Compactification of patterns by a singular convection or stress

A wide variety of propagating disturbances in physical systems are described by equations whose solutions lack a sharp propagating front. We demonstrate that presence of particular nonlinearities may induce such fronts. To exemplify this idea, we study both dissipative ut+∂xf(u)=uxx and dispersive ut+∂xf(u)+uxxx=0 patterns, and show that a weakly singular convection f(u)=-uα+um, 0<α<1<m, induces a sharp localization of fronts around the u=0 ground state. Notably, a sharp front also emerges in higher dimensional extensions: ut+∂x[f(u)+2u]=0 or in wave phenomena of the Boussinesq type: Ztt=•[F*(|Z|)Z]-4Z where F*(σ)=C2σ+f(σ).