  We investigate the phase behaviour of a two-dimensional colloidal model
system of ultra-soft particles on a substrate which varies periodically along
one spatial direction. Our calculations are based on mean-field density
functional theory for a system of particles interacting via an ultra-soft
potential, that is, the generalized exponential model with index four
(\mbox{GEM-4}). For suitable substrate periodicities (with commensurability
parameter $p=2$), we find a succession of phase transitions from a modulated
liquid to a locked smectic and then to a locked floating solid phase. The
appearance of a locked smectic phase is consistent with earlier theoretical
predictions and experiments for freezing of more repulsive systems on
structured surfaces (with $p=2$). However, the present ultra-soft system does
not display re-entrant melting. We here investigate the details of the density
distributions of the different phases, thereby supplementing earlier work on
GEM-4 systems with $p=1$ [Phys. Rev. E \textbf{101}, 012609 (2020)].
Interestingly, the observed succession of phase transitions can be triggered
through different paths along which physical control parameters are changed.
