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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2025-01-01
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| author | Tassos Bountis Julia Cantisán Jesús Cuevas-Maraver Jorge Eduardo Macías-Díaz Panayotis G. Kevrekidis |
| author_facet | Tassos Bountis Julia Cantisán Jesús Cuevas-Maraver Jorge Eduardo Macías-Díaz Panayotis G. Kevrekidis |
| author_sort | Tassos Bountis |
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| description | 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. |
| format | Article |
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| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-e7eeb98efbe144c5a6a328d0a40fc4262025-01-24T13:39:47ZengMDPI AGMathematics2227-73902025-01-0113222010.3390/math13020220On the Fractional Dynamics of Kinks in Sine-Gordon ModelsTassos Bountis0Julia Cantisán1Jesús Cuevas-Maraver2Jorge Eduardo Macías-Díaz3Panayotis G. Kevrekidis4Department of Mathematics, University of Patras, 26500 Patras, GreeceGrupo 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, SpainGrupo de Física No Lineal (FQM-280), Departamento de Física Aplicada I, Escuela Politécnica Superior, Universidad de Sevilla, C/ Virgen de África, 7, 41011 Sevilla, SpainDepartment of Mathematics and Didactics of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, 10120 Tallinn, EstoniaDepartment of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003-4515, USAIn 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.https://www.mdpi.com/2227-7390/13/2/220sine-Gordon equationkinksbreathersfractional derivativesCaputo derivativeRiesz derivative |
| spellingShingle | Tassos Bountis Julia Cantisán Jesús Cuevas-Maraver Jorge Eduardo Macías-Díaz Panayotis G. Kevrekidis On the Fractional Dynamics of Kinks in Sine-Gordon Models Mathematics sine-Gordon equation kinks breathers fractional derivatives Caputo derivative Riesz derivative |
| title | On the Fractional Dynamics of Kinks in Sine-Gordon Models |
| title_full | On the Fractional Dynamics of Kinks in Sine-Gordon Models |
| title_fullStr | On the Fractional Dynamics of Kinks in Sine-Gordon Models |
| title_full_unstemmed | On the Fractional Dynamics of Kinks in Sine-Gordon Models |
| title_short | On the Fractional Dynamics of Kinks in Sine-Gordon Models |
| title_sort | on the fractional dynamics of kinks in sine gordon models |
| topic | sine-Gordon equation kinks breathers fractional derivatives Caputo derivative Riesz derivative |
| url | https://www.mdpi.com/2227-7390/13/2/220 |
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