  We study the generalization of the dispersionless Kadomtsev - Petviashvili
(dKP) equation in n+1 dimensions and with nonlinearity of degree m+1, a model
equation describing the propagation of weakly nonlinear, quasi one dimensional
waves in the absence of dispersion and dissipation, and arising in several
physical contexts, like acoustics, plasma physics, hydrodynamics and nonlinear
optics. In 2+1 dimensions and with quadratic nonlinearity, this equation is
integrable through a novel IST, and it has been recently shown to be a
prototype model equation in the description of the two dimensional wave
breaking of localized initial data. In higher dimensions and with higher
nonlinearity, the generalized dKP equations are not integrable, but their
invariance under motions on the paraboloid allows one to construct in this
paper a family of exact solutions describing waves constant on their
paraboloidal wave front and breaking simultaneously in all points of it,
developing after breaking either multivaluedness or single valued discontinuous
shocks. Then such exact solutions are used to build the longtime behavior of
the solutions of the Cauchy problem, for small and localized initial data,
showing that wave breaking of small initial data takes place in the longtime
regime if and only if $m(n-1)\le 2$. At last, the analytic aspects of such a
wave breaking are investigated in detail in terms of the small initial data, in
both cases in which the solution becomes multivalued after breaking or it
develops a discontinuous shock. These results, contained in the 2012 master
thesis of one of the authors (FS), generalize those obtained by one of the
authors (PMS) and S.V.Manakov for the dKP equation in n+1 dimensions with
quadratic nonlinearity, and are obtained following the same strategy.
