Dynamics of Collapse of a Confined Bose Gas

Rigorous results on the nonlinear dynamics of a dilute Bose gas with a negative scattering length in an 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 equations in an external field we analyze the temporal evolution of the mean square radius of the gas cloud. In the 2D case the quantity undergoes harmonic oscillation with frequency $2\omega _0$ It implies that for a negative value of 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 collapsed after a finite time for a state with negative energy. A stringent condition for the collapse is also derived.