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Peakompactons: peaked compact nonlinear waves. (English) Zbl 1364.35306

Summary: This paper is meant as an accessible introduction to/tutorial on the analytical construction and numerical simulation of a class of nonstandard solitary waves termed 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-known compactons and unlike the soliton solutions of the Korteweg-de Vries equation, have finite support, i.e., they are of finite wavelength. However, unlike compactons, peakompactons are also peaked, 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.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35C08 Soliton solutions
35C07 Traveling wave solutions

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DLMF
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