On the Fractional Dynamics of Kinks in Sine-Gordon Models

On the Fractional Dynamics of Kinks in Sine-Gordon Models

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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML&quo...

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Bibliographic Details
Main Authors: Tassos Bountis, Julia Cantisán, Jesús Cuevas-Maraver, Jorge Eduardo Macías-Díaz, Panayotis G. Kevrekidis
Format: Article
Language:English
Published: MDPI AG 2025-01-01
Series:Mathematics
Subjects:
sine-Gordon equation
kinks
breathers
fractional derivatives
Caputo derivative
Riesz derivative
Online Access:https://www.mdpi.com/2227-7390/13/2/220
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Summary: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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> of the temporal derivative to that of a Caputo fractional type and found that, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>β</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>, this imposes a dissipative dynamical behavior on the coherent structures. We also examined the variation of a fractional Riesz order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> on the spatial derivative. Here, depending on whether this order was below or above the harmonic value <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>, 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.
ISSN:2227-7390

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