On one typical Einstein-Weyl equation: inverse spectral transform for theCauchy problem, longtime behaviour of the solutions and implicit solutions
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909561285246976 |
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| author | Yi, Ge Chen, Zikai Tian, Kelei Xu, Ying |
| author_facet | Yi, Ge Chen, Zikai Tian, Kelei Xu, Ying |
| contents | In this paper, we study one typical Einstein-Weyl equation. It arises from Ferapontov and Kruglikov's investigation on the integrability of several dispersionless partial differential equations and the geometry of their formal linearizations. First, by using Manakov-Santini IST (inverse spectral transform) method, we investigate its Cauchy problem including the direct problem from the initial data, the time evolution of the scattering and spectral data, and the inverse problem. Second, based on the nonlinear RH (Riemann-Hilbert) dressing, the longtime behaviour of the solutions is constructed. In addition, some implicit solutions are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On one typical Einstein-Weyl equation: inverse spectral transform for theCauchy problem, longtime behaviour of the solutions and implicit solutions Yi, Ge Chen, Zikai Tian, Kelei Xu, Ying Exactly Solvable and Integrable Systems In this paper, we study one typical Einstein-Weyl equation. It arises from Ferapontov and Kruglikov's investigation on the integrability of several dispersionless partial differential equations and the geometry of their formal linearizations. First, by using Manakov-Santini IST (inverse spectral transform) method, we investigate its Cauchy problem including the direct problem from the initial data, the time evolution of the scattering and spectral data, and the inverse problem. Second, based on the nonlinear RH (Riemann-Hilbert) dressing, the longtime behaviour of the solutions is constructed. In addition, some implicit solutions are presented. |
| title | On one typical Einstein-Weyl equation: inverse spectral transform for theCauchy problem, longtime behaviour of the solutions and implicit solutions |
| topic | Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2504.01361 |