Higher-order shallow water equations and the Camassa-Holm equation

The Korteweg-de Vries (KdV) equation has long been known to describe shallow water waves in an appropriate asymptotic limit. The Camassa-Holm (CH) equation, shown in 1993 [1] also to be completely integrable, allows solitons including peakons with discontinuous derivative. Consequently, its relevance to shallow water theory has been much doubted until the link was established [4] in 2004. Here, the perturbation procedure in terms of the standard amplitude ($\varepsilon$) and shallow depth ($\delta$) parameters shows that, while the KdV equation applies for $\delta^4 $«$ \varepsilon $«$ \delta$ with error term the larger of $\varepsilon^3$ and $\delta^6$,. However, the formal link to the CH equation imposes additional constraints on the coordinate ranges which are applicable. The derivation procedure is also extended to account for depth variations with bed slope $\mathcal{O}(\gamma\delta)$, provided that $\gamma $«$ \varepsilon$ and $\gamma $«$ \delta^2$ for $\delta^3 $«$ \varepsilon $«$ \delta$ and, within these parameter ranges, a theory for modulated waveforms is outlined. This utilizes expressions (in terms of elliptic functions) for the general travelling wave solutions to a (non-integrable) generalization of the CH equation. These include, as limiting cases, solitary waves with adjustable, but constrained, peak curvature (of $\mathcal{O}(\varepsilon \delta^2)$). The nonlinear dispersion relation for the periodic waveforms is illustrated and sample periodic waveforms are illustrated and compared to equivalent KdV approximations.