On the properties of compacton-anticompacton collisions

TL;DR

This paper investigates the collision dynamics of compactons and anticompactons in two different nonlinear equations, revealing distinct scattering behaviors and shock formation phenomena through numerical simulations.

Contribution

It provides a comparative analysis of compacton-anticompacton collisions in the RH and CSS equations using Padé discretization, highlighting their differing scattering outcomes.

Findings

CSS equation collisions resemble annihilation with wake dissipation

RH equation collisions lead to shock formation and blowup

Significant behavioral differences between the two equations' collision processes

Abstract

We study the properties of compacton-anticompacton collision processes. We compare and con- trast results for the case of compacton-anticompacton solutions of the K(l, p) Rosenau-Hyman (RH) equation for l = p = 2, with compacton-anticompacton solutions of the L(l,p) Cooper-Shepard- Sodano (CSS) equation for p = 1 and l = 3. This study is performed using a Pad\'e discretization of the RH and CSS equations. We find a significant difference in the behavior of compacton- anticompacton scattering. For the CSS equation, the scattering can be interpreted as "annihila- tion" as the wake left behind dissolves over time. In the RH equation, the numerical evidence is that multiple shocks form after the collision which eventually lead to "blowup" of the resulting waveform.