Evolution and Global Collapse of Trapped Bose Condensates under Variations of the Scattering Length

We develop the idea of selectively manipulating the condensate in a trapped Bose-condensed gas, without perturbing the thermal cloud. The idea is based on the possibility to modify the mean field interaction between atoms (scattering length) by nearly resonant incident light or by spatially uniform change of the trapping magnetic field. For the gas in the Thomas-Fermi regime we find analytical scaling solutions for the condensate wave function evolving under arbitrary variations of the scattering length $a$. The change of $a$ from positive to negative induces a global collapse of the condensate, and the final stages of the collapse will be governed by intrinsic decay processes.