Shock Waves and Compactons for Fifth-order Nonlinear Dispersion Equations

The following question is posed: to justify that the standing shock wave S−(x) = −signx = − { −1 for x < 0, 1 for x > 0, is a correct “entropy” solution of fifth-order nonlinear dispersion equations (NDEs), ut = −(uux)xxxx and ut = −(uuxxxx)x in R × R+. These two quasilinear degenerate PDEs are chosen as typical representatives, so other similar (2m+ 1)th-order NDEs with no divergence structure admit such shocks. As a related second problem, the opposite shock S+(x) = −S−(x) = signx is shown to be a non-entropy solution that gives rise to a continuous rarefaction wave for t > 0. Formation of shocks is also studied for the fifth-order in time NDE uttttt = (uux)xxxx. On the other hand, related NDEs are shown to admit smooth compactons, e.g., for ut = −(|u|ux)xxxx + |u|ux in R × R+, which are of changing sign. Nonnegative ones are nonexistent in general (not robust).