On the Fractional Dynamics of Kinks in sine-Gordon Models
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916562361909248 |
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| author | Bountis, T. Cantisán, J. Cuevas-Maraver, J. Macías-Díaz, J. E. Kevrekidis, P. G. |
| author_facet | Bountis, T. Cantisán, J. Cuevas-Maraver, J. Macías-Díaz, J. E. Kevrekidis, P. G. |
| contents | In the present work we explore 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 modify the order $β$ of the temporal derivative to that of a Caputo fractional type and find that, for $1<β<2$, this imposes a dissipative dynamical behavior on the coherent structures. We also examine the variation of a fractional Riesz order $α$ on the spatial derivative. Here, depending on whether this order is below or above the harmonic value $α=2$, we find, respectively, monotonically attracting kinks, or non-monotonic and potentially attracting or repelling kinks, with a saddle equilibrium separating the two. Finally, we also explore the interplay of the two derivatives, when both Caputo temporal and Riesz spatial derivatives are involved. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_18600 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Fractional Dynamics of Kinks in sine-Gordon Models Bountis, T. Cantisán, J. Cuevas-Maraver, J. Macías-Díaz, J. E. Kevrekidis, P. G. Pattern Formation and Solitons In the present work we explore 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 modify the order $β$ of the temporal derivative to that of a Caputo fractional type and find that, for $1<β<2$, this imposes a dissipative dynamical behavior on the coherent structures. We also examine the variation of a fractional Riesz order $α$ on the spatial derivative. Here, depending on whether this order is below or above the harmonic value $α=2$, we find, respectively, monotonically attracting kinks, or non-monotonic and potentially attracting or repelling kinks, with a saddle equilibrium separating the two. Finally, we also explore the interplay of the two derivatives, when both Caputo temporal and Riesz spatial derivatives are involved. |
| title | On the Fractional Dynamics of Kinks in sine-Gordon Models |
| topic | Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2411.18600 |