  In view of the many recent observations conducted by ALMA and SPHERE, it is
becoming clear that protoplanetary disks form planets in narrow annular gaps at
various distances from the central protostars before these protostars are
actually fully formed and the gaseous disks have concluded their
accretion/dispersal processes. This is in marked contrast to the many
multi-planet exoplanetary systems that do not conform to this pristine picture.
This major discrepancy calls for an explanation. We provide such an explanation
in this work, based on analytical solutions of the cylindrical isothermal
Lane-Emden equation with rotation which do not depend on boundary conditions.
These ``intrinsic'' solutions of the differential equation attract the
solutions of the Cauchy problem and force them to oscillate permanently. The
oscillations create density maxima in which dust and planetesimals are trapped
and they can form protoplanetary cores during the very early isothermal
evolution of such protoplanetary nebulae. We apply this model to our solar
nebula that formed in-situ a minimum of eleven protoplanetary cores that have
grown to planets which have survived undisturbed to the present day. We are
also in the process of applying the same model to the ALMA/DSHARP disks.
