  The dispersionless Kadomtsev-Petviashvili (dKP) equation
$(u_t+uu_x)_x=u_{yy}$ is one of the simplest nonlinear wave equations
describing two-dimensional shocks. To solve the dKP equation we use a
coordinate transformation inspired by the method of characteristics for the
one-dimensional Hopf equation $u_t+uu_x=0$. We show numerically that the
solutions to the transformed equation do not develop shocks. This permits us to
extend the dKP solution as the graph of a multivalued function beyond the
critical time when the gradients blow up. This overturned solution is
multivalued in a lip shape region in the $(x,y)$ plane, where the solution of
the dKP equation exists in a weak sense only, and a shock front develops. A
local expansion reveals the universal scaling structure of the shock, which
after a suitable change of coordinates corresponds to a generic cusp
catastrophe. We provide a heuristic derivation of the shock front position near
the critical point for the solution of the dKP equation, and study the solution
of the dKP equation when a small amount of dissipation is added. Using
multiple-scale analysis, we show that in the limit of small dissipation and
near the critical point of the dKP solution, the solution of the dissipative
dKP equation converges to a Pearcey integral. We test and illustrate our
results by detailed comparisons with numerical simulations of both the
regularized equation, the dKP equation, and the asymptotic description given in
terms of the Pearcey integral.
