Korteweg-de Vries hierarchy

In [10,11] (together with R. Schimming) we studied the explicit structure of the Korteweg-de Vries hierarchy. The Korteweg-de Vries hierarchy is an infinite-dimensional Hamiltonian system, whose flows are exactly the isospectral deformations of a Laplace type operator in one dimension, $L=-\partial_x^2+Q$, say on the circle M=S¹. The global heat kernel coefficients A_k are the spectral invariants and give the complete infinite sequence of the integrals of motion of the Korteweg-de Vries hierarchy. The right hand sides of the KdV hierarchy are differential polynomials in Q, which are known to be proportional to the diagonal values of the HMDS-coefficients $a_k^{\rm diag}$. We have found a new algorithm for the calculation of the heat kernel asymptotic expansion in one-dimension and established in this way the explicit structure of the whole KdV hierarchy. We proved that this algorithm is valid for the case of arbitrary matrix potential Q too.