Shock waves and compactons for fifth-order non-linear dispersion equations

Abstract

The followingfirst problemis posed:is a correct ‘entropy solution’ of the Cauchy problem for the fifth-order degenerate non-linear dispersion equations (NDEs), same as for the classic Euler oneut+uux= 0,These two quasi-linear degenerate partial differential equations (PDEs) are chosen as typical representatives; so other (2m+ 1)th-order NDEs of non-divergent form admit such shocks waves. As a relatedsecond problem, the opposite initial shockS+(x) = −S−(x) = signxis shown to be a non-entropy solution creating ararefaction wave, which becomesC∞for anyt> 0. Formation of shocks leads to non-uniqueness of any ‘entropy solutions’. Similar phenomena are studied for afifth-order in timeNDEuttttt= (uux)xxxxinnormal form.On the other hand, related NDEs, such asare shown to admit smoothcompactons, as oscillatorytravelling wavesolutions with compact support. The well-known non-negative compactons, which appeared in various applications (first examples by Dey, 1998,Phys. Rev.E, vol. 57, pp. 4733–4738, and Rosenau and Levy, 1999,Phys. Lett.A, vol. 252, pp. 297–306), are non-existent in general and are not robust relative to small perturbations of parameters of the PDE.