Symmetries in the fourth Painleve equation and Okamoto polynomials

Details Symmetries in the fourth Painleve equation and Okamoto polynomials Symmetries in the fourth Painleve equation and Okamoto polynomials

1997-08-19

arxiv.org/abs/q-alg/9708018v1

Mathematics

Quantum Algebra

25 pages, amslatex

Scientific paper

We propose a new representation of the fourth Painlev\'e equation in which the $A^{(1)}_2$-symmetries become clearly visible. By means of this representation, we clarify the internal relation between the fourth Painlev\'e equation and the modified KP hierarchy. We obtain in particular a complete description of the rational solutions of the fourth Painlev\'e equation in terms of Schur functions. This implies that the so-called Okamoto polynomials, which arise from the $\tau$-functions for rational solutions, are in fact expressible by the 3-reduced Schur functions.

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