  Quaternion-valued solutions to the non-commutative KdV equation are produced
using determinants. The solutions produced in this way are (breather) soliton
solutions, rational solutions, spatially periodic solutions and hybrids of
these three basic types. A complete characterization of the parameters that
lead to non-singular 1-soliton and periodic solutions is given. Surprisingly,
it is shown that such solutions are never singular when the solution is
essentially non-commutative. When a 1-soliton solution is combined with another
solution through an iterated Darboux transformation, the result behaves
asymptotically like a combination of different solutions. This ``non-linear
superposition principle'' is used to find a formula for the phase shift in the
general 2-soliton interaction. A concluding section compares these results with
other research on non-commutative soliton equations and lists some open
questions.
