A special class of Painlevé differential systems that generalize the elliptic functions

Abstract

The elliptic functions of order 2 may be viewed as the ratios Z2/Z1 of the solutions of a differential system of the general form: Zi3/8d2Zj/Zid u2=ZjRii−ZiRiji≠j,i,j=1,…,n, where n = 2, u is the independent variable, and the Rij are the homogeneous quadratic functions of the n variables Zi. The general solution of systems of this form automatically consists, for all n, of purely meromorphic functions. For arbitrary values of n, the system consists of n (n − 1) equations for (n − 1) inhomogeneous variables Zi/Z1, and the question arises as to whether there may exist systems of the above form, with n > 2, which are equivalent to a non-over-determined differential system of p equations, when n-p-1 appropriately chosen algebraic constraints are imposed on the dependent variables. In the present work, we show that such systems do exist, with n = 4 and n = 8, and are in fact special cases of the Dyson model (1968) of a spinning cloud of ellipsoidal shape, expanding adiabatically into a vacuum.