Symmetries in the fourth Painlevé equation and Okamoto polynomials

Abstract

Abstract The fourth Painlevé equation P IV is known to have symmetry of the affine Weyl group of type with respect to the Bäcklund transformations. We introduce a new representation of P IV , called the symmetric form , by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of P IV is given in terms of this representation. Through the symmetric form, it turns out that P IV is obtained as a similarity reduction of the 3-reduced modified KP hierarchy. It is proved in particular that the special polynomials for rational solutions P IV , called Okamoto polynomials , are expressible in terms of the 3-reduced Schur functions.