls to the central extension appearing in the N=1
superalgebras are established. We explain how the central extension can appear
in N=1 supersymmetry and explicitly obtain the central charge in various
models: the generalized Wess-Zumino models, and supersymmetric Yang-Mills
theories with or without matter. The BPS-saturated domain walls satisfy the
first-order differential equations which we call the creek equations, since
they formally coincide with the (complexified) equations of motion of an analog
high-viscosity fluid on a profile which is given by the superpotential of the
original problem. Some possible applications are considered.
