On a Theorem of Halphen and its Application to Integrable Systems

Abstract

We extend Halphen's theorem which characterizes solutions of certain nth-order differential equations with rational coefficients and meromorphic fundamental systems to a first-order n × n system of differential equations. As an application of this circle of ideas we consider stationary rational algebro-geometric solutions of the KdV hierarchy and illustrate some of the connections with completely integrable models of the Calogero-Moser type. In particular, our treatment recovers the complete characterization of the isospectral class of such rational KdV solutions in terms of a precise description of the Airault-McKean-Moser locus of their poles.

An explicit characterization of Calogero–Moser systems

“…However, a simple inductive argument using (A.1) proves that a rational function q unbounded near infinity cannot satisfy any of the stationary KdV equations (cf. [39]). …”

“…For an extension of Theorem 2.1 to first-order n × n systems and the explicit structure of the corresponding fundamental system of solutions we refer to our recent paper [39].…”

“…In this section we recall an application of Halphen's theorem to rational solutions of the KdV hierarchy recently presented in [39] and then extend these arguments to simply periodic and elliptic KdV potentials using corresponding theorems by Floquet and Picard. More precisely, we revisit stationary rational KdV potentials bounded near infinity (cf.…”

“…In fact, necessary and sufficient conditions on ζ for q in (1.1) to be a rational KdV solution are given by This result was first derived by Duistermaat and Grünbaum [24, p. 199] in 1986, as a by-product of their investigations of bispectral pairs of differential operators. An elementary alternative derivation of this result on the basis of Halphen's theorem, describing the structure of fundamental systems of solutions of differential equations with rational coefficients (and a growth restriction at infinity), and an explicit Frobenius-type analysis were recently provided in our paper [39]. For the convenience of the reader we will summarize these results in Section 2.…”

“…An elementary alternative derivation of this result on the basis of Halphen's theorem, describing the structure of fundamental systems of solutions of differential equations with rational coefficients (and a growth restriction at infinity), and an explicit Frobenius-type analysis were recently provided in our paper [39]. For the convenience of the reader we will summarize these results in Section 2.…”