Nonlinear Dispersion Equations: Smooth Deformations, Compactons, and Extensions to Higher Orders

The third-order nonlinear dispersion PDE, as the key model, (0.1) ut = (uux)xx in R × R+, is studied. Two Riemann’s problems for (0.1) with initial data S∓(x) = ∓signx, create the shock (u(x, t) ≡ S−(x)) and smooth rarefaction (for data S+) waves, [18]. The concept of “δ-entropy” solutions (a“δ-entropy test”) and others are developed for distinguishing shock and rarefaction waves by using stable smooth δ-deformations of discontinuous solutions. These are analogous to entropy solutions for scalar conservation laws such as ut + uux = 0, developed by Oleinik and Kruzhkov (in R N ) in the 1950-60s. The Rosenau–Hyman K(2, 2) (compacton) equation ut = (uux)xx + 4uux, which has a special importance for applications, is studied. Compactons as compactly supported travelling wave solutions are shown to pass the δ-entropy test. Shock and rarefaction waves are discussed for other NDEs such as ut = (u 2ux)xx, utt = (uux)xx, utt = uux, uttt = (uux)xx, ut = (uux)xxxxxx, etc. Dedicated to the memory of Professors O.A. Oleinik and S.N. Kruzhkov