Tau functions of solutions of soliton equations

In the holomorphic version of the inverse scattering method, we prove that the determinant of a Toeplitz-type Fredholm operator arising in the solution of the inverse problem is an entire function of the spatial variable for all potentials whose scattering data belong to a Gevrey class strictly less than 1. As a corollary, we establish that, up to a constant factor, every local holomorphic solution of the Korteweg–de Vries equation is the second logarithmic derivative of an entire function of the spatial variable. We discuss the possible order of growth of this entire function. Analogous results are given for all soliton equations of parabolic type.