  The transmissions of matter wave solitons through linear and nonlinear
inhomogeneities induced by the spatial variations of the trap and the
scattering length in Bose-Einstein condensates are investigated. New phenomena,
such as the enhanced transmission of a soliton through a linear trap by a
modulation of the scattering length, are exhibited. The theory is based on the
perturbed Inverse Scattering Transform for solitons, and we show that radiation
effects are important. Numerical simulations of the Gross-Pitaevskii equation
confirm the theoretical predictions.
