Collapse and stable self-trapping for Bose-Einstein condensates with 1/rb -type attractive interatomic interaction potential

Abstract

We consider dynamics of Bose-Einstein condensates with long-range attractive interaction proportional to $1/{r}^{b}$ and arbitrary angular dependence. It is shown exactly that collapse of a Bose-Einstein condensate without contact interactions is possible only for $b\ensuremath{\geqslant}2$. The case $b=2$ is critical and requires the number of particles to exceed a critical value to allow collapse. The critical collapse in that case is a strong one, trapping into the collapsing region a finite number of particles. The case $b>2$ is supercritical with an expected weak collapse which traps a rapidly decreasing number of particles during approach to collapse. For $b<2$ a singularity at $r=0$ is not strong enough to allow collapse but an attractive $1/{r}^{b}$ interaction admits stable self-trapping (a stable three-dimensional soliton solution) even in the absence of an external trapping potential.