Higher-order heat equation and the Gelfand-Dickey hierarchy

Abstract

In this paper we analyze the heat kernel of the equation ∂tv=±Lv, where L=∂xN+uN−2(x)∂xN−2+⋯+u0(x) is an N-th order differential operator and the ± 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 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 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 uj(x) vanish at x=∞. We also characterize these operators L as the rational solutions of the Gelfand-Dickey hierarchy with coefficients uj vanishing at x=∞, or as the rank-one solutions of the bispectral problem vanishing at ∞.