  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 $\infty$.
