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 g as with a nonlinearity of order (2k + 1) and scaling exponent (ni of the in teraction potential. Using the time-dependent variational analysis, we expl icitly show that for a particular combination of n, k and d, the generalise d GP equation has the universal monopole oscillation frequency 2 omega (0). We also find that the time-evolution of the width can be described univers ally by the same Hill's equation if the system satisfy that particular comb ination. We also obtain the condition for the exact self-similar solutions of the Gross-Pitaevskii equation. As an application, we discuss low-dimensi onal trapped Bose condensate state and Calogero model. (C) 2001 Published b y Elsevier Science B.V.