On the Fractional Dynamics of Kinks in sine-Gordon Models

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.