Stability and interaction of compactons in the sublinear KdV equation

TL;DR

This paper investigates the properties, stability, and interactions of compactons in a sublinear KdV equation, revealing their inelastic collisions and dual role as soliton-like and dispersive waves.

Contribution

It provides the first analytical proof of energetic stability for compactons in the sublinear KdV and explores their numerical dynamics and interactions.

Findings

Compactons are energetically stable under symmetric perturbations.

They interact inelastically but nearly restore their shapes after collisions.

Compactons serve as long-living structures and facilitate wave energy spreading.

Abstract

Compactons are studied in the framework of the Korteweg-de Vries (KdV) equation with the sublinear nonlinearity. Compactons represent localized bell-shaped waves of either polarity which propagate to the same direction as waves of the linear KdV equation. Their amplitude and width are inverse proportional to their speed. The energetic stability of compactons with respect to symmetric compact perturbations with the same support is proven analytically. Dynamics of compactons is studied numerically, including evolution of pulse-like disturbances and interactions of compactons of the same or opposite polarities. Compactons interact inelastically, though almost restore their shapes after collisions. Compactons play a two-fold role of the long-living soliton-like structures and of the small-scale waves which spread the wave energy.