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D. Pelinovsky

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

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J. Math. Phys. 39, 5377-5395 (1998)

**Abstract:**

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 tau-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.

**Keywords**:

KADOMTSEV-PETVIASHVILI HIERARCHY, KORTEWEG-DE VRIES EQUATION,
BOUSSINESQ EQUATION, NONLINEAR SCHRODINGER EQUATION,
SCATTERING OF PARTICLES, SOLITON SOLUTIONS, 2+1 DIMENSIONS,
WRONSKIAN FORM