Invariant Submanifolds of Darboux-KP Chain and Extension of the Discete KP Hierarchy

Abstract

Invariant submanifolds of the so-called Darboux-KP chain are investigated. It is shown that restriction of dynamics on some class of invariant submanifolds yields the extension of the discrete KP hierarchy, while the intersections leads to Lax pairs for a broad class of differential-difference systems with finite number of fields. Some attention is given to investigation of self-similar reductions. It is shown that self-similar 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 well known first discrete Painleve equation (dPI) corresponds to Volterra lattice hierarchy. It is written down equations which naturally generalize dPI in the sense that they have first Painleve transcedent in continuous limit.