Fredholm Determinants and the $\tau$ Function for the Kadomtsev–Petviashvili Hierarchy

Abstract

The “dressing method” of Zakharov and Shabat is applied to the theory of the $\tau$ function, vertex operators, and the bilinear identity obtained by Sato and his co-workers. The vertex operator identity relating the $\tau$ function to the Baker–Akhiezer function is obtained from their representations in terms of the Fredholm determinants and minors of the scattering operator appearing in the Gel'fand–Levitan–Marchenko equation. The bilinear identity is extended to wave functions analytic in a left half plane and is proved as a consequence of the inversion theorem and the convolution theorem for the Laplace transform.