ABSTRACT
Thus, we develop basic aspects of singularity and existence-uniqueness theory for odd-order nonlinear dispersion (or dispersive) PDEs, which, for short, we call NDEs. A canonical (meaning, here, simplest) model for us is the thirdorder quadratic NDE (the NDE-3)
ut = A(u) ≡ (uux)xx = uuxxx + 3uxuxx in IR× (0, T ), T > 0. (1) We pose for (1) the Cauchy problem with locally integrable initial data
u(x, 0) = u0(x) in IR. (2)
Often, we assume that u0 is bounded, smooth, and/or compactly supported. We will also deal with some initial-boundary value problems in (−L,L)×IR+, with Dirichlet and other boundary conditions at x = ±L. On integrable NDEs related to water wave theory. Concerning applications of equations such as (1), it is customary that various odd-order PDEs appear in classic theory of integrable PDEs, such as the KdV equation,
ut + uux = uxxx (3)
and
(Chapter 7), and the fifth-order KdV equation,
ut + uxxxxx + 30 u 2ux + 20 uxuxx + 10 uuxxx = 0,
and others from shallow water theory. The quasilinear Harry Dym equation
ut = u 3uxxx , (4)
which also belongs to the NDE-3 family, is one of the most exotic integrable soliton equations; see [174, § 4.7] for a survey and references therein. Integrable equation theory produced various hierarchies of quasilinear higherorder NDEs, such as the fifth-order Kawamoto equation [232]
ut = u 5uxxxxx + 5 u
4uxuxxxx + 10 u 5uxxuxxx.
