Similarity solutions and collapse in the attractive gross-pitaevskii equation

We analyze a generalized Gross-Pitaevskii (GP) equation involving a paraboloidal trap potential in D space dimensions and generalized to a nonlinearity of order 2n+1. For attractive coupling constants collapse of the particle density occurs for Dn>/=2 and typically to a delta function centered at the origin of the trap. By introducing a special variable for the spherically symmetric solutions, we show that all such solutions are self-similar close to the center of the trap. Exact self-similar solutions occur if, and only if, Dn=2, and for this case of Dn=2 we exhibit an exact but rather special D=1 analytical self-similar solution collapsing to a delta function which, however, recovers and collapses periodically, while the ordinary GP equation in two space dimensions also has a special solution with periodic delta function collapses and revivals of the density. The relevance of these various results to attractive Bose-Einstein condensation in spherically symmetric traps is discussed.