Third-Order Nonlinear Dispersive Equations: Shocks, Rarefaction, and Blowup Waves

Shock waves and blowup arising in third-order nonlinear dispersive equations are studied. The underlying model is the equation u(t) = (uu(x))(xx) in R x R+. It is shown that two basic Riemann problems for Eq. (0.1) with the initial data S--/+ (x) = -/+ sgnx exhibit a shock wave (u(x, t) = S-(x)) and a smooth rarefaction wave (for S+), respectively. Various blowing-up and global similarity solutions to Eq. (0.1) are constructed that demonstrate the fine structure of shock and rarefaction waves. A technique based on eigenfunctions and the nonlinear capacity is developed to prove the blowup of solutions. The analysis of Eq. (0.1) resembles the entropy theory of scalar conservation laws of the form u(t) + uu(x) = 0, which was developed by O.A. Oleinik and S. N. Kruzhkov (for equations in x is an element of R-N) in the 1950s-1960s.