Integrability, analyticity, isochrony, equilibria, small oscillations, and Diophantine relations: Results from the stationary Korteweg-de Vries hierarchy

Abstract

The isochronous variant is exhibited of the dynamical system corresponding to the Mth ordinary differential equation of the stationary Korteweg-de Vries (KdV) hierarchy. New Diophantine relations are thereby obtained, in the guise of matrices of arbitrary order having integer eigenvalues or equivalently of polynomials of arbitrary degree having integer zeros. Generalizations of these formulas to relations among rational functions are also obtained. The basic idea to arrive at such relations is not new, but the specific application reported in this paper is new, and it is likely to open the way to several analogous new findings.