Fredholm Determinants and the <i>τ</i> Function for the Kadomtsev-Petviashvili Hierarchy

Abstract

The "dressing method" of Zakharov and Shabat is applied to the theory of the τ function, vertex operators, and the bilinear identity obtained by Sato and his co-workers. The vertex operator identity relating the τ 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 GePfand-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.