A special solution technique: Further extensions

摘要

In a recent paper we gave a technique for identifying special solutions of hierarchies of ordinary differential equations (ODEs) and difference equations. In particular, we considered the phenomenon whereby members of one Painlevé hierarchy define solutions of members of a different Painlevé hierarchy, where both are related to the same (at least) bi-Hamiltonian completely integrable partial differential equation (PDE) or lattice hierarchy; we also considered the question of special integrals of further related Painlevé hierarchies. Here we consider three extensions of this approach. First of all, we consider the case of ODE hierarchies where the corresponding PDE hierarchy is only mono-Hamiltonian. Second, we show that our approach can also be extended in order to provide special solutions of PDEs. Our third extension is to the non-Hamiltonian case, where we consider hierarchies of ODEs related to the Burgers hierarchy. We also make the observation that our results can be phrased quite generally, and the systems of equations dealt with do not in fact need to be integrable.