  Gravitational coupling between protoplanetary discs and planets embedded in
them leads to the emergence of spiral density waves, which evolve into shocks
as they propagate through the disc. We explore the performance of a
semi-analytical framework for describing the nonlinear evolution of the global
planet-driven density waves, focusing on the low planet mass regime (below the
so-called thermal mass). We show that this framework accurately captures the
(quasi-)self-similar evolution of the wave properties expressed in terms of
properly rescaled variables, provided that certain theoretical inputs are
calibrated using numerical simulations (an approximate, first principles
calculation of the wave evolution based on the inviscid Burgers equation is in
qualitative agreement with simulations but overpredicts wave damping at the
quantitative level). We provide fitting formulae for such inputs, in
particular, the strength and global shape of the planet-driven shock accounting
for nonlinear effects. We use this nonlinear framework to theoretically compute
vortensity production in the disc by the global spiral shock and numerically
verify the accuracy of this calculation. Our results can be used for
interpreting observations of spiral features in discs, kinematic signatures of
embedded planets in CO line emission ("kinks"), and for understanding the
emergence of planet-driven vortices in protoplanetary discs.
