Higher-order heat equation and the Gelfand-Dickey hierarchy
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arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866929246683791360 |
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| author | Iliev, Plamen |
| author_facet | Iliev, Plamen |
| contents | In this paper we analyze the heat kernel of the equation $\partial_tv =\pm\mathcal{L} v$, where $\mathcal{L}=\partial_x^N+u_{N-2}(x)\partial_x^{N-2}+\cdots+u_0(x)$ is an $N$-th order differential operator and the $\pm$ sign on the right-hand side is chosen appropriately. Using formal pseudo-differential operators, we derive an explicit formula for Hadamard's coefficients in the expansion of the heat kernel in terms of the resolvent of $\mathcal{L}$. Combining this formula with soliton techniques and Sato's Grassmannian, we establish different properties of Hadamard's coefficients and relate them to the Gelfand-Dickey hierarchy. In particular, using the correspondence between commutative rings of differential operators and algebraic curves due to Burchnall-Chaundy and Krichever, we prove that the heat kernel consists of finitely many terms if and only if the operator $\mathcal{L}$ belongs to a rank-one commutative ring of differential operators whose spectral curve is rational with only one cusp-like singular point, and the coefficients $u_j(x)$ vanish at $x=\infty$. We also characterize these operators $\mathcal{L}$ as the rational solutions of the Gelfand-Dickey hierarchy with coefficients $u_j $ vanishing at $x=\infty$, or as the rank-one solutions of the bispectral problem vanishing at $\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_10857 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Higher-order heat equation and the Gelfand-Dickey hierarchy Iliev, Plamen Analysis of PDEs Mathematical Physics Algebraic Geometry Classical Analysis and ODEs In this paper we analyze the heat kernel of the equation $\partial_tv =\pm\mathcal{L} v$, where $\mathcal{L}=\partial_x^N+u_{N-2}(x)\partial_x^{N-2}+\cdots+u_0(x)$ is an $N$-th order differential operator and the $\pm$ sign on the right-hand side is chosen appropriately. Using formal pseudo-differential operators, we derive an explicit formula for Hadamard's coefficients in the expansion of the heat kernel in terms of the resolvent of $\mathcal{L}$. Combining this formula with soliton techniques and Sato's Grassmannian, we establish different properties of Hadamard's coefficients and relate them to the Gelfand-Dickey hierarchy. In particular, using the correspondence between commutative rings of differential operators and algebraic curves due to Burchnall-Chaundy and Krichever, we prove that the heat kernel consists of finitely many terms if and only if the operator $\mathcal{L}$ belongs to a rank-one commutative ring of differential operators whose spectral curve is rational with only one cusp-like singular point, and the coefficients $u_j(x)$ vanish at $x=\infty$. We also characterize these operators $\mathcal{L}$ as the rational solutions of the Gelfand-Dickey hierarchy with coefficients $u_j $ vanishing at $x=\infty$, or as the rank-one solutions of the bispectral problem vanishing at $\infty$. |
| title | Higher-order heat equation and the Gelfand-Dickey hierarchy |
| topic | Analysis of PDEs Mathematical Physics Algebraic Geometry Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2108.10857 |