On a generalized \(2+1\) dispersive water wave hierarchy (Q1596545)

Summary: 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.