Backlund transformations for the second Painleve hierarchy: a modified truncation approach

The second Painleve hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well known second Painleve equation, P-11. In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painleve analysis to ordinary differential equations. We extend these techniques in order to derive auto-Backlund transformations for the second Painleve hierarchy. We also derive a number of other Backlund transformations, including a Backlund transformation onto a hierarchy of P-34 equations, and a little known Backlund transformation for P-11 itself. We then use our results on Backlund transformations to obtain, for each member of the P-11 hierarchy, a sequence of special integrals.