  We look at simple BPS systems involving more than one field. We discuss the
conditions that have to be imposed on various terms in Lagrangians involving
many fields to produce BPS systems and then look in more detail at the simplest
of such cases. We analyse in detail BPS systems involving 2 interacting
Sine-Gordon like fields, both when one of them has a kink solution and the
second one either a kink or an antikink solution. We take their solitonic
static solutions and use them as initial conditions for their evolution in
Lorentz covariant versions of such models. We send these structures towards
themselves and find that when they interact weakly they can pass through each
other with a phase shift which is related to the strength of their interaction.
When they interact strongly they repel and reflect on each other. We use the
method of a modified gradient flow in order to visualize the solutions in the
space of fields.
