This paper considers the propagation of shallow-water solitary and nonlinearperiodic waves over a gradual slope with bottom friction in the framework of avariable-coefficient Korteweg-de Vries equation. We use the Whitham averagingmethod, using a recent development of this theory for perturbed integrableequations. This general approach enables us not only to improve known resultson the adiabatic evolution of isolated solitary waves and periodic wave trainsin the presence of variable topography and bottom friction, modeled by theChezy law, but also importantly, to study the effects of these factors on thepropagation of undular bores, which are essentially unsteady in the systemunder consideration. In particular, it is shown that the combined action ofvariable topography and bottom friction generally imposes certain globalrestrictions on the undular bore propagation so that the evolution of theleading solitary wave can be substantially different from that of an isolatedsolitary wave with the same initial amplitude. This non-local effect is due tononlinear wave interactions within the undular bore and can lead to anadditional solitary wave amplitude growth, which cannot be predicted in theframework of the traditional adiabatic approach to the propagation of solitarywaves in slowly varying media.
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