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 u_{t}+ partial differential_{x}f(u)=u_{xx} and dispersive u_{t}+ partial differential_{x}f(u)+u_{xxx}=0 patterns, and show that a weakly singular convection f(u)=-u;{alpha}+u;{m}, 0<alpha<1<m, induces a sharp localization of fronts around the u=0 ground state. Notably, a sharp front also emerges in higher dimensional extensions: u_{t}+ partial differential_{x}[f(u)+nabla;{2}u]=0 or in wave phenomena of the Boussinesq type: Z_{tt}=nabla.[F_{*}(|nablaZ|)nablaZ]-nabla;{4}Z where F_{*}(sigma)=C;{2}sigma+f(sigma).
Rosenau
Philip
P
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. rosenau@post.tau.ac.il