This essay is concerned with a class of higher-order models for the unidirectional propagation of small amplitude long waves on the surface of an ideal fluid derived in [3]. These models go beyond the classical first-order theory that goes back to Boussinesq [7] and Korteweg and de Vries [13] in the \(19^{th}\) century. In the water waves context, the latter models are proven to be good approximations of the two-dimensional Euler equations in regimes where their derivation is valid. However, the time scale of their validity extends only to about ten wavelengths or so. The second-order models considered here are formally accurate on the order of a hundred wavelengths. And they do not require auxiliary data beyond what the first-order models need. Nor are they computationally much more complicated than the first-order models. As a consequence, they appear to be worth extended study. Mathematical theory for the initial-value problems for these models begins already with [3] and is considerably improved in [4]. However, in the last-mentioned paper, which can countenance localized initial data as rough as the \(L^2(\mathbb R)\) -based Sobolev class \(H^1({\mathbb {R}})\) , there are some annoying restrictions if one wants the problem to be globally well-posed. These restrictions are herewith removed.
