Higherorder heat equation and the GelfandDickey hierarchy
Abstract
The goal of this paper is to study the heat kernel of the equation $\partial_tv =\pm\mathcal{L} v$, where $\mathcal{L}=\partial_x^N+u_{N2}(x)\partial_x^{N2}+\cdots+u_0(x)$ is an $N$th order differential operator and the $\pm$ sign on the righthand side is chosen appropriately. Using formal pseudodifferential 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 GelfandDickey hierarchy. In particular, using the correspondence between commutative rings of differential operators and algebraic curves due to BurchnallChaundy and Krichever, we prove that the heat kernel consists of finitely many terms if and only if the operator $\mathcal{L}$ belongs to a rankone commutative ring of differential operators whose spectral curve is rational with only one cusplike singular point, and the coefficients $u_j(x)$ vanish at $x=\infty$. Equivalently, we can characterize these operators $\mathcal{L}$ as the rational solutions of the GelfandDickey hierarchy with coefficients $u_j $ vanishing at $x=\infty$, or as the rankone solutions of the bispectral problem vanishing at $\infty$.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.10857
 Bibcode:
 2021arXiv210810857I
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 Mathematics  Algebraic Geometry;
 Mathematics  Classical Analysis and ODEs