  We study the evolution of a collisionally inhomogeneous matter wave in a
spatial gradient of the interaction strength. Starting with a Bose-Einstein
condensate with weak repulsive interactions in quasi-one-dimensional geometry,
we monitor the evolution of a matter wave that simultaneously extends into
spatial regions with attractive and repulsive interactions. We observe the
formation and the decay of soliton-like density peaks, counter-propagating
self-interfering wave packets, and the creation of cascades of solitons. The
matter-wave dynamics is well reproduced in numerical simulations based on the
nonpolynomial Schroedinger equation with three-body loss, allowing us to better
understand the underlying behaviour based on a wavelet transformation. Our
analysis provides new understanding of collapse processes for solitons, and
opens interesting connections to other nonlinear instabilities.
