A new derivation of the amplitude of asymptotic oscillatory tails of weakly delocalized solitons
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914766825455616 |
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| author | Fodor, Gyula Forgács, Péter Mushtaq, Muneeb |
| author_facet | Fodor, Gyula Forgács, Péter Mushtaq, Muneeb |
| contents | The computation of the amplitude, $α$, of asymptotic standing wave tails of weakly delocalized, stationary solutions in a fifth-order Korteweg-de Vries equation is revisited. Assuming the coefficient of the fifth order derivative term, $ε^2\ll1$, a new derivation of the ``beyond all orders in $ε$'' amplitude, $α$, is presented. It is shown by asymptotic matching techniques, extended to higher orders in $ε$, that the value of $α$ can be obtained from the asymmetry at the center of the unique solution exponentially decaying in one direction. This observation, complemented by some fundamental results of Hammersley and Mazzarino [Proc. R. Soc. Lond. A 424, 19 (1989)], not only sheds new light on the computation of $α$, but also greatly facilitates its numerical determination to a remarkable precision for so small values of $ε$, which are beyond the capabilities of standard numerical methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_15020 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A new derivation of the amplitude of asymptotic oscillatory tails of weakly delocalized solitons Fodor, Gyula Forgács, Péter Mushtaq, Muneeb High Energy Physics - Theory Pattern Formation and Solitons The computation of the amplitude, $α$, of asymptotic standing wave tails of weakly delocalized, stationary solutions in a fifth-order Korteweg-de Vries equation is revisited. Assuming the coefficient of the fifth order derivative term, $ε^2\ll1$, a new derivation of the ``beyond all orders in $ε$'' amplitude, $α$, is presented. It is shown by asymptotic matching techniques, extended to higher orders in $ε$, that the value of $α$ can be obtained from the asymmetry at the center of the unique solution exponentially decaying in one direction. This observation, complemented by some fundamental results of Hammersley and Mazzarino [Proc. R. Soc. Lond. A 424, 19 (1989)], not only sheds new light on the computation of $α$, but also greatly facilitates its numerical determination to a remarkable precision for so small values of $ε$, which are beyond the capabilities of standard numerical methods. |
| title | A new derivation of the amplitude of asymptotic oscillatory tails of weakly delocalized solitons |
| topic | High Energy Physics - Theory Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2404.15020 |