For fractional sine-Gordon, Caputo time order 1<β<2 damps kink motion, while Riesz space order above 2 creates non-monotonic tails and an unstable kink-antikink saddle.
Qureshi, Real life application of caputo fractional derivative for measles epidemiological autonomous dynamical system, Chaos, Solitons & Fractals 134 (2020) 109744
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On the Fractional Dynamics of Kinks in sine-Gordon Models
For fractional sine-Gordon, Caputo time order 1<β<2 damps kink motion, while Riesz space order above 2 creates non-monotonic tails and an unstable kink-antikink saddle.