Backlund transformations for new fourth Painleve hierarchies

摘要

We consider a system of equations defined using the Hamiltonian operator of the Bous-sinesq hierarchy, as well as two successive modifications thereof. We are able to reduce the order of these three systems and give Backlund transformations between the integrated equations. We also give auto-Backlund transformations for the two modified systems. Particular cases of two of the three equations considered correspond to generalized fourth Painleve hierarchies and are new; these are particular cases of the two modified systems. Thus we obtain auto-Backlund transformations for these new fourth Painleve hierarchies, as well as Backlund transformations between our hierarchies. Our results on reduction of order are also applicable in this special case, and include as a particular example a reduction of order for the scaling similarity reduction of the Boussinesq equation, a result which, remarkably, seems not to have been given previously.