Nonlinearization of the 3=3 Matrix Eigenvalue Problem Related to Coupled Nonlinear Schr¨odinger Equations

The nonlinearization method is extended to the investigation of coupled nonlinear Schr¨odinger equations associated with a 3=3 matrix eigenvalue problem, from which a new finite-dimensional Hamiltonian system is obtained by nonlinearization of the eigenvalue problem and its adjoint one. A scheme for generating involutive systems of conserved integrals is proposed, by which the finite-dimensional Hamiltonian system is further proved to be completely integrable in the Liouville sense.