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

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.

SOME RESULTS ON PARTIAL DIFFERENTIAL EQUATIONS AND ASIAN OPTIONS

We analyze partial differential equations arising in the evaluation of Asian options. The equations are strongly degenerate partial differential equations in three dimensions. We show that the solution of the no-arbitrage partial differential equation is sufficiently regular and standard numerical methods can be employed to approximate it.