  The possibility of generating multiple coherent modes in trapped Bose gases
is advanced. This requires the usage of several driving fields whose
frequencies are tuned close to the corresponding transition frequencies. A
general criterion is derived explaining when the driving fields, even being in
perfect resonance, cannot generate the topological coherent modes. This
criterion is termed the theorem of shape conservation. Bose-Einstein
condensates with generated coherent modes display a number of interesting
effects, such as: interference fringes, interference current, mode locking,
dynamic transition, critical phenomena, chaotic motion, harmonic generation,
parametric conversion, atomic squeezing, and entanglement production.
Approximate solutions, based on the averaging techniques, are found to be in
good agreement with direct numerical calculations for the Gross-Pitaevskii
equation.
