On the modified KP hierarchy: Tau functions, squared eigenfunction symmetries and additional symmetries

In this paper, the modified KP hierarchy in the Kupershmidt-Kiso version is investigated starting from the tau functions. There are two tau functions for the modified KP hierarchy. We firstly give some properties about the tau functions, including Hirota bilinear identities and Fay identities. And it is showed that these two functions can be viewed as two different tau functions of the KP hierarchy. Then after constructing the spectral representations of the eigenfunction and the adjoint eigenfunctions, the expressions of all the squared eigenfunction potentials in the modified KP hierarchy are obtained. Next, the study of the squared eigenfunction symmetries and the additional symmetries is considered, whose actions on the two tau functions are derived. At last, the Virasoro additional symmetries in the constrained cases are given, also with the actions on the tau functions.