Fractal structure of the soliton scattering for the graphene superlattice equation

TL;DR

This paper investigates the complex fractal structure of soliton scattering in a graphene superlattice modeled by a modified sine-Gordon equation, revealing resonance phenomena and supporting the resonant energy exchange theory.

Contribution

It provides the first numerical analysis of inelastic kink-antikink collisions in this system and characterizes the fractal resonance window structure.

Findings

Resonance windows exhibit fractal structures.

Collision outcomes depend on initial speeds and resonance conditions.

Results support the resonant energy exchange theory.

Abstract

The graphene superlattice equation, a modified sine-Gordon equation, governs the propagation of solitary electromagnetic waves in a graphene superlattice. This equation has kink solutions without explicit analytical expression, requiring the use of quadrature methods. The inelastic collision of kinks and antikinks with the same but opposite speed is studied numerically for the first time; after their interaction they escape to infinity when its speed is either larger than a critical value or it is inside a series of resonance windows; otherwise, they form a breather-like state that slowly decays by radiating energy. Here, the fractal structure of these resonance windows is characterized by using a multi-index notation and their main features are compared with the predictions of the resonant energy exchange theory showing good agreement. Our results can be interpreted as new evidence in favour of this theory.