On the Fractional Dynamics of Kinks in sine-Gordon Models

TL;DR

This paper investigates how fractional derivatives in time and space affect the dynamics of kinks in the sine-Gordon model, revealing dissipative behavior and different kink interactions depending on fractional orders.

Contribution

It introduces a fractional sine-Gordon model with Caputo and Riesz derivatives, analyzing their impact on kink dynamics and interactions, which is a novel extension of classical models.

Findings

Fractional time derivatives induce dissipative kink dynamics.

Spatial fractional derivatives alter kink attraction and repulsion behaviors.

Interplay of temporal and spatial fractional derivatives leads to complex kink interactions.

Abstract

In the present work we explore the dynamics of single kinks, kink-anti-kink pairs and bound states in the prototypical fractional Klein-Gordon example of the sine-Gordon equation. In particular, we modify the order $\beta$ of the temporal derivative to that of a Caputo fractional type and find that, for $1<\beta<2$, this imposes a dissipative dynamical behavior on the coherent structures. We also examine the variation of a fractional Riesz order $\alpha$ on the spatial derivative. Here, depending on whether this order is below or above the harmonic value $\alpha=2$, we find, respectively, monotonically attracting kinks, or non-monotonic and potentially attracting or repelling kinks, with a saddle equilibrium separating the two. Finally, we also explore the interplay of the two derivatives, when both Caputo temporal and Riesz spatial derivatives are involved.