A finite-dimensional integrable system and the r-matrix method

Abstract

The nonlinearization method is used on the soliton hierarchy associated with the 4×4 AKNS eigenvalue problem, from which a new finite-dimensional Hamiltonian system is obtained by nonlinearization of the eigenvalue problem and its ajoint one. A Lax representation is deduced for the system. The Lax operator admits the representation of an r-matrix. The 4N functionally independent and involutive integrals of motion are obtained via the r-matrix, which shows that the system is completely integrable in the Liouville sense.