"Goldfish'' equations for infinitely many particles
arXiv:2607.15237
The paper investigates extending the exactly solvable “goldfish” system of nonlinear ODEs from a finite number of particles to an infinite number, focusing on the analytic transition from polynomial coefficients to entire functions.
Abstract
The ``goldfish'' equations, so named because of their striking beauty, are a system of nonlinear ODE's, where 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 is infinite. The main difficulty involves the transition from polynomials to entire functions. Another approach using non-standard analysis, is left to future work.
14 pages, 1 figure