PAINLEVE CLASSIFICATION PROBLEMS FEATURING ESSENTIAL SINGULARITIES

In this article we construct and solve all Painleve-type differential equations of the second order and second degree that are built upon, i n a natural well-defined sense, the ''sn-log'' equation of Painleve, t he general integral of which admits a movable essential singularity (e lliptic function of a logarithm). This equation (which was studied by Painleve in the years 1893-1902) is frequently cited in the modern lit erature to elucidate various aspects of Painleve analysis and integrab ility of differential equations, especially the difficulty of detectin g essential singularities by local singularity analysis of differentia l equations. Our definition of the Painleve property permits movable e ssential singularities, provided there is no branching. While the esse ntial singularity presents no serious technical problems, we do need t o introduce new techniques for handling ''exotic'' Painleve equations, which are Painleve equations whose singular integrals admit movable b ranch points in the leading terms. We find that the corresponding full class of Painleve-type equations contains three, and only three, equa tions, which we denote SD-326-I, SD-326-II, and SD-326-III, each solva ble in terms of elliptic functions. The first is Painleve's own genera lization of his sn-log equation. The second and third are new, the thi rd being a 15-parameter exotic master equation. The appendices contain results (in general, without uniqueness proofs) of related Painleve c lassification problems, including full generalizations of two other se cond-degree equations discovered by Painleve, additional examples of e xotic Painleve equations and Painleve equations admitting movable esse ntial singularities, and third-order equations featuring sn-log and ot her essential singularities.