Long-time asymptotics of the Tzitzéica equation on the line
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914443740315648 |
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| author | Huang, Lin Wang, Deng-Shan Zhu, Xiaodong |
| author_facet | Huang, Lin Wang, Deng-Shan Zhu, Xiaodong |
| contents | In this paper, the renowned Riemann-Hilbert method is employed to investigate the initial value problem of Tzitzéica equation on the line. Initially, our analysis focuses on elucidating the properties of two reflection coefficients, which are determined by the initial values. Subsequently, leveraging these reflection coefficients, we construct a Riemann-Hilbert problem that is a powerful tool to articulate the solution of the Tzitzéica equation. Finally, the nonlinear steepest descent method is applied to the oscillatory Riemann-Hilbert problem, which enables us to delineate the long-time asymptotic behaviors of solutions to the Tzitzéica equation across various regions. Moreover, it is shown that the leading-order terms of asymptotic formulas match well with direct numerical simulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04999 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long-time asymptotics of the Tzitzéica equation on the line Huang, Lin Wang, Deng-Shan Zhu, Xiaodong Mathematical Physics Exactly Solvable and Integrable Systems In this paper, the renowned Riemann-Hilbert method is employed to investigate the initial value problem of Tzitzéica equation on the line. Initially, our analysis focuses on elucidating the properties of two reflection coefficients, which are determined by the initial values. Subsequently, leveraging these reflection coefficients, we construct a Riemann-Hilbert problem that is a powerful tool to articulate the solution of the Tzitzéica equation. Finally, the nonlinear steepest descent method is applied to the oscillatory Riemann-Hilbert problem, which enables us to delineate the long-time asymptotic behaviors of solutions to the Tzitzéica equation across various regions. Moreover, it is shown that the leading-order terms of asymptotic formulas match well with direct numerical simulations. |
| title | Long-time asymptotics of the Tzitzéica equation on the line |
| topic | Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2404.04999 |