Multi-dimensional compactons and compact vortices

Abstract

As though to compensate for the rarity of multidimensional integrable systems, non-integrable spatial extensions of many of the well known dispersive equations on the line exhibit a remarkable variety of solitary patterns unavailable in 1D. In the present work we elaborate upon the ubiquity of this effect and find it to be even more pronounced in compacton admitting systems which admit new patterns otherwise unavailable, in both the real (the sublinear ZK) and the complex realm (the sublinear NLS and complex KG). In the planar complex case, in addition to a countable number of radial solutions, for every spin number m we find both analytical and numerical evidence of a countable number of multi-modal compact vortices which have a strictly finite radius and, depending on m, either extend to the origin or vanish identically in a disk around it.