Peakompactons: Peaked compact nonlinear waves

This article is meant as an accessible introduction to/tutorial on the analytical construction and numerical simulation of a class of non-standard solitary waves termed \emph{peakompactons}. These peaked compactly supported waves arise as solutions to nonlinear evolution equations from a hierarchy of nonlinearly dispersive Korteweg--de Vries-type models. Peakompactons, like the now-well-know compactons and unlike the soliton solutions of the Korteweg--de Vries equation, have finite support, {\it i.e.}, they are of finite wavelength. However, unlike compactons, peakompactons are also peaked, {\it i.e.}, a higher spatial derivative suffers a jump discontinuity at the wave's crest. Here, we construct such solutions exactly by reducing the governing partial differential equation to a nonlinear ordinary differential equation and employing a phase-plane analysis. A simple, but reliable, finite-difference scheme is also designed and tested for the simulation of collisions of peakompactons. In addition to the peakompacton class of solutions, the general physical features of the so-called $K^\#(n,m)$ hierarchy of nonlinearly dispersive Korteweg--de Vries-type models are discussed as well.