  We present a family of nonautonomous bright and dark soliton solutions of
Bose-Einstein condensates with the time-dependent scattering length in an
expulsive parabolic potential. These solutions show that the amplitude, width,
and velocity of soliton can be manipulated by adjusting the atomic scattering
length via Feshbach resonance. For the cases of both attractive and repulsive
interaction, the total particles number is a conservation quantity, but the
peak (dip) density can be controlled by the Feshbach resonance parameter.
Especially, we investigate the modulation instability process in uniform
Bose-Einstein condensates with attractive interaction and nonvanishing
background, and clarify that the procedure of pattern formation is in fact the
superposition of the perturbed dark and bright solitary wave. At last, we give
the analytical expressions of nonautonomous dark one- and two-soliton solutions
for repulsive interaction, and investigate their properties analytically.
