An integrable pseudospherical equation with pseudo-peakon solutions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910750415519744 |
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| author | da Silva, Priscila Leal Freire, Igor Leite Filho, Nazime Sales |
| author_facet | da Silva, Priscila Leal Freire, Igor Leite Filho, Nazime Sales |
| contents | We study an integrable equation whose solutions define a triad of one-forms describing a surface with Gaussian curvature -1. We identify a local group of diffeomorphisms that preserve these solutions and establish conserved quantities. From the symmetries, we obtain invariant solutions that provide explicit metrics for the surfaces. These solutions are unbounded and often appear in mirrored pairs. We introduce the ``collage'' method, which uses conserved quantities to remove unbounded parts and smoothly join the solutions, leading to weak solutions consistent with the conserved quantities. As a result we get pseudo-peakons, which are smoother than Camassa-Holm peakons. Additionally, we apply a Miura-type transformation to relate our equation to the Degasperis-Procesi equation, allowing us to recover peakon and shock-peakon solutions for it from the solutions of the other equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01537 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An integrable pseudospherical equation with pseudo-peakon solutions da Silva, Priscila Leal Freire, Igor Leite Filho, Nazime Sales Mathematical Physics Analysis of PDEs Exactly Solvable and Integrable Systems 35C08, 35D30, 35E05, 53C21 We study an integrable equation whose solutions define a triad of one-forms describing a surface with Gaussian curvature -1. We identify a local group of diffeomorphisms that preserve these solutions and establish conserved quantities. From the symmetries, we obtain invariant solutions that provide explicit metrics for the surfaces. These solutions are unbounded and often appear in mirrored pairs. We introduce the ``collage'' method, which uses conserved quantities to remove unbounded parts and smoothly join the solutions, leading to weak solutions consistent with the conserved quantities. As a result we get pseudo-peakons, which are smoother than Camassa-Holm peakons. Additionally, we apply a Miura-type transformation to relate our equation to the Degasperis-Procesi equation, allowing us to recover peakon and shock-peakon solutions for it from the solutions of the other equation. |
| title | An integrable pseudospherical equation with pseudo-peakon solutions |
| topic | Mathematical Physics Analysis of PDEs Exactly Solvable and Integrable Systems 35C08, 35D30, 35E05, 53C21 |
| url | https://arxiv.org/abs/2409.01537 |