  In this paper we investigate the formation and propagation of singularities
for the system for one-dimensional Chaplygin gas. In particular, under suitable
assumptions we construct a physical solution with a new type of singularities
called "Delta-like" solution for this kind of quasilinear hyperbolic system
with linearly degenerate characteristics. By careful analysis, we study the
behavior of the solution in a neighborhood of a blowup point. The formation of
this new kind of singularities is due to the envelope of the different families
of characteristics instead of the same family of characteristics in the
traditional situation. This shows that the blowup phenomenon of solution for
the system with linearly degenerate characteristics is quite different from the
situation of shock formation for the system with genuinely nonlinear
characteristics. Different initial data can lead to kinds of different
Delta-like singularities: the Delta-like singularity with point-shape and
Delta-like singularity with line-shape.
