Non-isospectral scattering problems and truncation for hierarchies: Burgers and dispersive water waves

Non-isospectral scattering problems have been proven useful for several reasons, amongst them the information that they provide about Painlevé truncation for entire hierarchies of integrable partial differential equations (PDEs). We show in this paper how our approach provides in a very straightforward manner truncation results for two hierarchies in 1+1 dimensions, namely Burgers' hierarchy and the dispersive water wave hierarchy. Burgers' equation is well-known as a model for turbulence, and the dispersive water wave equations as a system governing shallow water waves. We also see how these results are easily extended to related hierarchies in 2+1 dimensions. Our results for the 2+1-dimensional Burgers' hierarchy, and for the 1+1 and 2+1-dimensional dispersive water wave hierarchies, are all new.