Invariant Submanifolds of the Darboux–Kadomtsev–Petviashvili Chain and an Extension of the Discrete Kadomtsev–Petviashvili Hierarchy

We investigate invariant submanifolds of the so-called Darboux–Kadomtsev–Petviashvili chain. We show that restricting the dynamics to a class of invariant submanifolds yields an extension of the discrete Kadomtsev–Petviashvili hierarchy and intersections of invariant submanifolds yield the Lax description of a wide class of differential-difference systems. We consider self-similar reductions. We show that self-similar substitutions result in purely discrete equations that depend on a finite set of parameters and in equations determining deformations w.r.t. these parameters. We present examples. In particular, we show that the well-known first discrete Painlevé equation corresponds to the Volterra chain hierarchy. We derive the equations naturally generalizing the first discrete Painlevé equation in the sense that all of them become the first Painlevé equation in the continuum limit.