Universality of monopole mode and time evolution of a d-dimensional trapped interacting Bose gas

We study a generalised Gross-Pitaevskii equation describing a d-dimensional harmonic trapped (with trap frequency omega (0)) weakly interacting Bose gas with a nonlinearity of order (2k + 1) and scaling exponent (ni of the interaction potential. Using the time-dependent variational analysis, we explicitly show that for a particular combination of n, k and d, the generalised GP equation has the universal monopole oscillation frequency 2 omega (0). We also find that the time-evolution of the width can be described universally by the same Hill's equation if the system satisfy that particular combination. We also obtain the condition for the exact self-similar solutions of the Gross-Pitaevskii equation. As an application, we discuss low-dimensional trapped Bose condensate state and Calogero model. (C) 2001 Published by Elsevier Science B.V.