  Optical response of free-electron gas leads to inherent nonlinear optical
behaviour of nanostructured plasmonic materials enabled via both strong local
field enhancements and inherent complex electron dynamics. We present a
comprehensive treatment of microscopic polarization of conduction electrons in
the time-domain using full hydrodynamic description which allows
self-consistent modelling of linear and nonlinear response and multiple
harmonics generation. The effects of convective acceleration, magnetic
contribution of the Lorenz force, quantum electron pressure, and nanostructures
boundaries have been taken into account leading to simultaneous appearance of
second and third harmonics. The developed method provides an ultimate approach
to investigate nonlinear responses of arbitrarily-shaped complex nano-scale
plasmonic structures and enables addressing their self-consistent nonlinear
dynamics.
