  We consider a scalar field model with a self-interaction potential that
possesses a discrete vacuum manifold. We point out that this model allows for
both topological as well as non-topological solitons. In (1+1) dimensions both
type of solutions have finite energy, while in (3+1) dimensions, the
topological solitons have finite energy per unit area only and correspond to
domain walls. Non-topological solitons with finite energy do exist in (3+1)
dimensions due to a non-trivial phase of the scalar field and an associated
U(1) symmetry of the model, though. We construct these so-called Q-ball
solutions numerically, point out the differences to previous studies with
different scalar field potentials and also discuss the influence of a minimal
coupling to both gravity as well as a U(1) gauge field. In this latter case,
the conserved Noether charge Q can be interpreted as the electric charge of the
solution.
