On a Generalized 2 + 1 Dispersive Water Wave Hierarchy

Abstract

We present a generalized non-isospectral dispersive water wave hierarchy in 2+1 dimensions. We characterize our entire hierarchy and its underlying linear problem using a single equation together with its corresponding non-isospectral scattering problem. This then allows a straightforward construction of linear problems for the entire generalized 2 + 1 hierarchy. Reductions of this hierarchy then yield new integrable hierarchies in 1+1 dimensions, and also new integrable hierarchies of ordinary differential equations, all together with their underlying linear problems. In particular, we obtain a generalized $P_{IV}-P_{II}$ hierarchy; this includes as special cases both a hierarchy of ODEs having the fourth Painlevé equation as first member, and also a hierarchy of ODEs having the second Painlevé equation as first member. All of these hierarchies of ordinary differential equations, as well as their underlying linear problems, are new; both the $P_{IV}$ hierarchy and the $P_{II}$ hierarchy obtained here are different from those which have previously been given.