On the Fractional Dynamics of Kinks in Sine-Gordon Models - Tassos Bountis; Julia Cantisán; Jesús Cuevas-Maraver; Jorge Eduardo Macías-Díaz; Panayotis G. Kevrekidis (Mathematics)
In the present work, we explored 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 modified the order <semantics>β</semantics> of the temporal derivative to that of a Caputo fractional type and found that, for <semantics>1<β<2</semantics>, this imposes a dissipative dynamical behavior on the coherent structures. We also examined the variation of a fractional Riesz order <semantics>α</semantics> on the spatial derivative. Here, depending on whether this order was below or above the harmonic value <semantics>α=2</semantics>, we found, respectively, monotonically attracting kinks, or non-monotonic and potentially attracting or repelling kinks, with a saddle equilibrium separating the two. Finally, we also explored the interplay of the two derivatives, when both Caputo temporal and Riesz spatial derivatives are involved.