"Goldfish'' equations for infinitely many particles

Abstract

The ``goldfish'' equations, so named because of their striking beauty, are a system of $N$ nonlinear ODE's, where $N$ is an arbitrary integer. They can be solved exactly in a very simple manner, by transforming them to free motion using a transformation involving the transition from the set of {\em coefficients\/} of a polynomial to that of its {\em zeroes}. This paper aims to explore the possibility of extending this solution to the case in which $N$ is infinite. The main difficulty involves the transition from polynomials to entire functions. Another approach using non-standard analysis, is left to future work.