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 PIV − PII 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 PIV hierarchy and the PII hierarchy obtained here are different from those which have previously been given.
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Andrew Pickering, Nalini Joshi, Pilar R. Gordoa, On a Generalized 2 + 1 Dispersive Water Wave Hierarchy Klein-Gordon equations is studied in the Sobolev space Hs = Hs(. Publ. Res. Inst. Math. Sci. 37 (2001), no. 3, pp. 327–347