Dynamics of collapse of a confined Bose gas

Rigorous results on the nonlinear dynamics of a dilute Bose gas in a harmonic magnetic trap are presented and sufficient conditions for the collapse of the system are formulated. By using the virial theorem for the Gross-Pitaevskii equation in an external field we analyze the temporal evolution of the mean square radius I = integral r(2) \psi\(2) dv of the gas cloud. in the 2D case this quantity undergoes harmonic oscillations with frequency 2 omega(o). This implies that for a negative scattering length and for a negative value of the energy of the system, the gas cloud will collapse after a finite time interval. For positive energy the cloud collapses if the initial conditions correspond to a large enough amplitude of oscillations. Stable oscillations with a small amplitude are possible. In the 3D case the system also collapses after a finite time for a state with negative energy. A stringent condition for the collapse is also derived.