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 non-linearity of order (2 k + 1) and scaling exponent (n) 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. 
Tarun Kanti Ghosh