Reality conditions for the KdV equation and exact quasi-periodic solutions in finite phase spaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910771615629312 |
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| author | Bernatska, Julia |
| author_facet | Bernatska, Julia |
| contents | In the present paper reality conditions for quasi-periodic solutions of the KdV equation are determined completely. As a result, solutions in the form of non-linear waves can be plotted and investigated.
The full scope of obtaining finite-gap solutions of the KdV equation is presented. It is proven that the multiply periodic $\wp_{1,1}$-function on the Jacobian variety of a hyperelliptic curve of arbitrary genus serves as the finite-gap solution, the genus coincides with the number of gaps. The subspace of the Jacobian variety where $\wp_{1,1}$, as well as other $\wp$-functions, are bounded and real-valued is found in any genus. This result covers every finite phase space of the KdV hierarchy, and can be extended to other completely integrable equations. A method of effective computation of this type of solutions is suggested, and illustrated in genera $2$ and $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10859 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Reality conditions for the KdV equation and exact quasi-periodic solutions in finite phase spaces Bernatska, Julia Exactly Solvable and Integrable Systems Mathematical Physics Algebraic Geometry 35Q53, 35B15, 30F10 In the present paper reality conditions for quasi-periodic solutions of the KdV equation are determined completely. As a result, solutions in the form of non-linear waves can be plotted and investigated. The full scope of obtaining finite-gap solutions of the KdV equation is presented. It is proven that the multiply periodic $\wp_{1,1}$-function on the Jacobian variety of a hyperelliptic curve of arbitrary genus serves as the finite-gap solution, the genus coincides with the number of gaps. The subspace of the Jacobian variety where $\wp_{1,1}$, as well as other $\wp$-functions, are bounded and real-valued is found in any genus. This result covers every finite phase space of the KdV hierarchy, and can be extended to other completely integrable equations. A method of effective computation of this type of solutions is suggested, and illustrated in genera $2$ and $3$. |
| title | Reality conditions for the KdV equation and exact quasi-periodic solutions in finite phase spaces |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Algebraic Geometry 35Q53, 35B15, 30F10 |
| url | https://arxiv.org/abs/2312.10859 |