Spin-orbit-coupled quantum gases are characterized by a rich
phase diagram, with the appearance of novel quantum phases
having intriguing static and dynamic features.
In this talk I shall discuss the effect of a periodic
potential generated by a one-dimensional optical lattice on
the properties of a S=1/2 spin-orbit-coupled Bose gas.
Because of the interplay between the modified band structure
and the two-body interaction the ground state exhibits a
mixed regime, where the condensate wave function
is given by a superposition of multiple Bloch-wave
components, and an unmixed one, in which the atoms occupy a
single Bloch state. Magnetic phase transitions between
polarized and unpolarized states can be induced by tuning
the Raman coupling and the lattice strength, with the
emergence of a quantum tricritical point whose existence
is a consequence of the spin-dependent interaction. At the
dynamic level, I shall illustrate the behavior of the phonon
and roton modes of the excitation spectrum,
pointing out the instabilities occurring when a phase
transition is approached.