The pth-order periodic solutions for a family of N-coupled nonlinear SchrOdingerequations

By using the solutions of an auxiliary Lam\'e equation, a direct algebraic method is proposed to construct the exact solutions of N-coupled nonlinear Schr?dinger equations. The abundant higher-order exact periodic solutions of a family of N-coupled nonlinear Schr?dinger equations are explicitly obtained with the aid of symbolic computation and they include corresponding envelope solitary and shock wave solutions.