Integrable Mappings Related to the Extended Discrete KP Hierarchy

Abstract

We investigate self-similar solutions of the extended discrete KP hierarchy. It is shown that corresponding ansatzes lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with respect to these parameters. Some examples are provided. In particular, it is shown that the well known discrete first Painleve equation (dPI) and its hierarchy arises as self-similar reduction of Volterra lattice hierarchy which in turn can be treated as a reduction of the extended discrete KP hierarchy. It is written down equations which naturally generalize dPI. It is shown that theses discrete systems describe B\"acklund transformations of Noumi-Yamada systems of type $A_{2(n-1)^{(1)}}$. We also consider Miura transformations relating different infinite- and finite-field integrable mappings. Simplest example of this kind of Miura transformations is given.