A class of coupled nonlinear Schrödinger equations and their relations to the complex Hamilton system

In this paper, the relation between the coupled nonlinear Schrodinger equation and the complex Hamilton system are discussed by means of the third-order spectral problem. Then the complete integrability of the complex Hamilton system associated with the coupled nonlinear Schrodinger equation is discussed by using the symplectic structure and the complex representation of the Poisson bracket. So the solutions to the coupled nonlinear Schrodinger equation are derived.