# Accelerating dynamical peakons and their behaviour

## TL;DR

This paper introduces a new class of time-dependent peakon solutions in nonlinear dispersive wave equations, revealing diverse dynamic behaviors such as oscillations, breather states, and blow-up phenomena.

## Contribution

It uncovers a novel type of dynamical peakons with time-dependent amplitude and speed, expanding the understanding of wave solutions in nonlinear dispersive equations.

## Key findings

- Discovery of time-dependent amplitude and speed in peakons
- Examples of peakons exhibiting oscillatory, dissipative, and blow-up behaviors
- Illustration of diverse wave phenomena in nonlinear dispersive equations

## Abstract

A wide class of nonlinear dispersive wave equations are shown to possess a novel type of peakon solution in which the amplitude and speed of the peakon are time-dependent. These novel dynamical peakons exhibit a wide variety of different behaviours for their amplitude, speed, and acceleration, including an oscillatory amplitude and constant speed which describes a peakon breather. Examples are presented of families of nonlinear dispersive wave equations that illust rate various interesting behaviours, such as asymptotic travelling-wave peakons, dissipating/anti-dissipating peakons, direction-reversing peakons, runaway and blow up peakons, among others.