A New Integrable N ‐Particle McMillan Map and Its Exact Solutions

Source abstract

ABSTRACT An integrable ‐particle generalization of the classical McMillan map with distinct particle‐dependent parameters is introduced. Its Liouville integrability is established through a discrete Lax representation and the associated classical ‐matrix structure. The map is further shown to provide a discrete auto‐Bäcklund transformation for the restricted Kaup–Newell flows and to arise from the nonlinearization of a Darboux transformation for the Kaup–Newell spectral problem. Using the associated hyperelliptic spectral curve and Baker–Akhiezer functions, explicit algebro‐geometric solutions for the multi‐particle orbits are derived in terms of Riemann theta functions. These results provide an exact solution framework for multi‐particle McMillan dynamics with particle‐dependent parameters and clarify its connection with the continuous Kaup–Newell hierarchy.