Rational solutions of the KP hierarchy and the dynamics of their poles. II. Construction of the degenerate polynomial solutions

A general approach to constructing the polynomial solutions satisfying various reductions of the Kadomtsev–Petviashvili (KP) hierarchy is described. Within this approach, the reductions of the KP hierarchy are equivalent to certain differential equations imposed on the τ-function of the hierarchy. In particular, the l-reduction and the k-constraint as well as their generalized counterparts are considered. A general construction of the rational solutions to these reductions is found and the particular solutions are explicitly derived for some typical examples including the KdV and Gardner equations, the Boussinesq and classical Boussinesq systems, the NLS and Yajima–Oikawa equations. It is shown that the degenerate rational solutions of the KP hierarchy are related to stationary manifolds of the Calogero–Moser (CM) hierarchy of dynamical systems. The scattering dynamics of interacting particles in the CM systems may become complicated due to an anomalously slow fractional-power rate of the particle motion along the stationary manifolds.