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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.06156 |
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| _version_ | 1866929275756609536 |
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| author | Matsutani, Shigeki |
| author_facet | Matsutani, Shigeki |
| contents | It is known that the elliptic function solutions of the nonlinear Schrödinger equation are reduced to the algebraic differential relation in terms of the Weierstrass sigma function, $\displaystyle{ \left[-{\frak{i}}\frac{\partial}{\partial t} +α\frac{\partial}{\partial u}\right]Ψ-\frac{1}{2} \frac{\partial^2}{\partial u^2}Ψ+(Ψ^* Ψ) Ψ= \frac12 (2β+α^2-3\wp(v))Ψ}$, where $Ψ(u;v, t):=\mathrm{e}^{αu+{\frak{i}}βt+c}$ $\displaystyle{\frac{\mathrm{e}^{-ζ(v)u}σ(u+v)}{σ(u)σ(v)}}$, its dual $Ψ^*(u; v,t)$, and certain complex numbers $α, β$ and $c$. In this paper, we generalize the algebraic differential relation to those of genera two and three in terms of the hyperelliptic sigma functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06156 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonlinear Schrödinger equation in terms of elliptic and hyperelliptic $σ$ functions Matsutani, Shigeki Exactly Solvable and Integrable Systems Algebraic Geometry It is known that the elliptic function solutions of the nonlinear Schrödinger equation are reduced to the algebraic differential relation in terms of the Weierstrass sigma function, $\displaystyle{ \left[-{\frak{i}}\frac{\partial}{\partial t} +α\frac{\partial}{\partial u}\right]Ψ-\frac{1}{2} \frac{\partial^2}{\partial u^2}Ψ+(Ψ^* Ψ) Ψ= \frac12 (2β+α^2-3\wp(v))Ψ}$, where $Ψ(u;v, t):=\mathrm{e}^{αu+{\frak{i}}βt+c}$ $\displaystyle{\frac{\mathrm{e}^{-ζ(v)u}σ(u+v)}{σ(u)σ(v)}}$, its dual $Ψ^*(u; v,t)$, and certain complex numbers $α, β$ and $c$. In this paper, we generalize the algebraic differential relation to those of genera two and three in terms of the hyperelliptic sigma functions. |
| title | Nonlinear Schrödinger equation in terms of elliptic and hyperelliptic $σ$ functions |
| topic | Exactly Solvable and Integrable Systems Algebraic Geometry |
| url | https://arxiv.org/abs/2403.06156 |