  We explore the structure around shell-crossing time of cold dark matter
protohaloes seeded by two or three crossed sine waves of various relative
initial amplitudes, by comparing Lagrangian perturbation theory (LPT) up to
10th order to high-resolution cosmological simulations performed with the
public Vlasov code ColDICE. Accurate analyses of the density, the velocity, and
related quantities such as the vorticity are performed by exploiting the fact
that ColDICE can follow locally the phase-space sheet at the quadratic level.
To test LPT predictions beyond shell-crossing, we employ a ballistic
approximation, which assumes that the velocity field is frozen just after
shell-crossing. In the generic case, where the amplitudes of the sine waves are
all different, high-order LPT predictions match very well the exact solution,
even beyond collapse. As expected, convergence slows down when going from
quasi-1D dynamics where one wave dominates over the two others, to the
axial-symmetric configuration, where all the amplitudes of the waves are equal.
It is also noticed that LPT convergence is slower when considering velocity
related quantities. Additionally, the structure of the system at and beyond
collapse given by LPT and the simulations agrees very well with singularity
theory predictions, in particular with respect to the caustic and vorticity
patterns that develop beyond collapse. Again, this does not apply to
axial-symmetric configurations, that are still correct from the qualitative
point of view, but where multiple foldings of the phase-space sheet produce
very high density contrasts, hence a strong backreaction of the gravitational
force.
