On the Fractional Dynamics of Kinks in Sine-Gordon Models

Tassos Bountis (), Julia Cantisán, Jesús Cuevas-Maraver, Jorge Eduardo Macías-Díaz and Panayotis G. Kevrekidis
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Tassos Bountis: Department of Mathematics, University of Patras, 26500 Patras, Greece
Julia Cantisán: Grupo de Física No Lineal (FQM-280), Departamento de Ciencias Integradas y Centro de Estudios Avanzados en Física, Matemáticas y Computación, Universidad de Huelva, 21071 Huelva, Spain
Jesús Cuevas-Maraver: Grupo de Física No Lineal (FQM-280), Departamento de Física Aplicada I, Escuela Politécnica Superior, Universidad de Sevilla, C/ Virgen de África, 7, 41011 Sevilla, Spain
Jorge Eduardo Macías-Díaz: Department of Mathematics and Didactics of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, 10120 Tallinn, Estonia
Panayotis G. Kevrekidis: Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003-4515, USA

Mathematics, 2025, vol. 13, issue 2, 1-16

Abstract: 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 β of the temporal derivative to that of a Caputo fractional type and found that, for 1 < β < 2 , this imposes a dissipative dynamical behavior on the coherent structures. We also examined the variation of a fractional Riesz order α on the spatial derivative. Here, depending on whether this order was below or above the harmonic value α = 2 , 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.

Keywords: sine-Gordon equation; kinks; breathers; fractional derivatives; Caputo derivative; Riesz derivative (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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