We develop two local energy-preserving integrators and a global energy-preserving integrator for the general multisymplectic Hamiltonian system. When applied to the 1D and multi-dimensional N-coupled nonlinear Schrödinger equations, the given schemes have the exact preservation of the local/global conservation law and are decoupled in the components ψ n, n= 1, 2,…, N, ie, each of the components can be solved independently. The decoupled feature is significant and helpful for overcoming the computational difficulty of the N-coupled (N≥ 3) nonlinear Schrödinger equations, especially of the multi-dimensional case. The composition method is employed to improve the accuracy of the schemes in time and the discrete fast Fourier transform is used to reduce the computational complexity. Several numerical experiments are carried out to exhibit the behaviors of the wave solutions. Numerical results confirm the theoretical results.