Algebro-geometric integration of the Boussinesq hierarchy
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908476504014848 |
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| author | Bernatska, Julia Skrypnyk, Taras |
| author_facet | Bernatska, Julia Skrypnyk, Taras |
| contents | We construct an integrable hierarchy of the Boussinesq equation using the Lie-algebraic approach of Holod-Flashka-Newell-Ratiu. We show that finite-gap hamiltonian systems of the hierarchy arise on coadjoint orbits in the loop algebra of $\mathfrak{sl}(3)$, and possess spectral curves from the family of $(3,3N\,{+}\,1)$-curves, $N\,{\in}\, \Natural$. Separation of variables leads to the Jacobi inversion problem on the mentioned curves, which is solved in terms of the corresponding multiply periodic functions. An exact finite-gap solution of the Boussinesq equation is obtained explicitly, and a conjecture on the reality conditions is made. The obtained solutions are computed for several spectral curves, and illustrated graphically. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_19179 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebro-geometric integration of the Boussinesq hierarchy Bernatska, Julia Skrypnyk, Taras Exactly Solvable and Integrable Systems Mathematical Physics 37J35, 17B80, 14H70 We construct an integrable hierarchy of the Boussinesq equation using the Lie-algebraic approach of Holod-Flashka-Newell-Ratiu. We show that finite-gap hamiltonian systems of the hierarchy arise on coadjoint orbits in the loop algebra of $\mathfrak{sl}(3)$, and possess spectral curves from the family of $(3,3N\,{+}\,1)$-curves, $N\,{\in}\, \Natural$. Separation of variables leads to the Jacobi inversion problem on the mentioned curves, which is solved in terms of the corresponding multiply periodic functions. An exact finite-gap solution of the Boussinesq equation is obtained explicitly, and a conjecture on the reality conditions is made. The obtained solutions are computed for several spectral curves, and illustrated graphically. |
| title | Algebro-geometric integration of the Boussinesq hierarchy |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics 37J35, 17B80, 14H70 |
| url | https://arxiv.org/abs/2507.19179 |