Stability and dynamical properties of Cooper-Shepard-Sodano compactons

TL;DR

This paper investigates the numerical stability and interaction dynamics of compactons in the CSS equation, demonstrating their stability and shape reemergence after collisions, with insights into the effects of parameters and numerical methods.

Contribution

It extends a Pade approximant method to study CSS compactons, analyzing their stability, interactions, and the influence of parameters and numerical viscosity.

Findings

CSS compactons are numerically stable.

Compactons reemerge with the same shape after collisions.

Lower artificial viscosity is needed for CSS compacton simulations.

Abstract

Extending a Pade approximant method used for studying compactons in the Rosenau-Hyman (RH) equation, we study the numerical stability of single compactons of the Cooper-Shepard-Sodano (CSS) equation and their pairwise interactions. The CSS equation has a conserved Hamiltonian which has allowed various approaches for studying analytically the nonlinear stability of the solutions. We study three different compacton solutions and find they are numerically stable. Similar to the collisions between RH compactons, the CSS compactons reemerge with same coherent shape when scattered. The time evolution of the small-amplitude ripple resulting after scattering depends on the values of the parameters $l$ and $p$ characterizing the corresponding CSS equation. The simulation of the CSS compacton scattering requires a much smaller artificial viscosity to obtain numerical stability, than in the case of RH compactons propagation.