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

Abstract

The Korteweg-de Vries (KdV) equation has long been known to describeshallow water waves in an appropriate asymptotic limit. TheCamassa-Holm (CH) equation, shown in 1993 [1] also to becompletely integrable, allows solitons including peakons withdiscontinuous derivative. Consequently, its relevance to shallowwater theory has been much doubted until the link was established[4] in 2004. Here, the perturbation procedure in terms ofthe 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 imposesadditional constraints on the coordinate ranges which areapplicable. The derivation procedure is also extended to accountfor 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, atheory for modulated waveforms is outlined. This utilizesexpressions (in terms of elliptic functions) for the generaltravelling wave solutions to a (non-integrable) generalization ofthe CH equation. These include, as limiting cases, solitary waveswith adjustable, but constrained, peak curvature (of$\mathcal{O}(\varepsilon \delta^2)$). The nonlinear dispersion relation forthe periodic waveforms is illustrated and sample periodic waveformsare illustrated and compared to equivalent KdV approximations.