Kadomtsev-Petviashvili hierarchies with non-formal pseudo-differential operators, non-formal solutions, and a Yang-Mills--like formulation
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916405480259584 |
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| author | Magnot, Jean-Pierre Reyes, Enrique G. |
| author_facet | Magnot, Jean-Pierre Reyes, Enrique G. |
| contents | We start from the classical Kadomtsev-Petviashvili hierarchy posed on formal pseudo-differential operators, and we produce two hierarchies of non-linear equations posed on non-formal pseudo-differential operators lying in the Kontsevich and Vishik's odd class, one of them with values in formal pseudo-differential operators. We prove that the corresponding Zakharov-Shabat equations hold in this context, and we express one of our hierarchies as the minimization of a class of Yang-Mills action functionals on a space of pseudo-differential connections whose curvature takes values in the Dixmier ideal. We finish by comparing our Kadomtsev-Petviashvili hierarchies in terms of the solutions that they produce to the KP-II equation: existence, uniqueness and formality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13771 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kadomtsev-Petviashvili hierarchies with non-formal pseudo-differential operators, non-formal solutions, and a Yang-Mills--like formulation Magnot, Jean-Pierre Reyes, Enrique G. Mathematical Physics High Energy Physics - Theory Differential Geometry 35Q51, 37K10, 37K25, 37K30, 58J40, 58B25, 47N20 We start from the classical Kadomtsev-Petviashvili hierarchy posed on formal pseudo-differential operators, and we produce two hierarchies of non-linear equations posed on non-formal pseudo-differential operators lying in the Kontsevich and Vishik's odd class, one of them with values in formal pseudo-differential operators. We prove that the corresponding Zakharov-Shabat equations hold in this context, and we express one of our hierarchies as the minimization of a class of Yang-Mills action functionals on a space of pseudo-differential connections whose curvature takes values in the Dixmier ideal. We finish by comparing our Kadomtsev-Petviashvili hierarchies in terms of the solutions that they produce to the KP-II equation: existence, uniqueness and formality. |
| title | Kadomtsev-Petviashvili hierarchies with non-formal pseudo-differential operators, non-formal solutions, and a Yang-Mills--like formulation |
| topic | Mathematical Physics High Energy Physics - Theory Differential Geometry 35Q51, 37K10, 37K25, 37K30, 58J40, 58B25, 47N20 |
| url | https://arxiv.org/abs/2409.13771 |