Zeros of rational functions and solvable nonlinear evolution equations

Abstract

Recently a convenient technique to relate the time evolution of the N zeros of a time-dependent polynomial pNz;t of degree N in the (complex) variable z to the time evolution of its N coefficients has been exploited to identify large classes of dynamical systems solvable by algebraic operations. These models also include N-body problems that evolve in the complex plane (or, equivalently, in the real Cartesian plane) according to systems of nonlinearly-coupled equations of motion of Newtonian type (“accelerations equal forces”). Many of these models feature remarkable properties: for instance, they are Hamiltonian and integrable and/or multiply periodic or even isochronous (featuring completely periodic solutions with a fixed period largely independent of the initial data), or asymptotically isochronous (featuring isochrony only up to corrections vanishing in the remote future). In this paper, an analogous technique is introduced that focuses instead on the time evolution of the N zeros of an appropriate class of time-dependent rational functions RNz;t, thereby opening large vistas of new dynamical systems solvable by algebraic operations and featuring remarkable properties. A few examples are reported.