Bäcklund transformations for fourth Painlevé hierarchies

Bäcklund transformations (BTs) for ordinary differential equations (ODEs), and in particular for hierarchies of ODEs, are a topic of great current interest. Here, we give an improved method of constructing BTs for hierarchies of ODEs. This approach is then applied to fourth Painlevé (PIV) hierarchies recently found by Gordoa et al. [Publ. Res. Inst. Math. Sci. (Kyoto) 37 (2001) 327–347]. We show how the known pattern of BTs for PIV can be extended to our PIV hierarchies. Remarkably, the BTs required to do this are precisely the Miura maps of the dispersive water wave hierarchy. We also obtain the important result that the fourth Painlevé equation has only one nontrivial fundamental BT, and not two such as is frequently stated.