Similarity Solutions and Collapse in the Attractive Gross-Pitaevskii Equation

Details Similarity Solutions and Collapse in the Attractive Gross-Pitaevskii Equation Similarity Solutions and Collapse in the Attractive Gross-Pitaevskii Equation

2000-01-06

arxiv.org/abs/cond-mat/0001059v2

Phys. Rev. E 62, 6224 (2000)

Physics

Condensed Matter

LaTeX, no figures, v.2(final); discussion of collapse issue extended

Scientific paper

10.1103/PhysRevE.62.6224

We analyse a generalised Gross-Pitaevskii equation involving a paraboloidal trap potential in $D$ space dimensions and generalised to a nonlinearity of order $2n+1$. For {\em attractive} coupling constants collapse of the particle density occurs for $Dn\ge 2$ and typically to a $\delta$-function centered at the origin of the trap. By introducing a new dynamical 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 G-P equation in 2 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.

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