On singular and sincerely singular compact patterns

Abstract

A third order dispersive equation ut+(um)x+1b[ua∇2ub]x=0 is used to explore two very different classes of compact patterns. In the first, the prevailing singularity at the edge induces traveling compactons, solitary waves with a compact support. In the second, the singularity induced at the perimeter of the initial excitation, entraps the dynamics within the domain's interior (nonetheless, certain very singular excitations may escape it). Here, overlapping compactons undergo interaction which may result in an interchange of their positions, or form other structures, all confined within their initial support. We conjecture, and affirm it empirically, that whenever the system admits more than one type of compactons, only the least singular compactons may be evolutionary. The entrapment due to singularities is also unfolded and confirmed numerically in a class of diffusive equations ut=uk∇2un with k>1 and n>0 with excitations entrapped within their initial support observed to converge toward a space–time separable structure. A similar effect is also found in a class of nonlinear Klein–Gordon Equations.