Transient asymptotics of the modified Camassa-Holm equation
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915804758409216 |
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| author | Xu, Taiyang Yang, Yiling Zhang, Lun |
| author_facet | Xu, Taiyang Yang, Yiling Zhang, Lun |
| contents | We investigate long time asymptotics of the modified Camassa-Holm equation in three transition zones under a nonzero background. The first transition zone lies between the soliton region and the first oscillatory region, the second one lies between the second oscillatory region and the fast decay region, and possibly, the third one, namely, the collisionless shock region, that bridges the first transition region and the first oscillatory region. Under a low regularity condition on the initial data, we obtain Painlevé-type asymptotic formulas in the first two transition regions, while the transient asymptotics in the third region involves the Jacobi theta function. We establish our results by performing a $\bar{\partial}$ nonlinear steepest descent analysis to the associated Riemann-Hilbert problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06950 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Transient asymptotics of the modified Camassa-Holm equation Xu, Taiyang Yang, Yiling Zhang, Lun Analysis of PDEs Mathematical Physics Exactly Solvable and Integrable Systems 35Q53, 37K15, 34M50, 35Q15, 35B40, 37K40, 33E17, 34M55 We investigate long time asymptotics of the modified Camassa-Holm equation in three transition zones under a nonzero background. The first transition zone lies between the soliton region and the first oscillatory region, the second one lies between the second oscillatory region and the fast decay region, and possibly, the third one, namely, the collisionless shock region, that bridges the first transition region and the first oscillatory region. Under a low regularity condition on the initial data, we obtain Painlevé-type asymptotic formulas in the first two transition regions, while the transient asymptotics in the third region involves the Jacobi theta function. We establish our results by performing a $\bar{\partial}$ nonlinear steepest descent analysis to the associated Riemann-Hilbert problem. |
| title | Transient asymptotics of the modified Camassa-Holm equation |
| topic | Analysis of PDEs Mathematical Physics Exactly Solvable and Integrable Systems 35Q53, 37K15, 34M50, 35Q15, 35B40, 37K40, 33E17, 34M55 |
| url | https://arxiv.org/abs/2308.06950 |