Compact patterns in a class of sublinear Gardner equations

We study a class of sublinear Gardner equations u_t+−u^α±mu^m_x+u_xxx=0, where 0<α<1<m, which exhibit two distinguished features; their solitary waves have a finite span, i.e., are compactons and, depending on their amplitude, may propagate either to the right or to the left, and thus undergo both chase and head-on interactions or, switch direction upon collision. All in all, their morphology, is vastly richer than encountered in the unidirectional KdV-like equations. The multi-dimensional extension u_t+−u^α±mu^m_x+(∇²u)_x=0, reveals an even richer landscape; each admissible velocity supports an entire spectrum of multi-nodal compactons.