Non-isospectral scattering problems: Painleve truncation for hierarchies

In a series of recent papers we have used partial differential equations (PDEs) of a certain form, having non-isospectral scattering problems, in the construction of hierarchies of integrable PDEs together with their underlying linear problems. Here we show how these ideas provide a key to obtaining truncation results for entire hierarchies of integrable PDEs. This new method allows us to obtain, in a very straightforward way, results first obtained by Weiss. The approach given here is readily extended to the various generalizations of the truncation process that appear in the literature