  Collisional dynamics of solitary matter waves of superfluid hard core bosons,
consisting of dark and bright non-linear waves as well as supersonic periodic
trains, reveals remarkable richness and coherence, with the phase of the
condensate playing a key role. Depending upon the condensate density and their
relative velocity, two distinctive collisional types emerge: the intuitively
expected repulsive collisions due to the hard core boson constraint; and, also
collisions in which they "pass through" one another. In addition to confirming
the soliton status of both bright and dark solitary waves, our studies reveal a
variety of multi-solitons including multiple families of breathers, that can be
produced and precisely controlled via quantum phase engineering.
